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acplot.spad
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acplot.spad
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)abbrev package REALSOLV RealSolvePackage
RealSolvePackage() : Exports == Implementation where
++ This package provides numerical solutions of systems of polynomial
++ equations for use in ACPLOT.
I ==> Integer
IE ==> IndexedExponents Symbol
L ==> List
NF ==> Float
P ==> Polynomial
RN ==> Fraction Integer
SY ==> Symbol
RFI ==> Fraction Polynomial Integer
LIFT ==> PolynomialCategoryLifting(IE, SY, RN, P RN, RFI)
SOLV ==> FloatingRealPackage Float
Exports ==> with
solve : (P RN, NF) -> L NF
++ solve(p, eps) finds the real zeroes of a
++ univariate rational polynomial p with precision eps.
solve : (P I, NF) -> L NF
++ solve(p, eps) finds the real zeroes of a univariate
++ integer polynomial p with precision eps.
realSolve : (L P I, L SY, NF) -> L L NF
++ realSolve(lp, lv, eps) = compute the list of the real
++ solutions of the list lp of polynomials with integer
++ coefficients with respect to the variables in lv,
++ with precision eps.
Implementation ==> add
prn2rfi : P RN -> RFI
prn2rfi p ==
map(x +-> x::RFI, x +-> (numer(x)::RFI)/(denom(x)::RFI), p)$LIFT
pi2rfi : P I -> RFI
pi2rfi p == p :: RFI
solve(p : P RN, eps : NF) == realRoots(prn2rfi p, eps)$SOLV
solve(p : P I, eps : NF) ==
realRoots(p :: RFI, eps)$SOLV
realSolve(lp, lv, eps) ==
realRoots(map(pi2rfi, lp)$ListFunctions2(P I, RFI), lv, eps)$SOLV
)abbrev domain ACPLOT PlaneAlgebraicCurvePlot
--% PlaneAlgebraicCurvePlot
++ Plot a NON-SINGULAR plane algebraic curve p(x, y) = 0.
++ Author: Clifton J. Williamson
++ Date Created: Fall 1988
++ Keywords: algebraic curve, non-singular, plot
++ Examples:
++ References:
PlaneAlgebraicCurvePlot() : Exports == Implementation where
B ==> Boolean
OUT ==> OutputForm
SY ==> Symbol
L ==> List
SEG ==> Segment
I ==> Integer
PI ==> PositiveInteger
RN ==> Fraction Integer
NF ==> Float
SF ==> DoubleFloat
P ==> Polynomial
UP ==> UnivariatePolynomial
SUP ==> SparseUnivariatePolynomial
FR ==> Factored
Pt ==> Point DoubleFloat
BoundaryPts ==> Record(left : L Pt, _
right : L Pt, _
bottom : L Pt, _
top : L Pt)
NewPtInfo ==> Record(newPt : Pt, _
type : String)
Corners ==> Record(minXVal : SF, _
maxXVal : SF, _
minYVal : SF, _
maxYVal : SF)
kinte ==> solve$RealSolvePackage()
rsolve ==> realSolve$RealSolvePackage()
Exports ==> PlottablePlaneCurveCategory with
makeSketch : (P I, SY, SY, SEG RN, SEG RN) -> %
++ makeSketch(p, x, y, a..b, c..d) creates an ACPLOT of the
++ curve \spad{p = 0} in the region {\em a <= x <= b, c <= y <= d}.
++ More specifically, 'makeSketch' plots a non-singular algebraic curve
++ \spad{p = 0} in an rectangular region {\em xMin <= x <= xMax},
++ {\em yMin <= y <= yMax}. The user inputs
++ \spad{makeSketch(p, x, y, xMin..xMax, yMin..yMax)}.
