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 \documentclass[dvipdfm]{book} \newcommand{\VolumeName}{Volume 8.1: Axiom Gallery} \input{bookheader.tex} \mainmatter \setcounter{chapter}{0} % Chapter 1 \chapter{General examples} These examples come from code that ships with Axiom in various input files. \section{Two dimensional functions} \newpage \subsection{A Simple Sine Function} {\center{\includegraphics[scale=0.50]{ps/v81equation123.eps}}} \begin{chunk}{equation123} draw(sin(11*x),x = 0..2*%pi) \end{chunk} \newpage \subsection{A Simple Sine Function, Non-adaptive plot} {\center{\includegraphics[scale=0.50]{ps/v81equation124.eps}}} \begin{chunk}{equation124} draw(sin(11*x),x = 0..2*%pi,adaptive == false,title == "Non-adaptive plot") \end{chunk} \newpage \subsection{A Simple Sine Function, Drawn to Scale} {\center{\includegraphics[scale=0.50]{ps/v81equation125.eps}}} \begin{chunk}{equation125} draw(sin(11*x),x = 0..2*%pi,toScale == true,title == "Drawn to scale") \end{chunk} \newpage \subsection{A Simple Sine Function, Polar Plot} {\center{\includegraphics[scale=0.50]{ps/v81equation126.eps}}} \begin{chunk}{equation126} draw(sin(11*x),x = 0..2*%pi,coordinates == polar,title == "Polar plot") \end{chunk} \newpage \subsection{A Simple Tangent Function, Clipping On} {\center{\includegraphics[scale=0.50]{ps/v81equation127.eps}}} \begin{chunk}{equation127} draw(tan x,x = -6..6,title == "Clipping on") \end{chunk} \newpage \subsection{A Simple Tangent Function, Clipping On} {\center{\includegraphics[scale=0.50]{ps/v81equation128.eps}}} \begin{chunk}{equation128} draw(tan x,x = -6..6,clip == false,title == "Clipping off") \end{chunk} \newpage \subsection{Tangent and Sine} {\center{\includegraphics[scale=0.50]{ps/v81equation101.eps}}} \begin{chunk}{equation101} f(x:DFLOAT):DFLOAT == sin(tan(x))-tan(sin(x)) draw(f,0..6) \end{chunk} \newpage \subsection{A 2D Sine Function in BiPolar Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation107.eps}}} \begin{chunk}{equation107} draw(sin(x),x=0.5..%pi,coordinates == bipolar(1$DFLOAT)) \end{chunk} \newpage \subsection{A 2D Sine Function in Elliptic Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation110.eps}}} \begin{chunk}{equation110} draw(sin(4*t/7),t=0..14*%pi,coordinates == elliptic(1$DFLOAT)) \end{chunk} \newpage \subsection{A 2D Sine Wave in Polar Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation118.eps}}} \begin{chunk}{equation118} draw(sin(4*t/7),t=0..14*%pi,coordinates == polar) \end{chunk} \section{Two dimensional curves} \newpage \subsection{A Line in Parabolic Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation114.eps}}} \begin{chunk}{equation114} h1(t:DFLOAT):DFLOAT == t h2(t:DFLOAT):DFLOAT == 2 draw(curve(h1,h2),-3..3,coordinates == parabolic) \end{chunk} \newpage \subsection{Lissajous Curve} {\center{\includegraphics[scale=0.50]{ps/v81equation102.eps}}} \begin{chunk}{equation102} i1(t:DFLOAT):DFLOAT == 9*sin(3*t/4) i2(t:DFLOAT):DFLOAT == 8*sin(t) draw(curve(i1,i2),-4*%pi..4*%pi,toScale == true, title == "Lissajous Curve") \end{chunk} \newpage \subsection{A Parametric Curve} {\center{\includegraphics[scale=0.50]{ps/v81equation129.eps}}} \begin{chunk}{equation129} draw(curve(sin(5*t),t),t = 0..2*%pi,title == "Parametric curve") \end{chunk} \newpage \subsection{A Parametric Curve in Polar Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation130.eps}}} \begin{chunk}{equation130} draw(curve(sin(5*t),t),t = 0..2*%pi,_ coordinates == polar,title == "Parametric polar curve") \end{chunk} \section{Three dimensional functions} \newpage \subsection{A 3D Constant Function in Elliptic Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation111.eps}}} \begin{chunk}{equation111} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,0..2*%pi,0..%pi,coordinates == elliptic(1$DFLOAT)) \end{chunk} \newpage \subsection{A 3D Constant Function in Oblate Spheroidal} {\center{\includegraphics[scale=0.50]{ps/v81equation113.eps}}} \begin{chunk}{equation113} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,-%pi/2..%pi/2,0..2*%pi,coordinates == oblateSpheroidal(1$DFLOAT)) \end{chunk} \newpage \subsection{A 3D Constant in Polar Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation119.eps}}} \begin{chunk}{equation119} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,0..2*%pi, 0..%pi,coordinates == polar) \end{chunk} \newpage \subsection{A 3D Constant in Prolate Spheroidal Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation120.eps}}} \begin{chunk}{equation120} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,-%pi/2..%pi/2,0..2*%pi,coordinates == prolateSpheroidal(1$DFLOAT)) \end{chunk} \newpage \subsection{A 3D Constant in Spherical Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation121.eps}}} \begin{chunk}{equation121} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,0..2*%pi,0..%pi,coordinates == spherical) \end{chunk} \newpage \subsection{A 2-Equation Space Function} {\center{\includegraphics[scale=0.50]{ps/v81equation104.eps}}} \begin{chunk}{equation104} f(x:DFLOAT,y:DFLOAT):DFLOAT == cos(x*y) colorFxn(x:DFLOAT,y:DFLOAT):DFLOAT == 1/(x**2 + y**2 + 1) draw(f,-3..3,-3..3, colorFunction == colorFxn,title=="2-Equation Space Curve") \end{chunk} \section{Three dimensional curves} \newpage \subsection{A Parametric Space Curve} {\center{\includegraphics[scale=0.50]{ps/v81equation131.eps}}} \begin{chunk}{equation131} draw(curve(sin(t)*cos(3*t/5),cos(t)*cos(3*t/5),cos(t)*sin(3*t/5)),_ t = 0..15*%pi,title == "Parametric curve") \end{chunk} \newpage \subsection{A Tube around a Parametric Space Curve} {\center{\includegraphics[scale=0.50]{ps/v81equation132.eps}}} \begin{chunk}{equation132} draw(curve(sin(t)*cos(3*t/5),cos(t)*cos(3*t/5),cos(t)*sin(3*t/5)),_ t = 0..15*%pi,tubeRadius == .15,title == "Tube around curve") \end{chunk} \newpage \subsection{A 2-Equation Cylindrical Curve} {\center{\includegraphics[scale=0.50]{ps/v81equation109.eps}}} \begin{chunk}{equation109} j1(t:DFLOAT):DFLOAT == 4 j2(t:DFLOAT):DFLOAT == t draw(curve(j1,j2,j2),-9..9,coordinates == cylindrical) \end{chunk} \section{Three dimensional surfaces} \newpage \subsection{A Icosahedron} {\center{\includegraphics[scale=0.50]{ps/v81icosahedron.eps}}} \begin{chunk}{Icosahedron} )se exp add con InnerTrigonometricManipulations exp(%i*2*%pi/5) FG2F % % -1 complexForm % norm % simplify % s:=sqrt % ph:=exp(%i*2*%pi/5) A1:=complex(1,0) A2:=A1*ph A3:=A2*ph A4:=A3*ph A5:=A4*ph ca1:=map(numeric , complexForm FG2F simplify A1) ca2:=map(numeric , complexForm FG2F simplify A2) ca3:=map(numeric ,complexForm FG2F simplify A3) ca4:=map(numeric ,complexForm FG2F simplify A4) ca5:=map(numeric ,complexForm FG2F simplify A5) B1:=A1*exp(2*%i*%pi/10) B2:=B1*ph B3:=B2*ph B4:=B3*ph B5:=B4*ph cb1:=map (numeric ,complexForm FG2F simplify B1) cb2:=map (numeric ,complexForm FG2F simplify B2) cb3:=map (numeric ,complexForm FG2F simplify B3) cb4:=map (numeric ,complexForm FG2F simplify B4) cb5:=map (numeric ,complexForm FG2F simplify B5) u:=numeric sqrt(s*s-1) p0:=point([0,0,u+1/2])@Point(SF) p1:=point([real ca1,imag ca1,0.5])@Point(SF) p2:=point([real ca2,imag ca2,0.5])@Point(SF) p3:=point([real ca3,imag ca3,0.5])@Point(SF) p4:=point([real ca4,imag ca4,0.5])@Point(SF) p5:=point([real ca5,imag ca5,0.5])@Point(SF) p6:=point([real cb1,imag cb1,-0.5])@Point(SF) p7:=point([real cb2,imag cb2,-0.5])@Point(SF) p8:=point([real cb3,imag cb3,-0.5])@Point(SF) p9:=point([real cb4,imag cb4,-0.5])@Point(SF) p10:=point([real cb5,imag cb5,-0.5])@Point(SF) p11:=point([0,0,-u-1/2])@Point(SF) space:=create3Space()$ThreeSpace DFLOAT polygon(space,[p0,p1,p2]) polygon(space,[p0,p2,p3]) polygon(space,[p0,p3,p4]) polygon(space,[p0,p4,p5]) polygon(space,[p0,p5,p1]) polygon(space,[p1,p6,p2]) polygon(space,[p2,p7,p3]) polygon(space,[p3,p8,p4]) polygon(space,[p4,p9,p5]) polygon(space,[p5,p10,p1]) polygon(space,[p2,p6,p7]) polygon(space,[p3,p7,p8]) polygon(space,[p4,p8,p9]) polygon(space,[p5,p9,p10]) polygon(space,[p1,p10,p6]) polygon(space,[p6,p11,p7]) polygon(space,[p7,p11,p8]) polygon(space,[p8,p11,p9]) polygon(space,[p9,p11,p10]) polygon(space,[p10,p11,p6]) makeViewport3D(space,title=="Icosahedron",style=="smooth") \end{chunk} \newpage \subsection{A 3D figure 8 immersion (Klein bagel)} {\center{\includegraphics[scale=0.50]{ps/v81kleinbagel.eps}}} \begin{chunk}{kleinbagel} r := 1 X(u,v) == (r+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*cos(u) Y(u,v) == (r+cos(u/2)*sin(v)-sin(u/2)*sin(2*v))*sin(u) Z(u,v) == sin(u/2)*sin(v)+cos(u/2)*sin(2*v) v3d:=draw(surface(X(u,v),Y(u,v),Z(u,v)),u=0..2*%pi,v=0..2*%pi,_ style=="solid",title=="Figure 8 Klein") colorDef(v3d,blue(),blue()) axes(v3d,"off") \end{chunk} From \verb|en.wikipedia.org/wiki/Klein_bottle|. The figure 8'' immersion (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a figure 8'' torus with a 180 degree M\"obius'' twist inserted. In this immersion, the self-intersection circle is a geometric circle in the x-y plane. The positive constant $r$ is the radius of this circle. The parameter $u$ gives the angle in the x-y plane, and $v$ specifies the position around the 8-shaped cross section. With the above parameterization the cross section is a 2:1 Lissajous curve. \subsection{A 2-Equation bipolarCylindrical Surface} {\center{\includegraphics[scale=0.50]{ps/v81equation108.eps}}} \begin{chunk}{equation108} draw(surface(u*cos(v),u*sin(v),u),u=1..4,v=1..2*%pi,_ coordinates == bipolarCylindrical(1$DFLOAT)) \end{chunk} \newpage \subsection{A 3-Equation Parametric Space Surface} {\center{\includegraphics[scale=0.50]{ps/v81equation105.eps}}} \begin{chunk}{equation105} n1(u:DFLOAT,v:DFLOAT):DFLOAT == u*cos(v) n2(u:DFLOAT,v:DFLOAT):DFLOAT == u*sin(v) n3(u:DFLOAT,v:DFLOAT):DFLOAT == v*cos(u) colorFxn(x:DFLOAT,y:DFLOAT):DFLOAT == 1/(x**2 + y**2 + 1) draw(surface(n1,n2,n3),-4..4,0..2*%pi, colorFunction == colorFxn) \end{chunk} \newpage \subsection{A 3D Vector of Points in Elliptic Cylindrical} {\center{\includegraphics[scale=0.50]{ps/v81equation112.eps}}} \begin{chunk}{equation112} U2:Vector Expression Integer := vector [0,0,1] x(u,v) == beta(u) + v*delta(u) beta u == vector [cos u, sin u, 0] delta u == (cos(u/2)) * beta(u) + sin(u/2) * U2 vec := x(u,v) draw(surface(vec.1,vec.2,vec.3),v=-0.5..0.5,u=0..2*%pi,_ coordinates == ellipticCylindrical(1$DFLOAT),_ var1Steps == 50,var2Steps == 50) \end{chunk} \newpage \subsection{A 3D Constant Function in BiPolar Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation106.eps}}} \begin{chunk}{equation106} m(u:DFLOAT,v:DFLOAT):DFLOAT == 1 draw(m,0..2*%pi, 0..%pi,coordinates == bipolar(1$DFLOAT)) \end{chunk} \newpage \subsection{A Swept in Parabolic Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation115.eps}}} \begin{chunk}{equation115} draw(surface(u*cos(v),u*sin(v),2*u),u=0..4,v=0..2*%pi,coordinates==parabolic) \end{chunk} \newpage \subsection{A Swept Cone in Parabolic Cylindrical Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation116.eps}}} \begin{chunk}{equation116} draw(surface(u*cos(v),u*sin(v),v*cos(u)),u=0..4,v=0..2*%pi,_ coordinates == parabolicCylindrical) \end{chunk} \newpage \subsection{A Truncated Cone in Toroidal Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation122.eps}}} \begin{chunk}{equation122} draw(surface(u*cos(v),u*sin(v),u),u=1..4,v=1..4*%pi,_ coordinates == toroidal(1$DFLOAT)) \end{chunk} \newpage \subsection{A Swept Surface in Paraboloidal Coordinates} {\center{\includegraphics[scale=0.50]{ps/v81equation117.eps}}} \begin{chunk}{equation117} draw(surface(u*cos(v),u*sin(v),u*v),u=0..4,v=0..2*%pi,_ coordinates==paraboloidal,var1Steps == 50, var2Steps == 50) \end{chunk} \chapter{Jenks Book images} \newpage \subsection{The Complex Gamma Function} {\center{\includegraphics[scale=0.50]{ps/v81complexgamma.eps}}} \begin{chunk}{complexgamma} gam(x:DoubleFloat,y:DoubleFloat):Point(DoubleFloat) == _ ( g:Complex(DoubleFloat):= Gamma complex(x,y) ; _ point [x,y,max(min(real g, 4), -4), argument g] ) v3d:=draw(gam, -%pi..%pi, -%pi..%pi, title == "Gamma(x + %i*y)", _ var1Steps == 100, var2Steps == 100, style=="smooth") \end{chunk} A 3-d surface whose height is the real part of the Gamma function, and whose color is the argument of the Gamma function. \newpage \subsection{The Complex Arctangent Function} {\center{\includegraphics[scale=0.50]{ps/v81complexarctangent.eps}}} \begin{chunk}{complexarctangent} atf(x:DoubleFloat,y:DoubleFloat):Point(DoubleFloat) == _ ( a := atan complex(x,y) ; _ point [x,y,real a, argument a] ) v3d:=draw(atf, -3.0..%pi, -3.0..%pi, style=="shade") rotate(v3d,210,-60) \end{chunk} The complex arctangent function. The height is the real part and the color is the argument. \chapter{Hyperdoc examples} Examples in this section come from the Hyperdoc documentation tool. These examples are accessed from the Basic Examples Draw section. \section{Two dimensional examples} \newpage \subsection{A function of one variable} {\center{\includegraphics[scale=0.50]{ps/v81equation001.eps}}} \begin{chunk}{equation001} draw(x*cos(x),x=0..30,title=="y = x*cos(x)") \end{chunk} This is one of the demonstration equations used in hypertex. It demonstrates a function of one variable. It draws $y = f(x)$ where $y$ is the dependent variable and $x$ is the independent variable. \newpage \subsection{A Parametric function} {\center{\includegraphics[scale=0.50]{ps/v81equation002.eps}}} \begin{chunk}{equation002} draw(curve(-9*sin(4*t/5),8*sin(t)),t=-5*%pi..5*%pi,title=="Lissajous") \end{chunk} This is one of the demonstration equations used in hypertex. It draw a parametrically defined curve $f1(t), f2(t)$ in terms of two functions $f1$ and $f2$ and an independent variable t. \newpage \subsection{A Polynomial in 2 variables} {\center{\includegraphics[scale=0.50]{ps/v81equation003.eps}}} \begin{chunk}{equation003} draw(y**2+7*x*y-(x**3+16*x) = 0,x,y,range==[-15..10,-10..50]) \end{chunk} This is one of the demonstration equations used in hypertex. Plotting the solution to $p(x,y) = 0$ where p is a polynomial in two variables $x$ and $y$. \section{Three dimensional examples} \newpage \subsection{A function of two variables} {\center{\includegraphics[scale=0.50]{ps/v81equation004.eps}}} \begin{chunk}{equation004} cf(x,y) == 0.5 draw(exp(cos(x-y)-sin(x*y))-2,x=-5..5,y=-5..5,_ colorFunction==cf,style=="smooth") \end{chunk} This is one of the demonstration equations used in hypertex. A function of two variables $z = f(x,y)$ where $z$ is the dependent variable and where $x$ and $y$ are the dependent variables. \newpage \subsection{A parametrically defined curve} {\center{\includegraphics[scale=0.50]{ps/v81equation005.eps}}} \begin{chunk}{equation005} f1(t) == 1.3*cos(2*t)*cos(4*t)+sin(4*t)*cos(t) f2(t) == 1.3*sin(2*t)*cos(4*t)-sin(4*t)*sin(t) f3(t) == 2.5*cos(4*t) cf(x,y) == 0.5 draw(curve(f1(t),f2(t),f3(t)),t=0..4*%pi,tubeRadius==.25,tubePoints==16,_ title=="knot",colorFunction==cf,style=="smooth") \end{chunk} This is one of the demonstration equations used in hypertex. This ia parametrically defined curve $f1(t), f2(t), f3(t)$ in terms of three functions $f1$, $f2$, and $f3$ and an independent variable $t$. \newpage \subsection{A parametrically defined surface} {\center{\includegraphics[scale=0.50]{ps/v81equation006.eps}}} \begin{chunk}{equation006} f1(u,v) == u*sin(v) f2(u,v) == v*cos(u) f3(u,v) == u*cos(v) cf(x,y) == 0.5 draw(surface(f1(u,v),f2(u,v),f3(u,v)),u=-%pi..%pi,v=-%pi/2..%pi/2,_ title=="surface",colorFunction==cf,style=="smooth") \end{chunk} This is one of the demonstration equations used in hypertex. This ia parametrically defined curve $f1(t), f2(t), f3(t)$ in terms of three functions $f1$, $f2$, and $f3$ and an independent variable $t$. \chapter{CRC Standard Curves and Surfaces} \section{Standard Curves and Surfaces} In order to have an organized and thorough evaluation of the Axiom graphics code we turn to the CRC Standard Curves and Surfaces \cite{Segg93} (SCC). This volume was written years after the Axiom graphics code was written so there was no attempt to match the two until now. However, the SCC volume will give us a solid foundation to both evaluate the features of the current code and suggest future directions. According to the SCC we can organize the various curves by the taxonomy: \begin{enumerate} \item[1] {\bf random} \begin{enumerate} \item[1.1] {\bf fractal} \item[1.2] {\bf gaussian} \item[1.3] {\bf non-gaussian} \end{enumerate} \item[2] {\bf determinate} \begin{enumerate} \item[2.1] {\bf algebraic} -- A polynomial is defined as a summation of terms composed of integral powers of $x$ and $y$. An algebraic curve is one whose implicit function $f(x,y)=0$ is a polynomial in $x$ and $y$ (after rationalization, if necessary). Because a curve is often defined in the explicit form $y=f(x)$ there is a need to distinguish rational and irrational functions of $x$. \begin{enumerate} \item[2.1.1] {\bf irrational} -- An irrational function of $x$ is a quotient of two polynomials, one or both of which has a term (or terms) with power $p/q$, where $p$ and $q$ are integers. \item[2.1.2] {\bf rational} -- A rational function of $x$ is a quotient of two polynomials in $x$, both having only integer powers. \begin{enumerate} \item[2.1.2.1] {\bf polynomial} \item[2.1.2.2] {\bf non-polynomial} \end{enumerate} \end{enumerate} \item[2.2] {\bf integral} -- Certain continuous functions are not expressible in algebraic or transcendental forms but are familiar mathematical tools. These curves are equal to the integrals of algebraic or transcendental curves by definition; examples include Bessel functions, Airy integrals, Fresnel integrals, and the error function. \item[2.3] {\bf transcendental} -- The transcendental curves cannot be expressed as polynomials in $x$ and $y$. These are curves containing one or more of the following forms: exponential ($e^x$), logarithmic ($\log(x)$), or trigonometric ($\sin(x)$, $\cos(x)$). \begin{enumerate} \item[2.3.1] {\bf exponential} \item[2.3.2] {\bf logarithmic} \item[2.3.3] {\bf trigonometric} \end{enumerate} \item[2.4] {\bf piecewise continuous} -- Other curves, except at a few singular points, are smooth and differentiable. The class of nondifferentiable curves have discontinuity of the first derivative as a basic attribute. They are often composed of straight-line segments. Simple polygonal forms, regular fractal curves, and turtle tracks are examples. \begin{enumerate} \item[2.4.1] {\bf periodic} \item[2.4.2] {\bf non-periodic} \item[2.4.3] {\bf polygonal} \begin{enumerate} \item[2.4.3.1] {\bf regular} \item[2.4.3.2] {\bf irregular} \item[2.4.3.3] {\bf fractal} \end{enumerate} \end{enumerate} \end{enumerate} \end{enumerate} \section{CRC graphs} \subsection{Functions with $x^{n/m}$} \newpage \subsubsection{Page 26 2.1.1} $y=cx^n$ $y-cx^n=0$ \begin{chunk}{p26-2.1.1.1-3} )clear all )set mes auto off f(c,x,n) == c*x^n lineColorDefault(green()) viewport1:=draw(f(1,x,1), x=-2..2, adaptive==true, unit==[1.0,1.0],_ title=="p26-2.1.1.1-3") graph2111:=getGraph(viewport1,1) lineColorDefault(blue()) viewport2:=draw(f(1,x,3), x=-2..2, adaptive==true, unit==[1.0,1.0]) graph2112:=getGraph(viewport2,1) lineColorDefault(red()) viewport3:=draw(f(1,x,5), x=-2..2, adaptive==true, unit==[1.0,1.0]) graph2113:=getGraph(viewport3,1) putGraph(viewport1,graph2112,2) putGraph(viewport1,graph2113,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp26-2.1.1.1-3.eps}}} \label{CRCp26-2.1.1.1-3} \index{figures!CRCp26-2.1.1.1-3} \newpage \begin{chunk}{p26-2.1.1.4-6} )clear all )set mes auto off f(c,x,n) == c*x^n lineColorDefault(green()) viewport1:=draw(f(1,x,2), x=-2..2, adaptive==true, unit==[1.0,1.0],_ title=="p26-2.1.1.4-6") graph2111:=getGraph(viewport1,1) lineColorDefault(blue()) viewport2:=draw(f(1,x,4), x=-2..2, adaptive==true, unit==[1.0,1.0]) graph2112:=getGraph(viewport2,1) lineColorDefault(red()) viewport3:=draw(f(1,x,6), x=-2..2, adaptive==true, unit==[1.0,1.0]) graph2113:=getGraph(viewport3,1) putGraph(viewport1,graph2112,2) putGraph(viewport1,graph2113,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp26-2.1.1.4-6.eps}}} \label{CRCp26-2.1.1.4-6} \index{figures!CRCp26-2.1.1.4-6} \newpage \subsubsection{Page 26 2.1.2} $\displaystyle y=\frac{c}{x^n}$ $yx^n-c=0$ \begin{chunk}{p26-2.1.2.1-3} )clear all f(c,x,n) == c/x^n lineColorDefault(green()) viewport1:=draw(f(0.01,x,1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p26-2.1.2.1-3") graph2111:=getGraph(viewport1,1) lineColorDefault(blue()) viewport2:=draw(f(0.01,x,3),x=-4..4,adaptive==true,unit==[1.0,1.0]) graph2122:=getGraph(viewport2,1) lineColorDefault(red()) viewport3:=draw(f(0.01,x,5),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2123:=getGraph(viewport3,1) putGraph(viewport1,graph2122,2) putGraph(viewport1,graph2123,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp26-2.1.2.1-3.eps}}} \label{CRCp26-2.1.2.1-3} \index{figures!CRCp26-2.1.2.1-3} \newpage Note that Axiom's plot of viewport2 disagrees with CRC \begin{chunk}{p26-2.1.2.4-6} )clear all f(c,x,n) == c/x^n lineColorDefault(green()) viewport1:=draw(f(0.01,x,4),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p26-2.1.2.4-6") graph2124:=getGraph(viewport1,1) lineColorDefault(blue()) viewport2:=draw(f(0.01,x,5),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2125:=getGraph(viewport2,1) lineColorDefault(red()) viewport3:=draw(f(0.01,x,6),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2126:=getGraph(viewport3,1) putGraph(viewport1,graph2125,2) putGraph(viewport1,graph2126,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp26-2.1.2.4-6.eps}}} \label{CRCp26-2.1.2.4-6} \index{figures!CRCp26-2.1.2.4-6} \newpage \subsubsection{Page 28 2.1.3} $\displaystyle y=cx^{n/m}$ $\displaystyle y-cx^{n/m}=0$ \begin{chunk}{p28-2.1.3.1-6} )clear all f(c,x,n,m) == c*x^(n/m) lineColorDefault(color(1)) viewport1:=draw(f(1,x,1,4),x=0..1,adaptive==true,unit==[1.0,1.0],_ title=="p28-2.1.3.1-6") graph2131:=getGraph(viewport1,1) lineColorDefault(color(2)) viewport2:=draw(f(1,x,1,2),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2132:=getGraph(viewport2,1) lineColorDefault(color(3)) viewport3:=draw(f(1,x,3,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2133:=getGraph(viewport3,1) lineColorDefault(color(4)) viewport4:=draw(f(1,x,5,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2134:=getGraph(viewport4,1) lineColorDefault(color(5)) viewport5:=draw(f(1,x,3,2),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2135:=getGraph(viewport5,1) lineColorDefault(color(6)) viewport6:=draw(f(1,x,7,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2136:=getGraph(viewport6,1) putGraph(viewport1,graph2132,2) putGraph(viewport1,graph2133,3) putGraph(viewport1,graph2134,4) putGraph(viewport1,graph2135,5) putGraph(viewport1,graph2136,6) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp28-2.1.3.1-6.eps}}} \label{CRCp28-2.1.3.1-6} \index{figures!CRCp28-2.1.3.1-6} \newpage \begin{chunk}{p28-2.1.3.7-10} )clear all f(c,x,n,m) == c*x^(n/m) lineColorDefault(color(1)) viewport1:=draw(f(1,x,1,3),x=0..1,adaptive==true,unit==[1.0,1.0],_ title=="p28-2.1.3.7-10") graph2137:=getGraph(viewport1,1) lineColorDefault(color(2)) viewport2:=draw(f(1,x,2,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2138:=getGraph(viewport2,1) lineColorDefault(color(3)) viewport3:=draw(f(1,x,4,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2139:=getGraph(viewport3,1) lineColorDefault(color(4)) viewport4:=draw(f(1,x,5,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph21310:=getGraph(viewport4,1) putGraph(viewport1,graph2138,2) putGraph(viewport1,graph2139,3) putGraph(viewport1,graph21310,4) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp28-2.1.3.7-10.eps}}} \label{CRCp28-2.1.3.7-10} \index{figures!CRCp28-2.1.3.7-10} \newpage \subsubsection{Page 28 2.1.4} $\displaystyle y=\frac{c}{x^{n/m}}$ $\displaystyle yx^{n/m}-c=0$ \begin{chunk}{p28-2.1.4.1-6} )clear all f(c,x,n,m) == c/x^(n/m) lineColorDefault(color(1)) viewport1:=draw(f(0.01,x,1,4),x=0..1,adaptive==true,unit==[1.0,1.0],_ title=="p28-2.1.4.1-6") graph2141:=getGraph(viewport1,1) lineColorDefault(color(2)) viewport2:=draw(f(0.01,x,1,2),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2142:=getGraph(viewport2,1) lineColorDefault(color(3)) viewport3:=draw(f(0.01,x,3,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2143:=getGraph(viewport3,1) lineColorDefault(color(4)) viewport4:=draw(f(0.01,x,5,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2144:=getGraph(viewport4,1) lineColorDefault(color(5)) viewport5:=draw(f(0.01,x,3,2),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2145:=getGraph(viewport5,1) lineColorDefault(color(6)) viewport6:=draw(f(0.01,x,7,4),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2146:=getGraph(viewport6,1) putGraph(viewport1,graph2142,2) putGraph(viewport1,graph2143,3) putGraph(viewport1,graph2144,4) putGraph(viewport1,graph2145,5) putGraph(viewport1,graph2146,6) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp28-2.1.4.1-6.eps}}} \label{CRCp28-2.1.4.1-6} \index{figures!CRCp28-2.1.4.1-6} \newpage \begin{chunk}{p28-2.1.4.7-10} )clear all f(c,x,n,m) == c/x^(n/m) lineColorDefault(color(1)) viewport1:=draw(f(1,x,1,3),x=0..1,adaptive==true,unit==[1.0,1.0],_ title=="p28-2.1.4.7-10") graph2147:=getGraph(viewport1,1) lineColorDefault(color(2)) viewport2:=draw(f(1,x,2,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2148:=getGraph(viewport2,1) lineColorDefault(color(3)) viewport3:=draw(f(1,x,4,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph2149:=getGraph(viewport3,1) lineColorDefault(color(4)) viewport4:=draw(f(1,x,5,3),x=0..1,adaptive==true,unit==[1.0,1.0]) graph21410:=getGraph(viewport4,1) putGraph(viewport1,graph2148,2) putGraph(viewport1,graph2149,3) putGraph(viewport1,graph21410,4) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp28-2.1.4.7-10.eps}}} \label{CRCp28-2.1.4.7-10} \index{figures!CRCp28-2.1.4.7-10} \newpage \subsection{Functions with $x^n$ and $(a+bx)^m$} \subsubsection{Page 30 2.2.1} $y=c(a+bx)$ $y-bcx-ac=0$ \begin{chunk}{p30-2.2.1.1-3} )clear all f(x,a,b,c) == c*(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.1.1-3") graph2211:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2212:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2213:=getGraph(viewport3,1) putGraph(viewport1,graph2212,2) putGraph(viewport1,graph2213,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.1.1-3.eps}}} \label{CRCp30-2.2.1.1-3} \index{figures!CRCp30-2.2.1.1-3} \newpage \subsubsection{Page 30 2.2.2} $y=c(a+bx)^2$ $y - cb^2x^2 - 2abcx - a^2c = 0$ \begin{chunk}{p30-2.2.2.