++ Here p is a polynomial in the variables x and y with
++ integer coefficients (p belongs to the domain
++ \spad{Polynomial Integer}). The case
++ where p is a polynomial in only one of the variables is
++ allowed. The variables x and y are input to specify the
++ the coordinate axes. The horizontal axis is the x-axis and
++ the vertical axis is the y-axis. The rational numbers
++ xMin, ..., yMax specify the boundaries of the region in
++ which the curve is to be plotted.
refine : (%, SF) -> %
++ refine(p, x) \undocumented{}
Implementation ==> add
import from PointPackage DoubleFloat
import from Plot
import from RealSolvePackage
import from L(Pt)
singValBetween? : (SF, SF, L SF) -> B
segmentInfo : (SF -> SF, SF, SF, L SF, L SF, L SF, SF, SF) -> _
Record(seg : SEG SF, left : SF, lowerVals : L SF, upperVals : L SF)
swapCoords : Pt -> Pt
samePlottedPt? : (Pt, Pt) -> B
findPtOnList:(Pt,L Pt) -> Union(Pt,"failed")
makeCorners : (SF, SF, SF, SF) -> Corners
getXMin : Corners -> SF
getXMax : Corners -> SF
getYMin : Corners -> SF
getYMax : Corners -> SF
SFPolyToUPoly : P SF -> SUP SF
RNPolyToUPoly : P RN -> SUP RN
coerceCoefsToSFs : P I -> P SF
coerceCoefsToRNs : P I -> P RN
RNtoSF : RN -> SF
RNtoNF : RN -> NF
SFtoNF : SF -> NF
listPtsOnHorizBdry : (P RN, SY, RN, NF, NF) -> L Pt
listPtsOnVertBdry : (P RN, SY, RN, NF, NF) -> L Pt
listPtsInRect : (L L NF, NF, NF, NF, NF) -> L Pt
ptsSuchThat? : (L L NF, L NF -> B) -> B
inRect? : (L NF, NF, NF, NF, NF) -> B
onHorzSeg? : (L NF, NF, NF, NF) -> B
onVertSeg? : (L NF, NF, NF, NF) -> B
newX : (L L NF, L L NF, NF, NF, NF, RN, RN) -> RN
newY : (L L NF, L L NF, NF, NF, NF, RN, RN) -> RN
makeOneVarSketch : (P I, SY, SY, RN, RN, RN, RN, SY) -> %
makeLineSketch : (P I, SY, SY, RN, RN, RN, RN) -> %
makeRatFcnSketch : (P I, SY, SY, RN, RN, RN, RN, SY) -> %
makeGeneralSketch : (P I, SY, SY, RN, RN, RN, RN) -> %
traceBranches : (P SF, P SF, P SF, SY, SY, Corners, SF, SF, PI, _
L Pt, BoundaryPts) -> L L Pt
dummyFirstPt : (Pt, P SF, P SF, SY, SY, L Pt, L Pt, L Pt, L Pt) -> Pt
listPtsOnSegment : (P SF, P SF, P SF, SY, SY, Pt, Pt, Corners, _
SF, SF, PI, L Pt, L Pt) -> L L Pt
listPtsOnLoop : (P SF, P SF, P SF, SY, SY, Pt, Corners, _
SF, SF, PI, L Pt, L Pt) -> L L Pt
computeNextPt : (P SF, P SF, P SF, SY, SY, Pt, Pt, Corners, _
SF, SF, PI, L Pt, L Pt) -> NewPtInfo
newtonApprox:(SUP SF, SF, SF, PI) -> Union(SF, "failed")
--% representation
Rep := Record(poly : P I, _
xVar : SY, _
yVar : SY, _
minXVal : RN, _
maxXVal : RN, _
minYVal : RN, _
maxYVal : RN, _
bdryPts : BoundaryPts, _
hTanPts : L Pt, _
vTanPts : L Pt, _
branches : L L Pt)
--% global constants
EPSILON : NF := 0.000001 -- precision to which realSolve finds roots
PLOTERR : SF := float(1, -3, 10)
-- maximum allowable difference in each coordinate when
-- determining if 2 plotted points are equal
--% global flags
NADA : String := "nothing in particular"
BDRY : String := "boundary point"
CRIT : String := "critical point"
BOTTOM : String := "bottom"
TOP : String := "top"
--% hacks
NFtoSF : NF -> SF
NFtoSF x == 0 + convert(x)$NF
--% points
makePt : (SF, SF) -> Pt
makePt(xx, yy) == point(l : L SF := [xx, yy])
swapCoords(pt) == makePt(yCoord pt, xCoord pt)