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.2.1-3") graph2221:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2222:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2223:=getGraph(viewport3,1) putGraph(viewport1,graph2222,2) putGraph(viewport1,graph2223,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.2.1-3.eps}}} \label{CRCp30-2.2.2.1-3} \index{figures!CRCp30-2.2.2.1-3} \newpage \subsubsection{Page 30 2.2.3} $y=c(a+bx)^3$ $y - b^3cx^3 - 3ab^2cx^2 - 3a^2bcx - a^3c = 0$ \begin{chunk}{p30-2.2.3.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.3.1-3") graph2231:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2232:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2233:=getGraph(viewport3,1) putGraph(viewport1,graph2232,2) putGraph(viewport1,graph2233,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.3.1-3.eps}}} \label{CRCp30-2.2.3.1-3} \index{figures!CRCp30-2.2.3.1-3} \newpage \subsubsection{Page 30 2.2.4} $y=cx(a+bx)$ $y - bcx^2 - acx = 0$ \begin{chunk}{p30-2.2.4.1-3} )clear all f(x,a,b,c) == c*x*(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.4.1-3") graph2241:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2242:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2243:=getGraph(viewport3,1) putGraph(viewport1,graph2242,2) putGraph(viewport1,graph2243,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.4.1-3.eps}}} \label{CRCp30-2.2.4.1-3} \index{figures!CRCp30-2.2.4.1-3} \newpage \subsubsection{Page 30 2.2.5} $y=cx(a+bx)^2$ $y - b^2cx^3 - 2abcx^2 - a^2cx = 0$ \begin{chunk}{p30-2.2.5.1-3} )clear all f(x,a,b,c) == c*x*(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.5.1-3") graph2251:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2252:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2253:=getGraph(viewport3,1) putGraph(viewport1,graph2252,2) putGraph(viewport1,graph2253,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.5.1-3.eps}}} \label{CRCp30-2.2.5.1-3} \index{figures!CRCp30-2.2.5.1-3} \newpage \subsubsection{Page 30 2.2.6} $y=cx(a+bx)^3$ $y - b^3cx^4 - 3ab^2cx^3 - 3a^2bcx^2 - a^3cx = 0$ \begin{chunk}{p30-2.2.6.1-3} )clear all f(x,a,b,c) == c*x*(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.6.1-3") graph2261:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2262:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2263:=getGraph(viewport3,1) putGraph(viewport1,graph2262,2) putGraph(viewport1,graph2263,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp30-2.2.6.1-3.eps}}} \label{CRCp30-2.2.6.1-3} \index{figures!CRCp30-2.2.6.1-3} \newpage \subsubsection{Page 32 2.2.7} $y=cx^2(a+bx)$ $y - bcx^3 - acx^2 = 0$ \begin{chunk}{p32-2.2.7.1-3} )clear all f(x,a,b,c) == c*x^2*(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p30-2.2.7.1-3") graph2271:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2272:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2273:=getGraph(viewport3,1) putGraph(viewport1,graph2272,2) putGraph(viewport1,graph2273,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.7.1-3.eps}}} \label{CRCp32-2.2.7.1-3} \index{figures!CRCp32-2.2.7.1-3} \newpage \subsubsection{Page 32 2.2.8} $y = cx^2(a+bx)^2$ $y - b^2cx^4 - 2abcx^3 - a^2cx^2 = 0$ \begin{chunk}{p32-2.2.8.1-3} )clear all f(x,a,b,c) == c*x^2*(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p32-2.2.8.1-3") graph2281:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2282:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2283:=getGraph(viewport3,1) putGraph(viewport1,graph2282,2) putGraph(viewport1,graph2283,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.8.1-3.eps}}} \label{CRCp32-2.2.8.1-3} \index{figures!CRCp32-2.2.8.1-3} \newpage \subsubsection{Page 32 2.2.9} $y = cx^2(a+bx)^3$ $y - b^3cx^5 - 3ab^2cx^4 - 3a^2bcx^3 - a^3cx^2 = 0$ \begin{chunk}{p32-2.2.9.1-3} )clear all f(x,a,b,c) == c*x^2*(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0],_ title=="p32-2.2.9.1-3") graph2291:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2292:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..1,adaptive==true,unit==[1.0,1.0]) graph2293:=getGraph(viewport3,1) putGraph(viewport1,graph2292,2) putGraph(viewport1,graph2293,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.9.1-3.eps}}} \label{CRCp32-2.2.9.1-3} \index{figures!CRCp32-2.2.9.1-3} \newpage \subsubsection{Page 32 2.2.10} $y = cx^3(a+bx)$ $y - bcx^4 - acx^3 = 0$ \begin{chunk}{p32-2.2.10.1-3} )clear all f(x,a,b,c) == c*x^3*(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p32-2.2.10.1-3") graph22101:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22102:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22103:=getGraph(viewport3,1) putGraph(viewport1,graph22102,2) putGraph(viewport1,graph22103,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.10.1-3.eps}}} \label{CRCp32-2.2.10.1-3} \index{figures!CRCp32-2.2.10.1-3} \newpage \subsubsection{Page 32 2.2.11} $y = cx^3(a+bx)^2$ $y - b^2cx^5 - 2abcx^4 - a^2cx^3 = 0$ \begin{chunk}{p32-2.2.11.1-3} )clear all f(x,a,b,c) == c*x^3*(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p32-2.2.11.1-3") graph22111:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22112:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22113:=getGraph(viewport3,1) putGraph(viewport1,graph22112,2) putGraph(viewport1,graph22113,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.11.1-3.eps}}} \label{CRCp32-2.2.11.1-3} \index{figures!CRCp32-2.2.11.1-3} \newpage \subsubsection{Page 32 2.2.12} $y = cx^3(a+bx)^3$ $y - b^3cx^6 - 3ab^2cx^5 - 3a^2bcx^4 - a^3cx^3 = 0$ \begin{chunk}{p32-2.2.12.1-3} )clear all f(x,a,b,c) == c*x^3*(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p32-2.2.12.1-3") graph22121:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22122:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,2.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22123:=getGraph(viewport3,1) putGraph(viewport1,graph22122,2) putGraph(viewport1,graph22123,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp32-2.2.12.1-3.eps}}} \label{CRCp32-2.2.12.1-3} \index{figures!CRCp32-2.2.12.1-3} \newpage \subsubsection{Page 34 2.2.13} $\displaystyle y = \frac{c}{(a+bx)}$ $ay + bxy - c = 0$ \begin{chunk}{p34-2.2.13.1-3} )clear all f(x,a,b,c) == c/(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.13.1-3") graph22131:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22132:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22133:=getGraph(viewport3,1) putGraph(viewport1,graph22132,2) putGraph(viewport1,graph22133,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.13.1-3.eps}}} \label{CRCp34-2.2.13.1-3} \index{figures!CRCp34-2.2.13.1-3} \newpage \subsubsection{Page 34 2.2.14} $\displaystyle y = \frac{c}{(a+bx)^2}$ $a^2y + 2abxy + b^2x^2y - c = 0$ \begin{chunk}{p34-2.2.14.1-3} )clear all f(x,a,b,c) == c/(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.14.1-3") graph22141:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22142:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22143:=getGraph(viewport3,1) putGraph(viewport1,graph22142,2) putGraph(viewport1,graph22143,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.14.1-3.eps}}} \label{CRCp34-2.2.14.1-3} \index{figures!CRCp34-2.2.14.1-3} \newpage \subsubsection{Page 34 2.2.15} $\displaystyle y = \frac{c}{(a+bx)^3}$ $a^3y + 2a^2bxy + 2ab^2x^2y + b^3x^3y - c = 0$ \begin{chunk}{p34-2.2.15.1-3} )clear all f(x,a,b,c) == c/(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.15.1-3") graph22151:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22152:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22153:=getGraph(viewport3,1) putGraph(viewport1,graph22152,2) putGraph(viewport1,graph22153,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.15.1-3.eps}}} \label{CRCp34-2.2.15.1-3} \index{figures!CRCp34-2.2.15.1-3} \newpage \subsubsection{Page 34 2.2.16} $\displaystyle y = \frac{cx}{(a+bx)}$ $ay + bxy - cx = 0$ \begin{chunk}{p34-2.2.16.1-3} )clear all f(x,a,b,c) == c*x/(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.16.1-3") graph22161:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22162:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22163:=getGraph(viewport3,1) putGraph(viewport1,graph22162,2) putGraph(viewport1,graph22163,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.16.1-3.eps}}} \label{CRCp34-2.2.16.1-3} \index{figures!CRCp34-2.2.16.1-3} \newpage \subsubsection{Page 34 2.2.17} $\displaystyle y = \frac{cx}{(a+bx)^2}$ $a^2y + 2abxy + b^2x^2y - cx = 0$ \begin{chunk}{p34-2.2.17.1-3} )clear all f(x,a,b,c) == c*x/(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.17.1-3") graph22171:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22172:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22173:=getGraph(viewport3,1) putGraph(viewport1,graph22172,2) putGraph(viewport1,graph22173,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.17.1-3.eps}}} \label{CRCp34-2.2.17.1-3} \index{figures!CRCp34-2.2.17.1-3} \newpage \subsubsection{Page 34 2.2.18} $\displaystyle y = \frac{cx}{(a+bx)^3}$ $a^3y + 3a^2bxy + 3ab^2x^2y + b^3x^3y - cx = 0$ \begin{chunk}{p34-2.2.18.1-3} )clear all f(x,a,b,c) == c*x/(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p34-2.2.18.1-3") graph22181:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22182:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22183:=getGraph(viewport3,1) putGraph(viewport1,graph22182,2) putGraph(viewport1,graph22183,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp34-2.2.18.1-3.eps}}} \label{CRCp34-2.2.18.1-3} \index{figures!CRCp34-2.2.18.1-3} \newpage \subsubsection{Page 36 2.2.19} $\displaystyle y = \frac{cx^2}{(a+bx)}$ $ay + bxy - cx^2 = 0$ \begin{chunk}{p36-2.2.19.1-3} )clear all f(x,a,b,c) == c*x^2/(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.19.1-3") graph22191:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22192:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22193:=getGraph(viewport3,1) putGraph(viewport1,graph22192,2) putGraph(viewport1,graph22193,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.19.1-3.eps}}} \label{CRCp36-2.2.19.1-3} \index{figures!CRCp36-2.2.19.1-3} \newpage \subsubsection{Page 36 2.2.20} $\displaystyle y = \frac{cx^2}{(a+bx)^2}$ $a^2y + 2abxy + b^2x^2y - cx^2 = 0$ \begin{chunk}{p36-2.2.20.1-3} )clear all f(x,a,b,c) == c*x^2/(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.20.1-3") graph22201:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22202:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22203:=getGraph(viewport3,1) putGraph(viewport1,graph22202,2) putGraph(viewport1,graph22203,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.20.1-3.eps}}} \label{CRCp36-2.2.20.1-3} \index{figures!CRCp36-2.2.20.1-3} \newpage \subsubsection{Page 36 2.2.21} $\displaystyle y = \frac{cx^2}{(a+bx)^3}$ $a^3y + 3a^2bxy + 3ab^2x^2y + b^3x^3y - cx^2 = 0$ \begin{chunk}{p36-2.2.21.1-3} )clear all f(x,a,b,c) == c*x^2/(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.21.1-3") graph22211:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22212:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22213:=getGraph(viewport3,1) putGraph(viewport1,graph22212,2) putGraph(viewport1,graph22213,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.21.1-3.eps}}} \label{CRCp36-2.2.21.1-3} \index{figures!CRCp36-2.2.21.1-3} \newpage \subsubsection{Page 36 2.2.22} $\displaystyle y = \frac{cx^3}{(a+bx)}$ $ay + bxy - cx^3 = 0$ \begin{chunk}{p36-2.2.22.1-3} )clear all f(x,a,b,c) == c*x^3/(a+b*x) lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.22.1-3") graph22221:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22222:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22223:=getGraph(viewport3,1) putGraph(viewport1,graph22222,2) putGraph(viewport1,graph22223,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.22.1-3.eps}}} \label{CRCp36-2.2.22.1-3} \index{figures!CRCp36-2.2.22.1-3} \newpage \subsubsection{Page 36 2.2.23} $\displaystyle y = \frac{cx^3}{(a+bx)^2}$ $a^2y + 2abxy + b^2x^2y - cx^3 = 0$ \begin{chunk}{p36-2.2.23.1-3} )clear all f(x,a,b,c) == c*x^3/(a+b*x)^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.23.1-3") graph22231:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22232:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22233:=getGraph(viewport3,1) putGraph(viewport1,graph22232,2) putGraph(viewport1,graph22233,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.23.1-3.eps}}} \label{CRCp36-2.2.23.1-3} \index{figures!CRCp36-2.2.23.1-3} \newpage \subsubsection{Page 36 2.2.24} $\displaystyle y = \frac{cx^3}{(a+bx)^3}$ $a^3y + 3a^2bxy + 3ab^2x^2y + b^3x^3y - cx^3 = 0$ \begin{chunk}{p36-2.2.24.1-3} )clear all f(x,a,b,c) == c*x^3/(a+b*x)^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p36-2.2.24.1-3") graph22241:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,3.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22242:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,4.0,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22243:=getGraph(viewport3,1) putGraph(viewport1,graph22242,2) putGraph(viewport1,graph22243,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp36-2.2.24.1-3.eps}}} \label{CRCp36-2.2.24.1-3} \index{figures!CRCp36-2.2.24.1-3} \newpage \subsubsection{Page 38 2.2.25} $\displaystyle y = \frac{c(a+bx)}{x}$ $xy - bcx - ca = 0$ \begin{chunk}{p38-2.2.25.1-3} )clear all f(x,a,b,c) == c*(a+b*x)/x lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.04),x=-0.5..0.5,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.25.1-3") graph22251:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.04),x=-0.5..0.5,adaptive==true,unit==[1.0,1.0]) graph22252:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.04),x=-0.5..0.5,adaptive==true,unit==[1.0,1.0]) graph22253:=getGraph(viewport3,1) putGraph(viewport1,graph22252,2) putGraph(viewport1,graph22253,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.25.1-3.eps}}} \label{CRCp38-2.2.25.1-3} \index{figures!CRCp38-2.2.25.1-3} \newpage \subsubsection{Page 38 2.2.26} $\displaystyle y = \frac{c(a+bx)^2}{x}$ $xy - b^2cx^2 - 2abcx - a^2c = 0$ \begin{chunk}{p38-2.2.26.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^2/x lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.26.1-3") graph22261:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22262:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22263:=getGraph(viewport3,1) putGraph(viewport1,graph22262,2) putGraph(viewport1,graph22263,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.26.1-3.eps}}} \label{CRCp38-2.2.26.1-3} \index{figures!CRCp38-2.2.26.1-3} \newpage \subsubsection{Page 38 2.2.27} $\displaystyle y = \frac{c(a+bx)^3}{x}$ $xy - b^3cx^3 - 3ab^2cx^2 - 3a^2bcx - a^3c = 0$ \begin{chunk}{p38-2.2.27.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^3/x lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.27.1-3") graph22271:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22272:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22273:=getGraph(viewport3,1) putGraph(viewport1,graph22272,2) putGraph(viewport1,graph22273,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.27.1-3.eps}}} \label{CRCp38-2.2.27.1-3} \index{figures!CRCp38-2.2.27.1-3} \newpage \subsubsection{Page 38 2.2.28} $\displaystyle y = \frac{c(a+bx)}{x^2}$ $x^2y - bcx - ca = 0$ \begin{chunk}{p38-2.2.28.1-3} )clear all f(x,a,b,c) == c*(a+b*x)/x^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.04),x=-1..1,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.28.1-3") graph22281:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.04),x=-1..1,adaptive==true,unit==[1.0,1.0]) graph22282:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.04),x=-1..1,adaptive==true,unit==[1.0,1.0]) graph22283:=getGraph(viewport3,1) putGraph(viewport1,graph22282,2) putGraph(viewport1,graph22283,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.28.1-3.eps}}} \label{CRCp38-2.2.28.1-3} \index{figures!CRCp38-2.2.28.1-3} \newpage \subsubsection{Page 38 2.2.29} $\displaystyle y = \frac{c(a+bx)^2}{x^2}$ $x^2y - b^2cx^2 - 2abcx - a^2c = 0$ \begin{chunk}{p38-2.2.29.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^2/x^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.29.1-3") graph22291:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22292:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22293:=getGraph(viewport3,1) putGraph(viewport1,graph22292,2) putGraph(viewport1,graph22293,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.29.1-3.eps}}} \label{CRCp38-2.2.29.1-3} \index{figures!CRCp38-2.2.29.1-3} \newpage \subsubsection{Page 38 2.2.30} $\displaystyle y = \frac{c(a+bx)^3}{x^2}$ $x^2y - b^3cx^3 - 3ab^2cx^2 - 3a^2bcx - a^3c = 0$ \begin{chunk}{p38-2.2.30.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^3/x^2 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.003),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p38-2.2.30.1-3") graph22301:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.003),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22302:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.003),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22303:=getGraph(viewport3,1) putGraph(viewport1,graph22302,2) putGraph(viewport1,graph22303,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp38-2.2.30.1-3.eps}}} \label{CRCp38-2.2.30.1-3} \index{figures!CRCp38-2.2.30.1-3} \newpage \subsubsection{Page 40 2.2.31} $\displaystyle y = \frac{c(a+bx)}{x^3}$ $x^3y - bcx - ca = 0$ \begin{chunk}{p40-2.2.31.1-3} )clear all f(x,a,b,c) == c*(a+b*x)/x^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.02),x=-4..4,adaptive==true,unit==[1.0,1.0],_ title=="p40-2.2.31.1-3") graph22311:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.02),x=-4..4,adaptive==true,unit==[1.0,1.0]) graph22312:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.02),x=-4..4,adaptive==true,unit==[1.0,1.0]) graph22313:=getGraph(viewport3,1) putGraph(viewport1,graph22312,2) putGraph(viewport1,graph22313,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp40-2.2.31.1-3.eps}}} \label{CRCp40-2.2.31.1-3} \index{figures!CRCp40-2.2.31.1-3} \newpage \subsubsection{Page 40 2.2.32} $\displaystyle y = \frac{c(a+bx)^2}{x^3}$ $x^3y - b^2cx^2 - 2abcx - a^2c = 0$ \begin{chunk}{p40-2.2.32.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^2/x^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p40-2.2.32.1-3") graph22321:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22322:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22323:=getGraph(viewport3,1) putGraph(viewport1,graph22322,2) putGraph(viewport1,graph22323,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp40-2.2.32.1-3.eps}}} \label{CRCp40-2.2.32.1-3} \index{figures!CRCp40-2.2.32.1-3} \newpage \subsubsection{Page 40 2.2.33} $\displaystyle y = \frac{c(a+bx)^3}{x^3}$ $x^3y - b^3cx^3 - 3ab^2cx^2 - 3a^2bcx - a^3c = 0$ \begin{chunk}{p40-2.2.33.1-3} )clear all f(x,a,b,c) == c*(a+b*x)^3/x^3 lineColorDefault(red()) viewport1:=draw(f(x,1.0,2.0,0.002),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p40-2.2.33.1-3") graph22331:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,4.0,0.002),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22332:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,6.0,0.002),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph22333:=getGraph(viewport3,1) putGraph(viewport1,graph22332,2) putGraph(viewport1,graph22333,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp40-2.2.33.1-3.eps}}} \label{CRCp40-2.2.33.1-3} \index{figures!CRCp40-2.2.33.1-3} \newpage \subsection{Functions with $a^2+x^2$ and $x^m$} \subsubsection{Page 42 2.3.1} $\displaystyle y=\frac{c}{(a^2+b^2)}$ $a^2y + x^2y - c = 0$ \begin{chunk}{p42-2.3.1.1-3} )clear all f(x,a,c) == c/(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p42-2.3.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.04),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp42-2.3.1.1-3.eps}}} \label{CRCp42-2.3.1.1-3} \index{figures!CRCp42-2.3.1.1-3} \newpage \subsubsection{Page 42 2.3.2} $\displaystyle y=\frac{cx}{(a^2+b^2)}$ $a^2y + x^2y - cx = 0$ \begin{chunk}{p42-2.3.2.1-3} )clear all f(x,a,c) == c*x/(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.3),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p42-2.3.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.3),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.3),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp42-2.3.2.1-3.eps}}} \label{CRCp42-2.3.2.1-3} \index{figures!CRCp42-2.3.2.1-3} \newpage \subsubsection{Page 42 2.3.3} $\displaystyle y=\frac{cx^2}{(a^2+b^2)}$ $a^2y + x^2y - cx^2 = 0$ \begin{chunk}{p42-2.3.3.1-3} )clear all f(x,a,c) == c*x^2/(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p42-2.3.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp42-2.3.3.1-3.eps}}} \label{CRCp42-2.3.3.1-3} \index{figures!CRCp42-2.3.3.1-3} \newpage \subsubsection{Page 42 2.3.4} $\displaystyle y=\frac{cx^3}{(a^2+b^2)}$ $a^2y + x^2y - cx^3 = 0$ \begin{chunk}{p42-2.3.4.1-3} )clear all f(x,a,c) == c*x^3/(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p42-2.3.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp42-2.3.4.1-3.eps}}} \label{CRCp42-2.3.4.1-3} \index{figures!CRCp42-2.3.4.1-3} \newpage \subsubsection{Page 44 2.3.5} $\displaystyle y=\frac{c}{x(a^2+b^2)}$ $a^2xy + x^3y - c = 0$ \begin{chunk}{p44-2.3.5.1-3} )clear all f(x,a,c) == c/(x*(a^2+x^2)) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p44-2.3.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp44-2.3.5.1-3.eps}}} \label{CRCp44-2.3.5.1-3} \index{figures!CRCp44-2.3.5.1-3} \newpage \subsubsection{Page 44 2.3.6} $\displaystyle y=\frac{c}{x^2(a^2+b^2)}$ $a^2x^2y + x^4y - c = 0$ \begin{chunk}{p44-2.3.6.1-3} )clear all f(x,a,c) == c/(x^2*(a^2+x^2)) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p44-2.3.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp44-2.3.6.1-3.eps}}} \label{CRCp44-2.3.6.1-3} \index{figures!CRCp44-2.3.6.1-3} \newpage \subsubsection{Page 44 2.3.7} $\displaystyle y=cx(a^2+x^2)$ $y - a^2cx - cx^3 = 0$ \begin{chunk}{p44-2.3.7.1-3} )clear all f(x,a,c) == c*x*(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p44-2.3.7.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp44-2.3.7.1-3.eps}}} \label{CRCp44-2.3.7.1-3} \index{figures!CRCp44-2.3.7.1-3} \newpage \subsubsection{Page 44 2.3.8} $\displaystyle y=cx^2(a^2+x^2)$ $y - a^2cx^2 - cx^4 = 0$ \begin{chunk}{p44-2.3.8.1-3} )clear all f(x,a,c) == c*x^2*(a^2+x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p44-2.3.8.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp44-2.3.8.1-3.eps}}} \label{CRCp44-2.3.8.1-3} \index{figures!CRCp44-2.3.8.1-3} \newpage \subsection{Functions with $a^2-x^2$ and $x^m$} \subsubsection{Page 46 2.4.1} $\displaystyle y=\frac{c}{(a^2-x^2)}$ $a^2y - x^2y - c = 0$ \begin{chunk}{p46-2.4.1.1-3} )clear all f(x,a,c) == c/(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.03),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p46-2.4.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.03),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.03),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp46-2.4.1.1-3.eps}}} \label{CRCp46-2.4.1.1-3} \index{figures!CRCp46-2.4.1.1-3} \newpage \subsubsection{Page 46 2.4.2} $\displaystyle y=\frac{cx}{(a^2-x^2)}$ $a^2y - x^2y - cx = 0$ \begin{chunk}{p46-2.4.2.1-3} )clear all f(x,a,c) == c*x/(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p46-2.4.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp46-2.4.2.1-3.eps}}} \label{CRCp46-2.4.2.1-3} \index{figures!CRCp46-2.4.2.1-3} \newpage \subsubsection{Page 46 2.4.3} $\displaystyle y=\frac{cx^2}{(a^2-x^2)}$ $a^2y - x^2y - cx^2 = 0$ \begin{chunk}{p46-2.4.3.1-3} )clear all f(x,a,c) == c*x^2/(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p46-2.4.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp46-2.4.3.1-3.eps}}} \label{CRCp46-2.4.3.1-3} \index{figures!CRCp46-2.4.3.1-3} \newpage \subsubsection{Page 46 2.4.4} $\displaystyle y=\frac{cx^3}{(a^2-x^2)}$ $a^2y - x^2y - cx^3 = 0$ \begin{chunk}{p46-2.4.4.1-3} )clear all f(x,a,c) == c*x^3/(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p46-2.4.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp46-2.4.4.1-3.eps}}} \label{CRCp46-2.4.4.1-3} \index{figures!CRCp46-2.4.4.1-3} \newpage \subsubsection{Page 48 2.4.5} $\displaystyle y=\frac{c}{x(a^2-x^2)}$ $a^2xy - x^3y - c = 0$ \begin{chunk}{p48-2.4.5.1-3} )clear all f(x,a,c) == c/(x*(a^2-x^2)) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.001),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p48-2.4.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.001),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.001),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp48-2.4.5.1-3.eps}}} \label{CRCp48-2.4.5.1-3} \index{figures!CRCp48-2.4.5.1-3} \newpage \subsubsection{Page 48 2.4.6} $\displaystyle y=\frac{c}{x^2(a^2-x^2)}$ $a^2x^2y - x^4y - c = 0$ \begin{chunk}{p48-2.4.6.1-3} )clear all f(x,a,c) == c/(x^2*(a^2-x^2)) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.0003),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p48-2.4.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.0003),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.0003),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp48-2.4.6.1-3.eps}}} \label{CRCp48-2.4.6.1-3} \index{figures!CRCp48-2.4.6.1-3} \newpage \subsubsection{Page 48 2.4.7} $\displaystyle y=cx(a^2-x^2)$ $y - a^2cx + cx^3 = 0$ \begin{chunk}{p48-2.4.7.1-3} )clear all f(x,a,c) == c*x*(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p48-2.4.7.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp48-2.4.7.1-3.eps}}} \label{CRCp48-2.4.7.1-3} \index{figures!CRCp48-2.4.7.1-3} \newpage \subsubsection{Page 48 2.4.8} $\displaystyle y=cx^2(a^2-x^2)$ $y - a^2cx^2 + cx^4 = 0$ \begin{chunk}{p48-2.4.8.1-3} )clear all f(x,a,c) == c*x^2*(a^2-x^2) lineColorDefault(red()) viewport1:=draw(f(x,0.2,4.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p48-2.4.8.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,4.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,4.