samePlottedPt?(p0, p1) ==
-- determines if p1 lies in a square with side 2 PLOTERR
-- centered at p0
x0 := xCoord p0; y0 := yCoord p0
x1 := xCoord p1; y1 := yCoord p1
(abs(x1-x0) < PLOTERR) and (abs(y1-y0) < PLOTERR)
findPtOnList(pt, pointList) ==
for point in pointList repeat
samePlottedPt?(pt, point) => return point
"failed"
--% corners
makeCorners(xMinSF, xMaxSF, yMinSF, yMaxSF) ==
[xMinSF, xMaxSF, yMinSF, yMaxSF]
getXMin(corners) == corners.minXVal
getXMax(corners) == corners.maxXVal
getYMin(corners) == corners.minYVal
getYMax(corners) == corners.maxYVal
--% coercions
SFPolyToUPoly(p) ==
-- 'p' is of type Polynomial, but has only one variable
zero? p => 0
monomial(leadingCoefficient p, totalDegree p) +
SFPolyToUPoly(reductum p)
RNPolyToUPoly(p) ==
-- 'p' is of type Polynomial, but has only one variable
zero? p => 0
monomial(leadingCoefficient p, totalDegree p) +
RNPolyToUPoly(reductum p)
coerceCoefsToSFs(p) ==
-- coefficients of 'p' are coerced to be DoubleFloat's
map(coerce, p)$PolynomialFunctions2(I, SF)
coerceCoefsToRNs(p) ==
-- coefficients of 'p' are coerced to be DoubleFloat's
map(coerce, p)$PolynomialFunctions2(I, RN)
RNtoSF(r) == coerce(r)@SF
RNtoNF(r) == coerce(r)@NF
SFtoNF(x) == convert(x)@NF
--% computation of special points
listPtsOnHorizBdry(pRN, y, y0, xMinNF, xMaxNF) ==
-- strict inequality here: corners on vertical boundary
pointList : L Pt := []
ySF := RNtoSF(y0)
f := eval(pRN, y, y0)
roots : L NF := kinte(f, EPSILON)
for root in roots repeat
if (xMinNF < root) and (root < xMaxNF) then
pointList := cons(makePt(NFtoSF root, ySF), pointList)
pointList
listPtsOnVertBdry(pRN, x, x0, yMinNF, yMaxNF) ==
pointList : L Pt := []
xSF := RNtoSF(x0)
f := eval(pRN, x, x0)
roots : L NF := kinte(f, EPSILON)
for root in roots repeat
if (yMinNF <= root) and (root <= yMaxNF) then
pointList := cons(makePt(xSF, NFtoSF root), pointList)
pointList
listPtsInRect(points, xMin, xMax, yMin, yMax) ==
pointList : L Pt := []
for point in points repeat
xx := first point; yy := second point
if (xMin<=xx) and (xx<=xMax) and (yMin<=yy) and (yy<=yMax) then
pointList := cons(makePt(NFtoSF xx, NFtoSF yy), pointList)
pointList
ptsSuchThat?(points, pred) ==
for point in points repeat
if pred point then return true
false
inRect?(point, xMinNF, xMaxNF, yMinNF, yMaxNF) ==
xx := first point; yy := second point
xMinNF <= xx and xx <= xMaxNF and yMinNF <= yy and yy <= yMaxNF
onHorzSeg?(point, xMinNF, xMaxNF, yNF) ==
xx := first point; yy := second point
yy = yNF and xMinNF <= xx and xx <= xMaxNF
onVertSeg?(point, yMinNF, yMaxNF, xNF) ==
xx := first point; yy := second point
xx = xNF and yMinNF <= yy and yy <= yMaxNF
newX(vtanPts, singPts, yMinNF, yMaxNF, xNF, xRN, horizInc) ==
xNewNF := xNF + RNtoNF horizInc
xRtNF := max(xNF, xNewNF); xLftNF := min(xNF, xNewNF)
foo : L NF -> B := x +-> inRect?(x, xLftNF, xRtNF, yMinNF, yMaxNF)
ptsSuchThat?(singPts, foo) =>
newX(vtanPts, singPts, yMinNF, yMaxNF, xNF, xRN, horizInc/2::RN)
goo : L NF -> B := x +-> onVertSeg?(x, yMinNF, yMaxNF, xNewNF)
ptsSuchThat?(vtanPts, goo) =>
newX(vtanPts, singPts, yMinNF, yMaxNF, xNF, xRN, horizInc/2::RN)
xRN + horizInc
newY(htanPts, singPts, xMinNF, xMaxNF, yNF, yRN, vertInc) ==
yNewNF := yNF + RNtoNF vertInc
yTopNF := max(yNF, yNewNF); yBotNF := min(yNF, yNewNF)
foo : L NF -> B := x +-> inRect?(x, xMinNF, xMaxNF, yBotNF, yTopNF)
ptsSuchThat?(singPts, foo) =>
newY(htanPts, singPts, xMinNF, xMaxNF, yNF, yRN, vertInc/2::RN)
goo : L NF -> B := x +-> onHorzSeg?(x, xMinNF, xMaxNF, yNewNF)
ptsSuchThat?(htanPts, goo) =>
newY(htanPts, singPts, xMinNF, xMaxNF, yNF, yRN, vertInc/2::RN)
yRN + vertInc
--% creation of sketches
makeSketch(p, x, y, xRange, yRange) ==
xMin := low(xRange); xMax := high(xRange)
yMin := low(yRange); yMax := high(yRange)
-- test input for consistency
xMax <= xMin =>
error "makeSketch: bad range for first variable"
yMax <= yMin =>
error "makeSketch: bad range for second variable"
varList := variables p
# varList > 2 =>
error "makeSketch: polynomial in more than 2 variables"
# varList = 0 =>
error "makeSketch: constant polynomial"
-- polynomial in 1 variable
# varList = 1 =>
(not member?(x, varList)) and (not member?(y, varList)) =>
error "makeSketch: bad variables"
makeOneVarSketch(p, x, y, xMin, xMax, yMin, yMax, first varList)
-- polynomial in 2 variables
(not member?(x, varList)) or (not member?(y, varList)) =>
error "makeSketch: bad variables"
totalDegree p = 1 =>
makeLineSketch(p, x, y, xMin, xMax, yMin, yMax)
-- polynomial is linear in one variable
-- y is a rational function of x
degree(p, y) = 1 =>
makeRatFcnSketch(p, x, y, xMin, xMax, yMin, yMax, y)
-- x is a rational function of y
degree(p, x) = 1 =>
makeRatFcnSketch(p, x, y, xMin, xMax, yMin, yMax, x)
-- the general case
makeGeneralSketch(p, x, y, xMin, xMax, yMin, yMax)
--% special cases
makeOneVarSketch(p, x, y, xMin, xMax, yMin, yMax, var) ==
-- the case where 'p' is a polynomial in only one variable
-- the graph consists of horizontal or vertical lines
if var = x then
minVal := RNtoNF xMin
maxVal := RNtoNF xMax
else
minVal := RNtoNF yMin
maxVal := RNtoNF yMax
lf : L Pt := []; rt : L Pt := []
bt : L Pt := []; tp : L Pt := []
htans : L Pt := []; vtans : L Pt := []
bran : L L Pt := []
roots := kinte(p, EPSILON)
sketchRoots : L SF := []
for root in roots repeat
if (minVal <= root) and (root <= maxVal) then
sketchRoots := cons(NFtoSF root, sketchRoots)
empty?(sketchRoots) =>
[p, x, y, xMin, xMax, yMin, yMax, [lf, rt, bt, tp], htans, vtans, bran]
if var = x then
yMinSF := RNtoSF yMin; yMaxSF := RNtoSF yMax
for rootSF in sketchRoots repeat
tp := cons(pt1 := makePt(rootSF, yMaxSF), tp)
bt := cons(pt2 := makePt(rootSF, yMinSF), bt)
branch : L Pt := [pt1, pt2]
bran := cons(branch, bran)
else
xMinSF := RNtoSF xMin; xMaxSF := RNtoSF xMax
for rootSF in sketchRoots repeat
rt := cons(pt1 := makePt(xMaxSF, rootSF), rt)
lf := cons(pt2 := makePt(xMinSF, rootSF), lf)
branch : L Pt := [pt1, pt2]
bran := cons(branch, bran)
[p, x, y, xMin, xMax, yMin, yMax, [lf, rt, bt, tp], htans, vtans, bran]
makeLineSketch(p, x, y, xMin, xMax, yMin, yMax) ==
-- the case where p(x, y) = a x + b y + c with a ~= 0, b ~= 0
-- this is a line which is neither vertical nor horizontal
xMinSF := RNtoSF xMin; xMaxSF := RNtoSF xMax
yMinSF := RNtoSF yMin; yMaxSF := RNtoSF yMax
-- determine the coefficients a, b, and c