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp48-2.4.8.1-3.eps}}} \label{CRCp48-2.4.8.1-3} \index{figures!CRCp48-2.4.8.1-3} \newpage \subsection{Functions with $a^3+x^3$ and $x^m$} \subsubsection{Page 50 2.5.1} $\displaystyle y=\frac{c}{(a^3+x^3)}$ $a^3y + x^3y - c = 0$ \begin{chunk}{p50-2.5.1.1-3} )clear all f(x,a,c) == c/(a^3+x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.4,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.1.1-3.eps}}} \label{CRCp50-2.5.1.1-3} \index{figures!CRCp50-2.5.1.1-3} \newpage \subsubsection{Page 50 2.5.2} $\displaystyle y=\frac{cx}{(a^3+x^3)}$ $a^3y + x^3y - cx = 0$ \begin{chunk}{p50-2.5.2.1-3} )clear all f(x,a,c) == c*x/(a^3+x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.2.1-3.eps}}} \label{CRCp50-2.5.2.1-3} \index{figures!CRCp50-2.5.2.1-3} \newpage \subsubsection{Page 50 2.5.3} $\displaystyle y=\frac{cx^2}{(a^3+x^3)}$ $a^3y + x^3y - cx^2 = 0$ \begin{chunk}{p50-2.5.3.1-3} )clear all f(x,a,c) == c*x^2/(a^3+x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.3.1-3.eps}}} \label{CRCp50-2.5.3.1-3} \index{figures!CRCp50-2.5.3.1-3} \newpage \subsubsection{Page 50 2.5.4} $\displaystyle y=\frac{cx^3}{(a^3+x^3)}$ $a^3y + x^3y - cx^3 = 0$ \begin{chunk}{p50-2.5.4.1-3} )clear all f(x,a,c) == c*x^3/(a^3+x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.02),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.4.1-3.eps}}} \label{CRCp50-2.5.4.1-3} \index{figures!CRCp50-2.5.4.1-3} \newpage \subsubsection{Page 50 2.5.5} $\displaystyle y=\frac{c}{x(a^3+x^3)}$ $a^3xy + x^4y - c = 0$ \begin{chunk}{p50-2.5.5.1-3} )clear all f(x,a,c) == c/(x*(a^3+x^3)) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.7,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.9,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.5.1-3.eps}}} \label{CRCp50-2.5.5.1-3} \index{figures!CRCp50-2.5.5.1-3} \newpage \subsubsection{Page 50 2.5.6} $\displaystyle y=cx(a^3+x^3)$ $y - a^3cx - cx^4 = 0$ \begin{chunk}{p50-2.5.6.1-3} )clear all f(x,a,c) == c*x*(a^3+x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p50-2.5.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.7,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.9,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp50-2.5.6.1-3.eps}}} \label{CRCp50-2.5.6.1-3} \index{figures!CRCp50-2.5.6.1-3} \newpage \subsection{Functions with $a^3-x^3$ and $x^m$} \subsubsection{Page 52 2.6.1} $\displaystyle y=\frac{c}{(a^3-x^3)}$ $a^3y - x^3y - c = 0$ \begin{chunk}{p52-2.6.1.1-3} )clear all f(x,a,c) == c/(a^3-x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.4,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.1.1-3.eps}}} \label{CRCp52-2.6.1.1-3} \index{figures!CRCp52-2.6.1.1-3} \newpage \subsubsection{Page 52 2.6.2} $\displaystyle y=\frac{cx}{(a^3-x^3)}$ $a^3y - x^3y - cx = 0$ \begin{chunk}{p52-2.6.2.1-3} )clear all f(x,a,c) == c*x/(a^3-x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.2.1-3.eps}}} \label{CRCp52-2.6.2.1-3} \index{figures!CRCp52-2.6.2.1-3} \newpage \subsubsection{Page 52 2.6.3} $\displaystyle y=\frac{cx^2}{(a^3-x^3)}$ $a^3y - x^3y - cx^2 = 0$ \begin{chunk}{p52-2.6.3.1-3} )clear all f(x,a,c) == c*x^2/(a^3-x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.3.1-3.eps}}} \label{CRCp52-2.6.3.1-3} \index{figures!CRCp52-2.6.3.1-3} \newpage \subsubsection{Page 52 2.6.4} $\displaystyle y=\frac{cx^3}{(a^3-x^3)}$ $a^3y - x^3y - cx^3 = 0$ \begin{chunk}{p52-2.6.4.1-3} )clear all f(x,a,c) == c*x^3/(a^3-x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.1,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.2),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.4.1-3.eps}}} \label{CRCp52-2.6.4.1-3} \index{figures!CRCp52-2.6.4.1-3} \newpage \subsubsection{Page 52 2.6.5} $\displaystyle y=\frac{c}{x(a^3-x^3)}$ $a^3xy - x^4y - c = 0$ \begin{chunk}{p52-2.6.5.1-3} )clear all f(x,a,c) == c/(x*(a^3-x^3)) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.7,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.9,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.5.1-3.eps}}} \label{CRCp52-2.6.5.1-3} \index{figures!CRCp52-2.6.5.1-3} \newpage \subsubsection{Page 52 2.6.6} $\displaystyle y=cx(a^3-x^3)$ $y - a^3cx + cx^4 = 0$ \begin{chunk}{p52-2.6.6.1-3} )clear all f(x,a,c) == c*x*(a^3-x^3) lineColorDefault(red()) viewport1:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p52-2.6.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.7,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.9,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp52-2.6.6.1-3.eps}}} \label{CRCp52-2.6.6.1-3} \index{figures!CRCp52-2.6.6.1-3} \newpage \subsection{Functions with $a^4+x^4$ and $x^m$} \subsubsection{Page 54 2.7.1} $\displaystyle y=\frac{c}{(a^4+x^4)}$ $a^4y + x^4y - c = 0$ \begin{chunk}{p54-2.7.1.1-3} )clear all f(x,a,c) == c/(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.3,0.007),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.007),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.007),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.1.1-3.eps}}} \label{CRCp54-2.7.1.1-3} \index{figures!CRCp54-2.7.1.1-3} \newpage \subsubsection{Page 54 2.7.2} $\displaystyle y=\frac{cx}{(a^4+x^4)}$ $a^4y + x^4y - cx = 0$ \begin{chunk}{p54-2.7.2.1-3} )clear all f(x,a,c) == c*x/(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.3,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.4,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.2.1-3.eps}}} \label{CRCp54-2.7.2.1-3} \index{figures!CRCp54-2.7.2.1-3} \newpage \subsubsection{Page 54 2.7.3} $\displaystyle y=\frac{cx^2}{(a^4+x^4)}$ $a^4y + x^4y - cx^2 = 0$ \begin{chunk}{p54-2.7.3.1-3} )clear all f(x,a,c) == c*x^2/(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.3,0.15),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.15),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.5,0.15),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.3.1-3.eps}}} \label{CRCp54-2.7.3.1-3} \index{figures!CRCp54-2.7.3.1-3} \newpage \subsubsection{Page 54 2.7.4} $\displaystyle y=\frac{cx^3}{(a^4+x^4)}$ $a^4y + x^4y - cx^3 = 0$ \begin{chunk}{p54-2.7.4.1-3} )clear all f(x,a,c) == c*x^3/(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.25),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.25),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.6,0.25),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.4.1-3.eps}}} \label{CRCp54-2.7.4.1-3} \index{figures!CRCp54-2.7.4.1-3} \newpage \subsubsection{Page 54 2.7.5} $\displaystyle y=\frac{cx^4}{(a^4+x^4)}$ $a^4y + x^4y - cx^4 = 0$ \begin{chunk}{p54-2.7.5.1-3} )clear all f(x,a,c) == c*x^4/(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.5.1-3.eps}}} \label{CRCp54-2.7.5.1-3} \index{figures!CRCp54-2.7.5.1-3} \newpage \subsubsection{Page 54 2.7.6} $\displaystyle y=cx(a^4+x^4)$ $y - a^4cx - cx^5 = 0$ \begin{chunk}{p54-2.7.6.1-3} )clear all f(x,a,c) == c*x*(a^4+x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.5,0.5),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p54-2.7.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,1.0,0.5),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.2,0.5),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp54-2.7.6.1-3.eps}}} \label{CRCp54-2.7.6.1-3} \index{figures!CRCp54-2.7.6.1-3} \newpage \subsection{Functions with $a^4-x^4$ and $x^m$} \subsubsection{Page 56 2.8.1} $\displaystyle y=\frac{c}{(a^4-x^4)}$ $a^4y - x^4y - c = 0$ \begin{chunk}{p56-2.8.1.1-3} )clear all f(x,a,c) == c/(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.4,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.1.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.6,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.1.1-3.eps}}} \label{CRCp56-2.8.1.1-3} \index{figures!CRCp56-2.8.1.1-3} \newpage \subsubsection{Page 56 2.8.2} $\displaystyle y=\frac{cx}{(a^4-x^4)}$ $a^4y - x^4y - cx = 0$ \begin{chunk}{p56-2.8.2.1-3} )clear all f(x,a,c) == c*x/(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.2.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.6,0.01),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.2.1-3.eps}}} \label{CRCp56-2.8.2.1-3} \index{figures!CRCp56-2.8.2.1-3} \newpage \subsubsection{Page 56 2.8.3} $\displaystyle y=\frac{cx^2}{(a^4-x^4)}$ $a^4y - x^4y - cx^2 = 0$ \begin{chunk}{p56-2.8.3.1-3} )clear all f(x,a,c) == c*x^2/(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.3.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.6,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.3.1-3.eps}}} \label{CRCp56-2.8.3.1-3} \index{figures!CRCp56-2.8.3.1-3} \newpage \subsubsection{Page 56 2.8.4} $\displaystyle y=\frac{cx^3}{(a^4-x^4)}$ $a^4y - x^4y - cx^3 = 0$ \begin{chunk}{p56-2.8.4.1-3} )clear all f(x,a,c) == c*x^3/(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.4.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.4,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.6,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.4.1-3.eps}}} \label{CRCp56-2.8.4.1-3} \index{figures!CRCp56-2.8.4.1-3} \newpage \subsubsection{Page 56 2.8.5} $\displaystyle y=\frac{cx^4}{(a^4-x^4)}$ $a^4y - x^4y - cx^4 = 0$ \begin{chunk}{p56-2.8.5.1-3} )clear all f(x,a,c) == c*x^4/(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.2,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.5.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.5,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,0.8,0.1),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.5.1-3.eps}}} \label{CRCp56-2.8.5.1-3} \index{figures!CRCp56-2.8.5.1-3} \newpage \subsubsection{Page 56 2.8.6} $\displaystyle y=cx(a^4-x^4)$ $y - a^4cx + cx^5 = 0$ \begin{chunk}{p56-2.8.6.1-3} )clear all f(x,a,c) == c*x*(a^4-x^4) lineColorDefault(red()) viewport1:=draw(f(x,0.4,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0],_ title=="p56-2.8.6.1-3") graph1:=getGraph(viewport1,1) lineColorDefault(green()) viewport2:=draw(f(x,0.8,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) lineColorDefault(blue()) viewport3:=draw(f(x,1.0,1.0),x=-2..2,adaptive==true,unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp56-2.8.6.1-3.eps}}} \label{CRCp56-2.8.6.1-3} \index{figures!CRCp56-2.8.6.1-3} \newpage \subsection{Functions with $(a+bx)^{1/2}$ and $x^m$} \subsubsection{Page 58 2.9.1} Parabola $\displaystyle y=c(a+bx)^{1/2}$ $y^2 - bc^2x - ac^2 = 0$ \begin{chunk}{p58-2.9.1.1-3} )clear all f1(x,y) == y^2 - x/8 - 1/2 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.1.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == y^2 - x - 1/2 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == y^2 - 2*x - 1/2 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.1.1-3.eps}}} \label{CRCp58-2.9.1.1-3} \index{figures!CRCp58-2.9.1.1-3} \newpage \subsubsection{Page 58 2.9.2} Trisectrix of Catalan... fails "singular pts in region of sketch" $\displaystyle y=cx(a+bx)^{1/2}$ $y^2 - bc^2x^3 - ac^2x^2 = 0$ \begin{chunk}{p58-2.9.2.1-3} )clear all f1(x,y) == y^2 - x^3/2 - x^2/2 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.2.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == y^2 - x^3 - x^2/2 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == y^2 - 2*x^3 - x^2/2 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} %{\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.2.1-3.eps}}} \label{CRCp58-2.9.2.1-3} \index{figures!CRCp58-2.9.2.1-3} \newpage \subsubsection{Page 58 2.9.3} fails with "singular pts in region of sketch" $\displaystyle y=cx^2(a+bx)^{1/2}$ $y^2 - bc^2x^5 - ac^2x^4 = 0$ \begin{chunk}{p58-2.9.3.1-3} )clear all f1(x,y) == y^2 - x^5/2 - x^4/2 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.3.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == y^2 - x^5 - x^4/2 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == y^2 - 2*x^5 - x^4/2 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} %{\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.3.1-3.eps}}} \label{CRCp58-2.9.3.1-3} \index{figures!CRCp58-2.9.3.1-3} \newpage \subsubsection{Page 58 2.9.4} $\displaystyle y=c(a+bx)^{1/2}/x$ $x^2y^2 - c^2bx - c^2a = 0$ \begin{chunk}{p58-2.9.4.1-3} )clear all f1(x,y) == x^2*y^2 - x/50 - 1/50 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.4.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == x^2*y^2 - x/25 - 1/50 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == x^2*y^2 - 2*x/25 -1/50 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.4.1-3.eps}}} \label{CRCp58-2.9.4.1-3} \index{figures!CRCp58-2.9.4.1-3} \newpage \subsubsection{Page 58 2.9.5} $\displaystyle y=c(a+bx)^{1/2}/x^2$ $x^4y^2 - c^2bx - c^2a = 0$ \begin{chunk}{p58-2.9.5.1-3} )clear all f1(x,y) == x^2*y^2 - x/200 - 1/200 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.5.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == x^2*y^2 - x/100 - 1/200 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == x^2*y^2 - x/50 - 1/200 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-2.0..2.0,-2.0..2.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.5.1-3.eps}}} \label{CRCp58-2.9.5.1-3} \index{figures!CRCp58-2.9.5.1-3} \newpage \subsubsection{Page 58 2.9.6} $\displaystyle y=c/(a+bx)^{1/2}$ $ay^2 + bxy^2 - c^2 = 0$ \begin{chunk}{p58-2.9.6.1-3} )clear all f1(x,y) == (x + 1)*y^2 - 1/4 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p58-2.9.6.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == (2*x+1)*y^2 - 1/4 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == (4*x+1)*y^2 - 1/4 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp58-2.9.6.1-3.eps}}} \label{CRCp58-2.9.6.1-3} \index{figures!CRCp58-2.9.6.1-3} \newpage \subsubsection{Page 60 2.9.7} singular pts in region of sketch $\displaystyle y=\frac{cx}{(a+bx)^{1/2}}$ $ay^2 + bxy^2 - c^2x^2 = 0$ \begin{chunk}{p60-2.9.7.1-3} )clear all f1(x,y) == (x + 1)*y^2 - x^2 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p60-2.9.7.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == (2*x+1)*y^2 - x^2 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == (4*x+1)*y^2 - x^2 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} %{\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp60-2.9.7.1-3.eps}}} \label{CRCp60-2.9.7.1-3} \index{figures!CRCp60-2.9.7.1-3} \newpage \subsubsection{Page 60 2.9.8} singular pts in region of sketch $\displaystyle y=\frac{cx^2}{(a+bx)^{1/2}}$ $ay^2 + bxy^2 - c^2x^4 = 0$ \begin{chunk}{p60-2.9.8.1-3} )clear all f1(x,y) == (x + 1)*y^2 - x^4 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p60-2.9.8.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == (2*x+1)*y^2 - x^4 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == (4*x+1)*y^2 - x^4 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} %{\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp60-2.9.8.1-3.eps}}} \label{CRCp60-2.9.8.1-3} \index{figures!CRCp60-2.9.8.1-3} \newpage \subsubsection{Page 60 2.9.9} $\displaystyle y=\frac{c}{x(a+bx)^{1/2}}$ $ax^2y^2 + bx^3y^2 - c^2 = 0$ \begin{chunk}{p60-2.9.9.1-3} )clear all f1(x,y) == (4/5*x^3 + x^2)*y*2 - 1/25 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p60-2.9.9.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == (x^3 + x^2)*y^2 - 1/25 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == (6/5*x^3 + x^2)*y^2 - 1/25 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp60-2.9.9.1-3.eps}}} \label{CRCp60-2.9.9.1-3} \index{figures!CRCp60-2.9.9.1-3} \newpage \subsubsection{Page 60 2.9.10} $\displaystyle y=\frac{c}{x^2(a+bx)^{1/2}}$ $ax^4y^2 + bx^5y^2 - c^2 = 0$ \begin{chunk}{p60-2.9.10.1-3} )clear all f1(x,y) == (4/5*x^5 + x^4)*y*2 - 1/100 lineColorDefault(red()) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p60-2.9.10.1-3") graph1:=getGraph(viewport1,1) f2(x,y) == (x^5 + x^4)*y^2 - 1/100 lineColorDefault(green()) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == (6/5*x^5 + x^4)*y^2 - 1/100 lineColorDefault(blue()) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp60-2.9.10.1-3.eps}}} \label{CRCp60-2.9.10.1-3} \index{figures!CRCp60-2.9.10.1-3} \newpage \subsubsection{Page 60 2.9.11} $\displaystyle y=\frac{cx^{1/2}}{(a+bx)^{1/2}}$ $y^2 - ac^2x - bc^2x^2 = 0$ \begin{chunk}{p60-2.9.11.1-6} )clear all f1(x,y) == y^2 + 2*x^2 - 2*x lineColorDefault(color(1)) viewport1:=draw(f1(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0],title=="p60-2.9.11.1-6") graph1:=getGraph(viewport1,1) f2(x,y) == y^2 + 3*x^2 - 2*x lineColorDefault(color(2)) viewport2:=draw(f2(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph2:=getGraph(viewport2,1) f3(x,y) == y^2 + 4*x^2 - 2*x lineColorDefault(color(3)) viewport3:=draw(f3(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph3:=getGraph(viewport3,1) f4(x,y) == y^2 - x^2 - 3*x/10 lineColorDefault(color(4)) viewport4:=draw(f4(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph4:=getGraph(viewport4,1) f5(x,y) == y^2 - x^2 - x/2 lineColorDefault(color(5)) viewport5:=draw(f5(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph5:=getGraph(viewport5,1) f6(x,y) == y^2 - x^2 -7*x/10 lineColorDefault(color(6)) viewport6:=draw(f6(x,y)=0,x,y,range==[-4.0..4.0,-4.0..4.0],adaptive==true,_ unit==[1.0,1.0]) graph6:=getGraph(viewport6,1) putGraph(viewport1,graph2,2) putGraph(viewport1,graph3,3) putGraph(viewport1,graph4,4) putGraph(viewport1,graph5,5) putGraph(viewport1,graph6,6) units(viewport1,1,"on") points(viewport1,1,"off") points(viewport1,2,"off") points(viewport1,3,"off") makeViewport2D(viewport1) \end{chunk} {\center{\includegraphics[height=8cm,width=8cm]{ps/v81crcp60-2.9.11.1-6.eps}}} \label{CRCp60-2.9.11.1-6} \index{figures!CRCp60-2.9.11.1-6} \chapter{Pasta by Design} This is a book\cite{Lege11} that combines a taxonomy of pasta shapes with the Mathematica equations that realize those shapes in three dimensions. We implemented examples from this book as a graphics test suite for Axiom. \newpage \section{Acini Di Pepe} {\center{\includegraphics[scale=0.50]{ps/v81acinidipepe.eps}}} \begin{chunk}{Acini Di Pepe} X(i,j) == 15*cos(i/60*%pi) Y(i,j) == 15*sin(i/60*%pi) Z(i,y) == j cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..120,j=0..30,_ style=="smooth",title=="Acini Di Pepe",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") \end{chunk} The smallest member of the {\sl postine minute} (tiny pasta) family, {\sl acini de pepe} (peppercorns) are most suited to consommes (clear soups), with the occasional addition of croutons and diced greens. Made of durum wheat flour and eggs, acini di pepe are commonly used in the Italian-American wedding soup'', a broth of vegetables and meat. \newpage \section{Agnolotti} {\center{\includegraphics[scale=0.50]{ps/v81agnolotti.eps}}} \begin{chunk}{Agnolotti} X(i,j) == (10*sin((i/120)*%pi)^(0.5) + _ (1/400)*sin(((3*j)/10)*%pi)) * _ cos(((19*j)/2000)*%pi+0.03*%pi) Y(i,j) == (10*sin((i/120)*%pi) + _ (1/400)*cos(((30*j)/10)*%pi)) * _ sin(((19*j)/2000)*%pi+0.03*%pi) Z(i,j) == 5*cos((i/120)*%pi)^5 * sin((j/100)*%pi) - _ 5*sin((j/100)*%pi) * cos((i/120)*%pi)^200 cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..60,j=0..100,_ style=="smooth",title=="Agnolotti",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,0.6,0.6,0.6) \end{chunk} These shell-like {\sl ravioli} from Piedmont, northern italy, are fashioned from small pieces of flattened dough made of wheat flour and egg, and are often filled with braised veal, port, vegetables or cheese. The true agnolotto should feature a crinkled edge, cut using a fluted pasta wheel. Recommended with melted butter and sage \newpage \section{Anellini} {\center{\includegraphics[scale=0.50]{ps/v81anellini.eps}}} \begin{chunk}{Anellini} X(i,j) == cos(0.01*i*%pi) Y(i,j) == 1.1*sin(0.01*i*%pi) Z(i,j) == 0.05*j canvas := createThreeSpace() cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 makeObject(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..8,space==canvas,_ colorFunction==cf,style=="smooth") makeObject(surface(X(i,j)/1.4,Y(i,j)/1.4,Z(i,j)),i=0..200,j=0..8,_ space==canvas,colorFunction==cf,style=="smooth") vp:=makeViewport3D(canvas,style=="smooth",title=="Anellini") colorDef(vp,yellow(),yellow()) axes(vp,"off") zoom(vp,1.1,1.1,1.1) \end{chunk} The diminutive {\sl onellini} (small rings) are part of the extended {\sl postine minute} (tiny pasta) clan. Their thickness varies between only 1.15 and 1.20 mm, and the are therefore usually found in light soups together with croutons and thinly sliced vegetables. This pasta may also be found served in a {\sl timballo} (baked pasta dish). \newpage \section{Bucatini} {\center{\includegraphics[scale=0.50]{ps/v81bucatini.eps}}} \begin{chunk}{Bucatini} X(i,j) == 0.3*cos(i/30*%pi) Y(i,j) == 0.3*sin(i/30*%pi) Z(i,j) == j/45 canvas := createThreeSpace() cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 makeObject(surface(X(i,j),Y(i,j),Z(i,j)),i=0..60,j=0..90,space==canvas,_ colorFunction==cf,style=="smooth") makeObject(surface(X(i,j)/2,Y(i,j)/2,Z(i,j)),i=0..60,j=0..90,space==canvas,_ colorFunction==cf,style=="smooth") vp:=makeViewport3D(canvas,style=="smooth",title=="Bucatini") colorDef(vp,yellow(),yellow()) axes(vp,"off") zoom(vp,2.0,2.0,2.0) \end{chunk} {\sl Bucatini} (pierced) pasta is commonly served as a {\sl pastasciutta} (pasta boiled, drained, and dished up with a sauce, rather than in broth). Its best known accompaniment is {\sl amatriciana}: a hearty traditional sauce made with dried port, Pecorino Romano and tomato sauce, and named after the medieval town of Amatrice in central Italy. \newpage \section{Buccoli} {\center{\includegraphics[scale=0.50]{ps/v81buccoli.eps}}} \begin{chunk}{Buccoli} X(i,j) == (0.7 + 0.2*sin(21*j/250 * %pi))*cos(i/20*%pi) Y(i,j) == (0.7 + 0.2*sin(21*j/250 * %pi))*sin(i/20*%pi) Z(i,j) == 39.0*i/1000. + 1.5*sin(j/50*%pi) cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..25,_ style=="smooth",title=="Buccoli",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) \end{chunk} A spiral-shaped example from the {\sl pasta corta} (short pasta) family, and of rather uncertain pedigree, {\sl buccoli} are suitable in a mushroom and sausage dish. They are also excellent with a tomato aubergine, pesto, and ricotta salad. \newpage \section{Calamaretti} {\center{\includegraphics[scale=0.50]{ps/v81calamaretti.eps}}} \begin{chunk}{Calamaretti} X(i,j) == cos(i/75*%pi) + 0.1*cos(j/40*%pi) + 0.1*cos(i/75*%pi + j/40*%pi) Y(i,j) == 1.2*sin(i/75*%pi) + 0.2*sin(j/40*%pi) Z(i,j) == j/10 cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..150,j=0..20,_ style=="smooth",title=="Calamaretti",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,1.1,1.1,1.1) \end{chunk} Literally little squids'', {\sl calamaretti} are small ring-shaped pasta cooked as {\sl pastasciutta} (pasta boiled and drained) then dished up with a tomato-, egg-, or cheese-based sauce. Their shape means that {\sl calamaretti} hold both chunky and thin sauces equally well. Fittingly, they are often served with seafood. \newpage \section{Cannelloni} {\center{\includegraphics[scale=0.50]{ps/v81cannelloni.eps}}} \begin{chunk}{Cannelloni} X(i,j) == (1+j/100)*cos(i*%pi/55) + 0.5*cos(j*%pi/100) + _ 0.1*cos(i*%pi/55+j*%pi/125) Y(i,j) == 1.3*sin(i*%pi/55) + 0.3*sin(j*%pi/100) Z(i,j) == 7.*j/50. cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..110,j=0..50,_ style=="smooth",title=="Cannelloni",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) \end{chunk} Made with wheat flour, eggs, and olive oil, {\sl cannelloni} (big tubes) originate as strips of pasta shaped into perfect cylinders, which can be stuffed with meat, vegetables, or ricotta. The stuffed {\sl cannelloni} are covered with a creamy {besciamella} sauce, a sprinkling of Parmigiano-Reggiano cheese and then oven-baked. \newpage \section{Cannolicchi Rigati} {\center{\includegraphics[scale=0.50]{ps/v81cannolicchirigati.eps}}} \begin{chunk}{Cannolicchi Rigati} X(i,j) == 8*cos(i*%pi/70) + 0.2*cos(2*i*%pi/7) + 5*cos(j*%pi/100) Y(i,j) == 8*sin(i*%pi/70) + 0.2*sin(2*i*%pi/7) + 4*sin(j*%pi/100) Z(i,j) == 6.0*j/5.0 cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..140,j=0..50,_ style=="smooth",title=="Cannolicchi Rigati",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,2,7) zoom(v3d,3.0,3.0,3.0) \end{chunk} Known as little tubes'', {\sl cannolicchi} exist both in a {\sl rigati} (grooved) and {\sl lisci} (smooth) form. These hollow {\sl pasta corta} (short pasta) come in various diameters and are often served with seafood. {\sl Cannolicchi} hail from Campania in southern Italy. \newpage \section{Capellini} {\center{\includegraphics[scale=0.50]{ps/v81capellini.eps}}} \begin{chunk}{Capellini} X(i,j) == 0.05*cos(2*i*%pi/15) + 0.6*cos(j*%pi/100) Y(i,j) == 0.05*sin(2*i*%pi/15) + 0.5*sin(j*%pi/100) Z(i,j) == 7.0*j/100.0 cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..15,j=0..100,_ style=="smooth",title=="Capellini",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,3.0,3.0,3.0) \end{chunk} An extra-fine rod-like pasta {\sl capellini} (thin hair) may be served in a light broth, but also combine perfectly with butter, nutmeg, or lemon. This variety (or its even more slender relative, {\sl capelli d'angelo} (angel hair) is sometimes used to form the basis of an unusual sweet pasta dish, made with lemons and almonds, called {\sl torta ricciolina}. \newpage \section{Cappelletti} {\center{\includegraphics[scale=0.50]{ps/v81cappelletti.eps}}} \begin{chunk}{Cappelletti} X(i,j) == (0.1 + sin(((3*i)/160)*%pi)) * cos(((2.3*j)/120)*%pi) Y(i,j) == (0.1 + sin(((3*i)/160)*%pi)) * sin(((2.3*j)/120)*%pi) Z(i,j) == 0.1 + (1/400.)*j + (0.3 - 0.231*(i/40.)) * cos((i/20.)*%pi) cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..40,j=0..120,_ style=="smooth",title=="Cappelletti",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,1.1,1.1,1.1) \end{chunk} This pasta is customarily served as the first course of a traditional north Italian Christmas meal, dished up in a chicken brodo (broth). Typically, it is the children of a houseold who prepare the cappelletti (little hats) on Christmas Eve, filling the pasta parcels (made from wheat flour and fresh eggs) with mixed meats or soft cheeses, such as ricotta. \newpage \section{Casarecce} {\center{\includegraphics[scale=0.50]{ps/v81casarecce.eps}}} \begin{chunk}{Casarecce} X(i,j) == _ if (i <= 30)_ then 0.5*cos(j*%pi/30)+0.5*cos((2*i+j+16)/40*%pi) _ else cos(j*%pi/40)+0.5*cos(j*%pi/30)+0.5*sin((2*i-j)/40*%pi) Y(i,j) == _ if (i <= 30)_ then 0.5*sin(j*%pi/30)+0.5*sin((2*i+j+16)/40*%pi) _ else sin(j*%pi/40)+0.5*sin(j*%pi/30)+0.5*cos((2*i-j)/40*%pi) Z(i,j) == j/4.0 cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..60,j=0..60,_ style=="smooth",title=="Casarecce",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,3.0,3.0,3.0) \end{chunk} Easily identified by their unique s-shaped cross-section {\sl casarecce} (home-made) are best cooked as {\sl postasciutta} (pasta boiled, drained and dished up with a sauce). Often {\sl casarecce} are served with a classic {\sl ragu} and topped with a sprinkle of pepper and Parmigiano-Reggiano cheese. \newpage \section{Castellane} {\center{\includegraphics[scale=0.50]{ps/v81castellane.eps}}} \begin{chunk}{Castellane} X(i,j) == ((0.3*sin(j*%pi/120)*abs(cos((j+3)*%pi/6)) + _ i^2/720.*(sin(2*j*%pi/300)^2+0.1) + 0.3)) * cos(7*i*%pi/150) Y(i,j) == ((0.3*sin(j*%pi/120)*abs(cos((j+3)*%pi/6)) + _ i^2/720.*(sin(2*j*%pi/300)^2+0.1) + 0.3)) * sin(7*i*%pi/150) Z(i,j) == 12*cos(j*%pi/120) cf(x:DFLOAT,y:DFLOAT):DFLOAT == 1.