a := ground(coefficient(p, x, 1)) :: SF
b := ground(coefficient(p, y, 1)) :: SF
c := ground(coefficient(coefficient(p, x, 0), y, 0)) :: SF
lf : L Pt := []; rt : L Pt := []
bt : L Pt := []; tp : L Pt := []
htans : L Pt := []; vtans : L Pt := []
branch : L Pt := []; bran : L L Pt := []
-- compute x coordinate of point on line with y = yMin
xBottom := (- b*yMinSF - c)/a
-- compute x coordinate of point on line with y = yMax
xTop := (- b*yMaxSF - c)/a
-- compute y coordinate of point on line with x = xMin
yLeft := (- a*xMinSF - c)/b
-- compute y coordinate of point on line with x = xMax
yRight := (- a*xMaxSF - c)/b
-- determine which of the above 4 points are in the region
-- to be plotted and list them as a branch
if (xMinSF < xBottom) and (xBottom < xMaxSF) then
bt := cons(pt := makePt(xBottom, yMinSF), bt)
branch := cons(pt, branch)
if (xMinSF < xTop) and (xTop < xMaxSF) then
tp := cons(pt := makePt(xTop, yMaxSF), tp)
branch := cons(pt, branch)
if (yMinSF <= yLeft) and (yLeft <= yMaxSF) then
lf := cons(pt := makePt(xMinSF, yLeft), lf)
branch := cons(pt, branch)
if (yMinSF <= yRight) and (yRight <= yMaxSF) then
rt := cons(pt := makePt(xMaxSF, yRight), rt)
branch := cons(pt, branch)
bran := cons(branch, bran)
[p, x, y, xMin, xMax, yMin, yMax, [lf, rt, bt, tp], htans, vtans, bran]
singValBetween?(xCurrent, xNext, xSingList) ==
for xVal in xSingList repeat
(xCurrent < xVal) and (xVal < xNext) => return true
false
segmentInfo(f, lo, hi, botList, topList, singList, minSF, maxSF) ==
repeat
-- 'current' is the smallest element of 'topList' and 'botList'
-- 'currentFrom' records the list from which it was taken
if empty?(topList) then
if empty?(botList) then
return [segment(lo, hi), hi, [], []]
else
current := first botList
botList := rest botList
currentFrom := BOTTOM
else
if empty?(botList) then
current := first topList
topList := rest topList
currentFrom := TOP
else
bot := first botList
top := first topList
if bot < top then
current := bot
botList := rest botList
currentFrom := BOTTOM
else
current := top
topList := rest topList
currentFrom := TOP
-- 'nxt' is the next smallest element of 'topList'
-- and 'botList'
-- 'nextFrom' records the list from which it was taken
if empty?(topList) then
if empty?(botList) then
return [segment(lo, hi), hi, [], []]
else
nxt := first botList
botList := rest botList
nextFrom := BOTTOM
else
if empty?(botList) then
nxt := first topList
topList := rest topList
nextFrom := TOP
else
bot := first botList
top := first topList
if bot < top then
nxt := bot
botList := rest botList
nextFrom := BOTTOM
else
nxt := top
topList := rest topList
nextFrom := TOP
if currentFrom = nextFrom then
if singValBetween?(current, nxt, singList) then
return [segment(lo, current), nxt, botList, topList]
else
val := f((nxt - current)/2::SF)
if (val <= minSF) or (val >= maxSF) then
return [segment(lo, current), nxt, botList, topList]
else
if singValBetween?(current, nxt, singList) then
return [segment(lo, current), nxt, botList, topList]
makeRatFcnSketch(p, x, y, xMin, xMax, yMin, yMax, depVar) ==
-- the case where p(x, y) is linear in x or y
-- Thus, one variable is a rational function of the other.