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..60,j=0..120,_ style=="smooth",title=="Castellane",colorFunction==cf) colorDef(v3d,yellow(),yellow()) axes(v3d,"off") viewpoint(v3d,0.0,0.0,45.0) viewpoint(v3d,5,5,0) zoom(v3d,1.5,1.5,1.5) \end{chunk} The manufacturer Bailla has recently created this elegant pasta shape. Accordking to its maker, they were originally called {\sl paguri} (hermit crabs) but renamed {\sl castellane} (castle dwellers). The sturdy form and rich nutty taste of {\sl castellane} stand up to hearty meals and full-flavoured sauces. \newpage \section{Cavatappi} {\center{\includegraphics[scale=0.50]{ps/v81cavatappi.eps}}} \begin{chunk}{Cavatappi} X(i,j) == (3+2*cos(i*%pi/35)+0.1*cos(2*i*%pi/7))*cos(j*%pi/30) Y(i,j) == (3+2*cos(i*%pi/35)+0.1*cos(2*i*%pi/7))*sin(j*%pi/30) Z(i,j) == 3+2*sin(i*%pi/35)+0.1*sin(2*i*%pi/7)+j/6.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..70,j=0..150,_ style=="smooth",title=="Cavatappi") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,3.0,3.0,3.0) \end{chunk} Perfect with chunky sauces made from lamb or pork, {\sl cavatappi} (corkscrews) are 36 mm-long, hollow helicoidal tubes. As well as an accompaniment to creamy sauces, such as {\sl boscaiola} (woodsman's) sauce, they are also often used in oven-baked cheese-topped dishes, or in salads with pesto. \newpage \section{Cavatelli} {\center{\includegraphics[scale=0.50]{ps/v81cavatelli.eps}}} \begin{chunk}{Cavatelli} A(i) == 0.5*cos(i*%pi/100) B(i,j) == j/60.*sin(i*%pi/100) X(i,j) == 3*(1-sin(A(i)*2*%pi))*cos(A(i)*%pi+0.9*%pi) Y(i,j) == 3*sin(A(i)*2*%pi)*sin(A(i)*%pi+0.63*%pi) Z(i,j) == 4*B(i,j)*(5-sin(A(i)*%pi)) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..30,_ style=="smooth",title=="Cavatelli") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,5,50) zoom(v3d,1.5,1.5,1.5) \end{chunk} Popular in the south of Italy, and related in shape to the longer twisted {\sl casareccia}, {\sl cavatelli} can be served {\sl alla puttanesca} (with a sauce containing chilli, garlic, capers, and anchovies). They can also be added to a salad with olive oil, sauteed crushed garlic and a dusting of soft cheese. \newpage \section{Chifferi Rigati} {\center{\includegraphics[scale=0.50]{ps/v81chifferirigati.eps}}} \begin{chunk}{Chifferi Rigati} X(i,j) == (0.45+0.3*cos(i*%pi/100)+0.005*cos(2*i*%pi/5)) * cos(j*%pi/45) + _ 0.15*(j/45.0)^10*cos(i*%pi/100)^3 Y(i,j) == (0.35+j/300.0)*sin(i*%pi/100) + 0.005*sin(2*i*%pi/5) Z(i,j) == (0.4+0.3*cos(i*%pi/100))*sin(j*%pi/45) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..45,_ style=="smooth",title=="Chifferi Rigati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,90,180) \end{chunk} This pasta - available in both {\sl rigoti} (grooved) and {\sl lisci} (smooth) forms - is typically cooked in broth or served in {\sl rogu alla bolognese}, though {\sl chifferi rigati} also make an excellent addition to salads with carrot, red pepper and courgette. {\sl Chifferi rigoti} bear a resemblance to, and the term is a transliteration of, the Austrian 'kipfel' sweet. \newpage \section{Colonne Pompeii} {\center{\includegraphics[scale=0.50]{ps/v81colonnepompeii.eps}}} \begin{chunk}{Colonne Pompeii} X0(i,j) == _ if (j <= 50) _ then 2*cos(i*%pi/20) _ else 2*cos(i*%pi/20)*cos(-j*%pi/25) Y0(i,j) == _ if (j <= 50) _ then 0.0 _ else 2*cos(i*%pi/20)*sin(-j*%pi/25) + 3*sin((j-50)*%pi/200) Z0(i,j) == _ if (j <= 50) _ then sin(i*%pi/20)+12 _ else sin(i*%pi/20)+6.0*j/25.0 X1(i,j) == _ if (j <= 200) _ then 2*cos(i*%pi/20)*cos(-j*%pi/25+2*%pi/3) _ else 2*cos(i*%pi/20)*sin(-28*%pi/3) Y1(i,j) == _ if (j <= 200) _ then 2*cos(i*%pi/20)*sin(-j*%pi/20 + 2*%pi/3) + 3*sin(j*%pi/200) _ else 2*cos(i*%pi/20)*sin(-28*%pi/3) Z1(i,j) == _ if (j <= 200) _ then 12+sin(i*%pi/20)+6.0*j/25.0 _ else sin(i*%pi/20)+60 X2(i,j) == _ if (j <= 200) _ then 2*cos(i*%pi/20)*cos(-j*%pi/25+4*%pi/3) _ else 2*cos(i*%pi/20)*sin(-28*%pi/3) Y2(i,j) == _ if (j <= 200) _ then 2*cos(i*%pi/200)*sin(-j*%pi/25+4*%pi/3)+3*sin(j*%pi/200) _ else 2*cos(i*%pi/20)*sin(-28*%pi/3) vsp:=createThreeSpace() makeObject(surface(X0(i,j),Y0(i,j),Z0(i,j)),i=0..10,j=0..250,space==vsp) makeObject(surface(X1(i,j),Y1(i,j),Z1(i,j)),i=0..10,j=0..250,space==vsp) makeObject(surface(X2(i,j),Y2(i,j),Z1(i,j)),i=0..10,j=0..250,space==vsp) vp:=makeViewport3D(vsp,title=="Colonne Pompeii",style=="smooth") colorDef(vp,yellow(),yellow()) axes(vp,"off") zoom(vp,3.0,3.0,3.0) \end{chunk} This ornate pasta (originally from Campania, southern Italy) is similar in shape to {\sl fusilloni} (a large {\sl fusilli}) but is substantially longer. {\sl Colonne Pompeii} (columns of Pompeii) are best served with a seasoning of fresh basil, pine nuts, finely sliced garlic and olive oil, topped with a sprinkling of freshly grated Parmigiano-Reggiano. \newpage \section{Conchiglie Rigate} {\center{\includegraphics[scale=0.50]{ps/v81conchiglierigate.eps}}} \begin{chunk}{Conchiglie Rigate} A(i,j) == 0.25*sin(j*%pi/250)*cos((6*j+25)/25*%pi) B(i,j) == ((40.0-i)/40.0)*(0.3+sin(j*%pi/250))*%pi C(i,j) == 2.5*cos(j*%pi/125)+2*sin((40-i)*%pi/80)^10 * _ sin(j*%pi/250)^10*sin(j*%pi/125+1.5*%pi) X(i,j) == A(i,j)+cos(j*%pi/125)+(5+30*sin(j*%pi/250))*sin(B(i,j)) * _ sin(i/40*(0.1*(1.1+sin(j*%pi/250)^5))*%pi) Y(i,j) == A(i,j)+(5+30*sin(j*%pi/250))*cos(B(i,j)) * _ sin(i/40*(0.1*(1.1+sin(j*%pi/250)^5))*%pi) + C(i,j) Z(i,j) == 25.0*cos(j*%pi/250) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..40,j=0..250,_ style=="smooth",title=="Conchiglie Rigate") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,45,45) zoom(v3d,1.5,1.5,1.5) \end{chunk} Shaped like their namesake, {\sl conchiglie} (shells) exist in both {\sl rigate} (grooved) and {\sl lisce} (smooth) forms. Suited to light tomato sauces, ricotta cheese or {\sl pesto genovese}, {\sl conchiglie} hold flavourings in their grooves and cunningly designed shell. Smaller versions are used in soups, while larger shells are more commonly served with a sauce. \newpage \section{Conchigliette Lisce} {\center{\includegraphics[scale=0.50]{ps/v81conchigliettelisce.eps}}} \begin{chunk}{Conchigliette Lisce} A(i,j) == (60.0-i)/60.0*(0.5+sin(j*%pi/60))*%pi B(i,j) == i/60.0*(0.1*(1.1+sin(j*%pi/60)^5))*%pi C(i,j) == 2.5*cos(j*%pi/30)+2*sin((60-i)*%pi/120)^10 * _ sin(j*%pi/60)^10*sin((j+45)*%pi/30) X(i,j) == (5+30*sin(j*%pi/60))*sin(A(i,j))*sin(B(i,j))+cos(j*%pi/30) Y(i,j) == (5+30*sin(j*%pi/60))*cos(A(i,j))*sin(B(i,j))+C(i,j) Z(i,j) == 25*cos(j*%pi/60) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..60,j=0..60,_ style=="smooth",title=="Conchigliette Lisce") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,45,45) zoom(v3d,1.5,1.5,1.5) \end{chunk} Typically found in central and southern Italy (notably Campania), {\sl conchigliette lisce} (small smooth shells) can be served in soups such as {\sl minestrone}. Alternatively, these shells can accompany a meat- or vegetable-based sauce. \newpage \section{Conchiglioni Rigate} {\center{\includegraphics[scale=0.50]{ps/v81conchiglionirigate.eps}}} \begin{chunk}{Conchiglioni Rigate} A(i,j) == 0.25*sin(j*%pi/200)*cos((j+4)*%pi/4) B(i,j) == i/40.0*(0.1+0.1*sin(j*%pi/200)^6)*%pi C(i,j) == 2.5*cos(j*%pi/100)+3*sin((40-i)*%pi/80)^10 * _ sin(j*%pi/200)^10*sin((j-150)*%pi/100) X(i,j) == A(i,j)+(10+30*sin(j*%pi/200)) * _ sin((40.0-i)/40*(0.3+sin(j*%pi/200)^3)*%pi) * _ sin(B(i,j))+cos(j*%pi/100) Y(i,j) == A(i,j)+(10+30*sin(j*%pi/200)) * _ cos((40.0-i)/40*(0.3+sin(j*%pi/200)^3)*%pi) * _ sin(B(i,j))+C(i,j) Z(i,j) == 30.0*cos(j*%pi/200) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..40,j=0..200,_ style=="smooth",title=="Conchiglioni Rigati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,45,45) zoom(v3d,1.5,1.5,1.5) \end{chunk} The shape of {\sl conchiglioni rigate} (large ribbed shells) is ideal for holding sauces and fillings (either fish or meat based) and the pasta can be baked in the oven, or placed under a grill and cooked as a gratin. {\sl Conchiglioni rigati} are often served in the Italian-American dish {\sl pasta primavera} (pasta in spring sauce) alongside crisp spring vegetables. \newpage \section{Corallini Lisci} {\center{\includegraphics[scale=0.50]{ps/v81corallinilisci.eps}}} \begin{chunk}{Corallini Lisci} X(i,j) == 0.8*cos(i*%pi/50) Y(i,j) == 0.8*sin(i*%pi/50) Z(i,j) == 3.0*j/50.0 vsp:=createThreeSpace() makeObject(surface(X(i,j),Y(i,j),Z(i,j)),i=0..140,j=0..70,space==vsp) makeObject(surface(X(i,j)/2,Y(i,j)/2,Z(i,j)),i=0..140,j=0..70,space==vsp) vp:=makeViewport3D(vsp,style=="smooth",title=="Corallini Lisci") colorDef(vp,yellow(),yellow()) axes(vp,"off") zoom(vp,2.0,2.0,2.0) \end{chunk} Members of the {\sl postine minute} (tiny pasta) group, {\sl corallini lisci} (small smooth coral) are so called because their pierced appearance resembles the coral beads worn as jewelry in Italy. Their small size (no larger than 3.5 mm in diameter) means that {\sl corallini} are best cooked in broths, such as Tuscan white bean soup. \newpage \section{Creste Di Galli} {\center{\includegraphics[scale=0.50]{ps/v81crestedigalli.eps}}} \begin{chunk}{Creste Di Galli} A(i) == ((1+sin((1.5+i)*%pi))/2)^5 B(i,j) == 0.3*sin(A(i/140.)*%pi+0.5*%pi)^1000*cos(j*%pi/70.0) C(i,j) == 0.3*cos(A(i/140.)*%pi)^1000*sin(j*%pi/70.0) X(i,j) == (0.5+0.3*cos(A(i/140.)*2*%pi))*cos(j*%pi/70.0) + _ 0.15*(j/70.0)^10*cos(A(i/140.0)*2*%pi)^3 + B(i,j) Y(i,j) == 0.35*sin(A(i/140.0)*2*%pi) + 0.15*j/70.0*sin(A(i/140.0)*2*%pi) Z(i,j) == (0.4+0.3*cos(A(i/140.0)*2*%pi))*sin(j*%pi/70.0) + C(i,j) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..140,j=0..70,_ style=="smooth",title=="Creste Di Galli") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,1.5,1.5,1.5) \end{chunk} Part of the {\sl pasta ripiena} (filled pasta) family, {\sl creste di galli} (coxcombs) are identical to {\sl galletti} except for the crest, which is smooth rather than crimped. They may be stuffed, cooked and served in a simple {\sl marinara} (mariner's) sauce, which contains tomato, garlic, and basil. \newpage \section{Couretti} {\center{\includegraphics[scale=0.50]{ps/v81couretti.eps}}} \begin{chunk}{Couretti} X(i,j) == 2*cos(i*%pi/150)-cos(i*%pi/75)-sin(i*%pi/300)^150 - _ (abs(cos(i*%pi/300)))^5 Y(i,j) == 2*sin(i*%pi/150)-sin(i*%pi/75) Z(i,j) == j/10.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..300,j=0..10,_ style=="smooth",title=="Couretti") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") \end{chunk} A romantically shaped scion of the {\sl postine minute} (tiny pasta) clan, {\sl cuoretti} (tiny hearts) are minuscule. In fact, along with {\sl acini di pepe}, they are one of the smallest forms of pasta. Like all {\sl postine} they may be served in soup, such as cream of chicken. \newpage \section{Ditali Rigati} {\center{\includegraphics[scale=0.50]{ps/v81ditalirigati.eps}}} \begin{chunk}{Ditali Rigati} X(i,j) == cos(i*%pi/100) + 0.03*cos((7*i)*%pi/40) + 0.25*cos(j*%pi/50) Y(i,j) == 1.1 * sin(i*%pi/100) + 0.03*sin((7*i)*%pi/40) + 0.25*sin(j*%pi/50) Z(i,j) == j/10.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..25,_ style=="smooth",title=="Ditali Rigati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,1.5,1.5,1.5) \end{chunk} Another speciality of the Campania region of southern Italy, {\sl ditali rigati} (grooved thimbles) are compact and typically less than 10 mm long. Like other {\sl pastine}, they are usually found in soups such as {\sl pasta e patate}. Their stocky shape makes them a sustaining winter snack, as well as an excellent addition to salads. \newpage \section{Fagottini} {\center{\includegraphics[scale=0.50]{ps/v81fagottini.eps}}} \begin{chunk}{Fagottini} A(i,j) == (0.8 + sin(i/100*%pi)^8 - 0.8 * cos(i/25*%pi))^1.5 + _ 0.2 + 0.2 * sin(1/100*%pi) B(i,j) == (0.9 + cos(i/100*%pi)^8 - 0.9 * cos(i/25*%pi + 0.03*%pi))^1.5 + _ 0.3 * cos(i/100*%pi) C(i,j) == 4 - ((4*j)/500)*(1+cos(i/100*%pi)^8 - 0.8*cos(i/25*%pi))^1.5 X(i,j) == cos(i/100*%pi) * _ (A(i,j) * sin(j/100*%pi)^8 + _ 0.6 * (2 + sin(i/100*%pi)^2) * sin(j/50*%pi)^2) Y(i,j) == sin(i/100*%pi) * _ (B(i,j) * sin(j/100*%pi)^8 + _ 0.6 * (2 + cos(i/100*%pi)^2) * sin(j/50*%pi)^2) Z(i,j) == (1 + sin(j/100*%pi - 0.5*%pi)) * _ (C(i,j) * ((4*j)/500) * _ (1 + sin(i/100*%pi)^8 - 0.8 * cos(i/25*%pi))^1.5) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..50,_ style=="smooth",title=="Fagottini") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") \end{chunk} A notable member of the pasta ripiena (filled pasta) family, fagottini (little purses) are made from circles of durum wheat dough. A spoonful of ricotta, steamed vegetables or even stewed fruit is placed on the dough, and the corners are then pinched together to form a bundle. These packed dumplings are similar to ravioli, only larger. \newpage \section{Farfalle} {\center{\includegraphics[scale=0.50]{ps/v81farfalle.eps}}} \begin{chunk}{Farfalle} A(i) == sin((7*i+16)*%pi/40) B(i,j) == (7.0*j/16.0)+4*sin(i*%pi/80)*sin((j-10)*%pi/120) C(i,j) == 10*cos((i+80)*%pi/80)*sin((j+110)*%pi/100)^9 D(i,j) == (7.0*j/16.0)-4*sin(i*%pi/80)-A(i)*sin((10-j)*%pi/20) -- E(i,j) was never defined. We guess at a likely function - close but wrong E(i,j) == _ if ((20 <= i) and (i <= 60)) _ then 7*sin((i+40)*%pi/40)^3*sin((2*j*%pi)/10+1.1*%pi)^9 _ else C(i,j) F(i) == _ if ((20 <= i) and (i <= 60)) _ then 7*sin((i+40)*%pi/40)^3*sin((j+110)*%pi/100)^9 _ else C(i,j) X(i,j) == (3.0*i)/8.0+F(i) Y(i,j) == _ if ((10 <= j) and (j <= 70)) _ then B(i,j)-4*sin(i*%pi/80)*sin((70-j)*%pi/120) _ else if (j <= 10) _ then D(i,j) _ else E(i,j) Z(i,j) == 3*sin((i+10)*%pi/20)*sin(j*%pi/80)^1.5 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..80,j=0..80,_ style=="smooth",title=="Farfalle") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,0.5,0.5,0.5) \end{chunk} A mixture of durum-wheat flour, eggs, and water, {\sl farfalle} (butterflies) come from the Emilia-Romagnaand Lombardy regions of northern Italy. They are best served in a rich {\sl carbonara} sauce (made with cream, eggs, and bacon). Depending on se, {\sl farfalle} might be accompanied by green peas and chicken or ham. \newpage \section{Farfalline} {\center{\includegraphics[scale=0.50]{ps/v81farfalline.eps}}} \begin{chunk}{Farfalline} A(i) == 30*cos(i*%pi/125)+0.5*cos((6*i)*%pi/25) B(i) == 30*sin(i*%pi/125)+0.5*sin((6*i)*%pi/25) X(i,j) == cos(3*A(i)*%pi/100) Y(i,j) == 0.5*sin((3*A(i))*%pi/100)*(1+sin(j*%pi/100)^10) Z(i,j) == B(i)*j/500.