-- Therefore, we may use the 2-dimensional function plotting
-- package. The only problem is determining the intervals on
-- on which the function is to be plotted.
--!! corners: e.g. upper left corner is on graph with y' > 0
factoredP := p :: FR P I
numberOfFactors(factoredP) > 1 =>
error "reducible polynomial" --!! sketch each factor
dpdx := differentiate(p, x)
dpdy := differentiate(p, y)
pRN := coerceCoefsToRNs p
xMinSF := RNtoSF xMin; xMaxSF := RNtoSF xMax
yMinSF := RNtoSF yMin; yMaxSF := RNtoSF yMax
xMinNF := RNtoNF xMin; xMaxNF := RNtoNF xMax
yMinNF := RNtoNF yMin; yMaxNF := RNtoNF yMax
-- 'p' is of degree 1 in the variable 'depVar'.
-- Thus, 'depVar' is a rational function of the other variable.
num := -coefficient(p, depVar, 0)
den := coefficient(p, depVar, 1)
numUPolySF := SFPolyToUPoly(coerceCoefsToSFs(num))
denUPolySF := SFPolyToUPoly(coerceCoefsToSFs(den))
-- this is the rational function
f : SF -> SF := s +-> elt(numUPolySF, s)/elt(denUPolySF, s)
-- values of the dependent and independent variables
if depVar = x then
indVarMin := yMin; indVarMax := yMax
indVarMinNF := yMinNF; indVarMaxNF := yMaxNF
indVarMinSF := yMinSF; indVarMaxSF := yMaxSF
depVarMin := xMin; depVarMax := xMax
depVarMinSF := xMinSF; depVarMaxSF := xMaxSF
else
indVarMin := xMin; indVarMax := xMax
indVarMinNF := xMinNF; indVarMaxNF := xMaxNF
indVarMinSF := xMinSF; indVarMaxSF := xMaxSF
depVarMin := yMin; depVarMax := yMax
depVarMinSF := yMinSF; depVarMaxSF := yMaxSF
-- Create lists of critical points.
htanPts := rsolve([p, dpdx], [x, y], EPSILON)
vtanPts := rsolve([p, dpdy], [x, y], EPSILON)
htans := listPtsInRect(htanPts, xMinNF, xMaxNF, yMinNF, yMaxNF)
vtans := listPtsInRect(vtanPts, xMinNF, xMaxNF, yMinNF, yMaxNF)
-- Create lists which will contain boundary points.
lf : L Pt := []; rt : L Pt := []
bt : L Pt := []; tp : L Pt := []
-- Determine values of the independent variable at the which
-- the rational function has a pole as well as the values of
-- the independent variable for which there is a point on the
-- upper or lower boundary.
singList : L SF :=
roots : L NF := kinte(den, EPSILON)
outList : L SF := []
for root in roots repeat
if (indVarMinNF < root) and (root < indVarMaxNF) then
outList := cons(NFtoSF root, outList)
sort((x, y) +-> x < y, outList)
topList : L SF :=
roots : L NF := kinte(eval(pRN, depVar, depVarMax), EPSILON)
outList : L SF := []
for root in roots repeat
if (indVarMinNF < root) and (root < indVarMaxNF) then
outList := cons(NFtoSF root, outList)
sort((x, y) +-> x < y, outList)
botList : L SF :=
roots : L NF := kinte(eval(pRN, depVar, depVarMin), EPSILON)
outList : L SF := []
for root in roots repeat
if (indVarMinNF < root) and (root < indVarMaxNF) then
outList := cons(NFtoSF root, outList)
sort((x, y) +-> x < y, outList)
-- We wish to determine if the graph has points on the 'left'
-- and 'right' boundaries, so we compute the value of the
-- rational function at the lefthand and righthand values of
-- the dependent variable. If the function has a singularity
-- on the left or right boundary, then 'leftVal' or 'rightVal'
-- is given a dummy value which will convince the program that
-- there is no point on the left or right boundary.