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..250,j=0..50,_ style=="smooth",title=="Farfalline") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") rotate(v3d,45,45) zoom(v3d,1.5,1.5,1.5) \end{chunk} The small size of this well-known member of the {\sl postine minute} (tiny pasta) lineage means that {\sl farfalline} (tiny butterflies), are suitable for light soups, such as {\sl pomodori e robiolo} (a mixture of tomato and soft cheese). A crimped pasta cutter and a central pinch create the iconic shape. \newpage \section{Farfalloni} {\center{\includegraphics[scale=0.50]{ps/v81farfalloni.eps}}} \begin{chunk}{Farfalloni} A(i,j) == 10*cos((i+70)*%pi/70)*sin((2*j)*%pi/175+1.1*%pi)^9 B(j) == 0.3*sin((6-j)*%pi/7+0.4*%pi) C(i,j) == _ if ((17 <= i) and (i <= 52)) _ then 7*sin((i+35)*%pi/35)^3*sin((2*j*%pi)/175+1.1*%pi)^9 _ else A(i,j) D(i,j) == (j/2.0)+4*sin(i*%pi/70)*sin((j-10)*%pi/100) - _ 4*sin(i*%pi/70)*sin((60-j)*%pi/100) E(i,j) == (j/2.0)+4*sin(i*%pi/70)+0.3*sin((2*i+2.8)*%pi/7)*sin((j-60)*%pi/20) F(i,j) == (j/2.0)-4*sin(i*%pi/70)-0.3*sin((2*i+2.8)*%pi/7)*sin((10-j)*%pi/20) X(i,j) == (3.0*i)/7.0+C(i,j) Y(i,j) == _ if ((10 <= j) and (j <= 60)) _ then D(i,j) _ else if (j <= 10) _ then F(i,j) _ else E(i,j) Z(i,j) == 3*sin((2*i+17.5)*%pi/35.)*sin(j*%pi/70)^1.5 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..70,j=0..70,_ style=="smooth",title=="Farfalloni") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.5,2.5,2.5) \end{chunk} Like {\sl farfalle}, {\sl farfalloni} (large butterflies) are well matched by a tomato- or butter-based sauce with peas and ham. They are also perfect with marrow vegetables such as roast courgette or pureed pumpkin, topped with Parmigiano-Reggiano and a sprinkling of {\sl noce moscata} (nutmeg). \newpage \section{Festonati} {\center{\includegraphics[scale=0.50]{ps/v81festonati.eps}}} \begin{chunk}{Festonati} X(i,j) == 5*cos(i*%pi/50)+0.5*cos(i*%pi/50)*(1+sin(j*%pi/100)) + _ 0.5*cos((i+25)*%pi/25)*(1+sin(j*%pi/5)) Y(i,j) == 5*sin(i*%pi/50)+0.5*sin(i*%pi/50)*(1+sin(j*%pi/100)) + _ 0.5*cos(i**%pi/25)*(1+sin(j*%pi/5)) Z(i,j) == j/2.0+2*sin((3*i+25)*%pi/50) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..100,j=0..100,_ style=="smooth",title=="Festonati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,3.0,3.0,3.0) \end{chunk} This smooth member of the {\sl pasta corta} (short pasta) family is named after 'festoons' (decorative lengths of fabric with the rippled profile of a garland). {\sl Festonati} can be served with grilled aubergine or home-grown tomatoes, topped with grated scamorza, fresh basil, olive oil, garlic, and red chilli flakes. \newpage \section{Fettuccine} {\center{\includegraphics[scale=0.50]{ps/v81fettuccine.eps}}} \begin{chunk}{Fettuccine} X(i,j) == 1.8*sin((4*i)*%pi/375) Y(i,y) == 1.6*cos((6*i)*%pi/375)*sin((3*i)*%pi/750) Z(i,j) == i/75.0 + j/20.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..150,j=0..10,_ style=="smooth",title=="Fettuccine") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") \end{chunk} This famous {\sl pasta lunga} (long pasta) is made with durum-wheat flour, water, and in the case of {\sl fettuccine alluovo}, eggs ideally within days of laying. {\sl Fettuccine} (little ribbons) hail from the Lazio region. Popular in many dishes, they are an ideal accompaniment to {\sl Alfredo} sauce, a rich mix of cream, parmesan, garlic, and parsley. \newpage \section{Fiocchi Rigati} {\center{\includegraphics[scale=0.50]{ps/v81fiocchirigati.eps}}} \begin{chunk}{Fiocchi Rigati} A(i,j) == 10*cos((i+80)*%pi/80)*sin((j+110)*%pi/100)^9 B(i,j) == 35.0*j/80.0+4*sin(i*%pi/80)*sin((j-10)*%pi/120) X(i,j) == _ if ((20 <= i) and (i <= 60)) _ then 7*sin((i+40)*%pi/40)^3*sin((j+110)*%pi/100)^9 + 30.0*i/80.0 _ else A(i,j) + 30.0*i/80.0 Y(i,j) == B(i,j)-4*sin(i*%pi/80)*sin((70-j)*%pi/120) Z(i,j) == 3*sin((1+10)*%pi/20)*sin(j*%pi/80)^1.5-0.7*((sin(3*j*%pi/8)+1)/2)^4 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..80,j=0..80,_ style=="smooth",title=="Fiocchi Rigati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) \end{chunk} A distant relative of the {\sl farfalle} family, {\sl fiocchi rigati} (grooved flakes) are smaller than either {\sl farfalloni} or {\sl farfalle}, but larger than {\sl farfalline}. Their corrugated surface collects more sauce than a typical {\sl farfalle}. For a more unusual disk, {\sl fiocchi rigati} can be served in a tomato and vodka sauce. \newpage \section{Fisarmoniche} {\center{\includegraphics[scale=0.50]{ps/v81fisarmoniche.eps}}} \begin{chunk}{Fisarmoniche} X(i,j) == (1.5+3*(i/70.0)^5+4*sin(j*%pi/200)^50)*cos(4*i*%pi/175) Y(i,j) == (1.5+3*(i/70.0)^5+4*sin(j*%pi/200)^50)*sin(4*i*%pi/175) Z(i,j) == j/50.0+cos(3*i*%pi/14)*sin(j*%pi/1000) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..70,j=0..1000,_ style=="smooth",title=="Fisarmoniche") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) \end{chunk} Named after the accordion - whose bellows their bunched profiles recall - {\sl fisarmoniche} are perfect for capturing thick sauces, which cling to their folds. This sturdy pasta is said to have been invented in the fifteenth century, in the Italian town of Loreto in the Marche, east central Italy. \newpage \section{Funghini} {\center{\includegraphics[scale=0.50]{ps/v81funghini.eps}}} \begin{chunk}{Funghini} A(i,j) == 5*cos(i*%pi/150)+0.05*cos(i*%pi/3)*sin(j*%pi/60)^2000 B(i,j) == j/30.0*(5*sin(i*%pi/150)+0.05*sin(i*%pi/3)) C(i,j) == j/10.0*(2*sin(i*%pi/150)+0.05*sin(i*%pi/3)) D(i,j) == _ if (i <= 150) _ then B(i,j) _ else if (j <= 10) _ then C(i,j) _ else 2*sin(i*%pi/150)+0.05*sin(i*%pi/6) X(i,j) == 0.05*cos(A(i,j)*%pi/5)+0.3*cos(A(i,j)*%pi/5)*sin(3*D(i,j)*%pi/50)^2 Y(i,j) == 0.01*sin(A(i,j)*%pi/5)+0.3*sin(A(i,j)*%pi/5)*sin(3*D(i,j)*%pi/50)^2 Z(i,j) == 0.25*sin((D(i,j)+3)*%pi/10) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..300,j=0..30,_ style=="smooth",title=="Funghini") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") \end{chunk} The modest dimensions of this {\sl pastine minute} (tiny pasta) make {\sl funghini} (little mushrooms) especially suitable for soups, such as a {\sl minestrone} made from chopped and sauteed celeriac. \newpage \section{Fusilli} {\center{\includegraphics[scale=0.50]{ps/v81fusilli.eps}}} \begin{chunk}{Fusilli} X(i,j) == 6*cos((3*i+10)*%pi/100)*cos(j*%pi/25) Y(i,j) == 6*sin((3*i+10)*%pi/100)*cos(j*%pi/25) Z(i,j) == (3.0*i)/20.0+2.5*cos((j+12.5)*%pi/25) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..200,j=0..25,_ style=="smooth",title=="Fusilli") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) \end{chunk} A popular set from the {\sl pasta corta} (short pasta) family, {\sl fusilli} (little spindles) were originally made by quickly wrapping a {\sl spaghetto} around a large needle. Best served as {\sl pastasciutta} (pasta boild and drained) with a creamy sauce containing slices of spicy sausage. \newpage \section{Fusilli al Ferretto} {\center{\includegraphics[scale=0.50]{ps/v81fusillialferretto.eps}}} \begin{chunk}{Fusilli al Ferretto} A(i,j) == 6.0*i/7.0+15*cos(j*%pi/20) X(i,j) == (3+1.5*sin(i*%pi/140)^0.5*sin(j*%pi/20))*sin(13*i*%pi/280) + _ 5*sin(2*A(i,j)*%pi/135) Y(i,j) == (3+1.5*sin(i*%pi/140)^0.5*sin(j*%pi/20))*cos(13*i*%pi/280) Z(i,j) == A(i,j) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..140,j=0..40,_ style=="smooth",title=="Fusilli al Ferretto") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,3.0,3.0,3.0) \end{chunk} To create this Neapolitan variety of {\sl fusilli}, a small amount of durum-wheat flour is kneaded and placed along a {\sl ferretto} (small iron stick) that is then rolled between the hands to create a thick irregular twist of dough. The shape is removed and left to dry on a wicker tray known as a {\sl spasa}. {\sl Fusilli al ferretto} are best dished up with a lamb {\sl ragu}. \newpage \section{Fusilli Capri} {\center{\includegraphics[scale=0.50]{ps/v81fusillicapri.eps}}} \begin{chunk}{Fusilli Capri} X(i,j) == 6*cos(j*%pi/50)*cos((i+2.5)*%pi/25) Y(i,j) == 6*cos(j*%pi/50)*sin((i+2.5)*%pi/25) Z(i,j) == 2.0*i/3.0 + 14*cos((j+25)*%pi/50) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..150,j=0..50,_ style=="smooth",title=="Fusilli Capri") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.5,2.5,2.5) \end{chunk} A longer and more compact regional adaptation of {\sl fusilli}, {\sl fusilli Capri} are suited to a hearty {\sl ragu} of lamb or por sausages, or may also be combined with rocket and lemon to form a lighter dish. \newpage \section{Fusilli Lunghi Bucati} {\center{\includegraphics[scale=0.50]{ps/v81fusillilunghibucati.eps}}} \begin{chunk}{Fusilli Lunghi Bucati} A(i,j) == 10+cos(i*%pi/10)+2*cos((j+10)*%pi/10)+10*cos((j+140)*%pi/160) B(i,j) == 20+cos(i*%pi/10)+2*cos((j+10)*%pi/10) C(i,j) == (j+10.0)*%pi/10.0 D(i,j) == i*%pi/10.0 E(i,j) == 7+20*sin((j-20)*%pi/160) F(i,j) == 70*(0.1-(j-180.0)/200.0) X(i,j) == _ if ((20 <= j) and (j <= 180)) _ then A(i,j) _ else if (j <= 20) _ then cos(D(i,j))+2*cos(C(i,j)) _ else B(i,j) Y(i,j) == _ if ((20 <= j) and (j <= 180)) _ then sin(D(i,j))+2*sin(C(i,j)) _ else sin(D(i,j))+2*sin(C(i,j)) Z(i,j) == _ if ((20 <= j) and (j <= 180)) _ then E(i,j) _ else if (j <= 20) _ then ((7.0*j)/20.0) _ else F(i,j) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..20,j=0..200,_ style=="smooth",title=="Fusilli Lunghi Bucati") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.5,2.5,2.5) \end{chunk} A distinctive member of the extended {\sl fusilli} clan, {\sl fusilli lunghi bucati} (long pierced {\sl fusilli}) originated in Campania, southern Italy, and have a spring-like profile. Like all {\sl fusilli} they are traditionally consumed with a meat-based {\sl ragu}, but may also be combined with thick vegetable sauces and baked in an oven. \newpage \section{Galletti} {\center{\includegraphics[scale=0.50]{ps/v81galletti.eps}}} \begin{chunk}{Galletti} A(i) == ((1+sin(i*%pi+1.5*%pi))/2)^5 B(i,j) == 0.4*sin(A(i/140)*%pi+0.5*%pi)^1000*cos(j*%pi/70) C(i,j) == 0.15*sin(A(i/140)*%pi+0.5*%pi)^1000*cos(j*%pi/7) D(i,j) == 0.4*cos(A(i/140)*%pi)^1000*sin(j*%pi/70) X(i,j) == (0.5+0.3*cos(A(i/140)*2*%pi))*cos(j*%pi/70) + _ 0.15*(j/70.0)^10*cos(A(i/140)*2*%pi)^3 + B(i,j) Y(i,j) == 0.35*sin(A(i/140)*2*%pi)+0.15*(j/70.0)*sin(A(i/140)*2*%pi)+C(i,j) Z(i,j) == (0.4+0.3*cos(A(i/140)*2*%pi))*sin(j*%pi/70) + D(i,j) v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..140,j=0..70,_ style=="smooth",title=="Galletti") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,1.5,1.5,1.5) \end{chunk} According to their maker, Barilla, the origin of {\sl galletti} (small cocks) is uncertain, but their shape recalls that of the {\sl chifferi} with the addtion of an undulating crest. {\sl Galletti} are usually served in tomato sauces, but combine wqually well with a {\sl boscaiola} (woodsman's) sauce of mushrooms. \newpage \section{Garganelli} {\center{\includegraphics[scale=0.50]{ps/v81garganelli.eps}}} \begin{chunk}{Garganelli} A(i,j) == (i-25.0)/125.0*j B(i,j) == (i-25.0)/125.0*(150.0-j) C(i,j) == _ if ((j <= 75) or (i <= 25)) _ then A(i,j) _ else if ((J >= 75) or (i <= 25)) _ then B(i,j) _ else if ((j >= 75) or (i >= 25)) _ then B(i,j) _ else A(i,j) X(i,j) == 0.1*cos(j*%pi/3)+(3+sin(C(i,j)*%pi/60))*cos(7*C(i,j)*%pi/60) Y(i,j) == 0.1*sin(j*%pi/3)+(3+sin(C(i,j)*%pi/60))*sin(7*C(i,j)*%pi/60) Z(i,j) == 6.0*j/25.0+C(i,j)/4.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..50,j=0..150,_ style=="smooth",title=="Garganelli") colorDef(v3d,yellow(),yellow()) axes(v3d,"off") zoom(v3d,2.0,2.0,2.0) viewpoint(v3d,-5,0,0) \end{chunk} A grooved {\sl pasta corta} (short pasta), similar to {\sl maccheroni} but with pointed slanting ends, {\sl garganelli} are shaped like the gullet of a chicken ('{\sl garganel}' in the northern Italian Emiliano-Romagnolo dialect). Traditionally cooked in broth, {\sl garganelli} are also sometimes served in hare sauce with chopped bacon. \newpage \section{Gemelli} {\center{\includegraphics[scale=0.50]{ps/v81gemelli.eps}}} \begin{chunk}{Gemelli} X(i,j) == 6*cos(j*1.9*%pi/50+0.55*%pi)*cos(3.0*i/25.0) Y(i,j) == 6*cos(j*1.9*%pi/50+0.55*%pi)*sin(3.0*i/25.0) Z(i,j) == 8*sin(j*1.9*%pi/50+0.55*%pi)+3.0*i/4.0 v3d:=draw(surface(X(i,j),Y(i,j),Z(i,j)),i=0..100,j=0..50,_ style=="smooth",title=="Gemelli") colorDef(v3d,yellow(),yellow()) axes(v3d,"off")