denUPolyRN := RNPolyToUPoly(coerceCoefsToRNs(den))
if elt(denUPolyRN, indVarMin) = 0$RN then
leftVal := depVarMinSF - (abs(depVarMinSF) + 1$SF)
else
leftVal := f(indVarMinSF)
if elt(denUPolyRN, indVarMax) = 0$RN then
rightVal := depVarMinSF - (abs(depVarMinSF) + 1$SF)
else
rightVal := f(indVarMaxSF)
-- Now put boundary points on the appropriate lists.
if depVar = x then
if (xMinSF < leftVal) and (leftVal < xMaxSF) then
bt := cons(makePt(leftVal, yMinSF), bt)
if (xMinSF < rightVal) and (rightVal < xMaxSF) then
tp := cons(makePt(rightVal, yMaxSF), tp)
for val in botList repeat
lf := cons(makePt(xMinSF, val), lf)
for val in topList repeat
rt := cons(makePt(xMaxSF, val), rt)
else
if (yMinSF < leftVal) and (leftVal < yMaxSF) then
lf := cons(makePt(xMinSF, leftVal), lf)
if (yMinSF < rightVal) and (rightVal < yMaxSF) then
rt := cons(makePt(xMaxSF, rightVal), rt)
for val in botList repeat
bt := cons(makePt(val, yMinSF), bt)
for val in topList repeat
tp := cons(makePt(val, yMaxSF), tp)
bran : L L Pt := []
-- Determine segments on which the rational function is to
-- be plotted.
if (depVarMinSF < leftVal) and (leftVal < depVarMaxSF) then
lo := indVarMinSF
else
if empty?(topList) then
if empty?(botList) then
return [p, x, y, xMin, xMax, yMin, yMax, [lf, rt, bt, tp], _
htans, vtans, bran]
else
lo := first botList
botList := rest botList
else
if empty?(botList) then
lo := first topList
topList := rest topList
else
bot := first botList
top := first topList
if bot < top then
lo := bot
botList := rest botList
else
lo := top
topList := rest topList
hi := 0$SF -- @#$%^&* compiler
if (depVarMinSF < rightVal) and (rightVal < depVarMaxSF) then
hi := indVarMaxSF
else
if empty?(topList) then
if empty?(botList) then
error "makeRatFcnSketch: plot domain"
else
hi := last botList
botList := remove(hi, botList)
else
if empty?(botList) then
hi := last topList
topList := remove(hi, topList)
else
bot := last botList
top := last topList
if bot > top then
hi := bot
botList := remove(hi, botList)
else
hi := top
topList := remove(hi, topList)
if (depVar = x) then
(minSF := xMinSF; maxSF := xMaxSF)
else
(minSF := yMinSF; maxSF := yMaxSF)
segList : L SEG SF := []
repeat
segInfo := segmentInfo(f, lo, hi, botList, topList, singList, _
minSF, maxSF)
segList := cons(segInfo.seg, segList)
lo := segInfo.left
botList := segInfo.lowerVals
topList := segInfo.upperVals
if lo = hi then break
for segment in segList repeat
RFPlot : Plot := plot(f, segment)
curve := first(listBranches(RFPlot))
if depVar = y then
bran := cons(curve, bran)
else
bran := cons(map(swapCoords, curve), bran)
[p, x, y, xMin, xMax, yMin, yMax, [lf, rt, bt, tp], htans, vtans, bran]
--% the general case
makeGeneralSketch(pol, x, y, xMin, xMax, yMin, yMax) ==
--!! corners of region should not be on curve
--!! enlarge region if necessary
factoredPol := pol :: FR P I
flist := factorList factoredPol
# flist > 1 =>
error "reducible polynomial" --!! sketch each factor
p := first(flist).factor
dpdx := differentiate(p, x); dpdy := differentiate(p, y)
xMinNF := RNtoNF xMin; xMaxNF := RNtoNF xMax
yMinNF := RNtoNF yMin; yMaxNF := RNtoNF yMax
-- compute singular points; error if singularities in region
singPts := rsolve([p, dpdx, dpdy], [x, y], EPSILON)
foo : L NF -> B := s +-> inRect?(s, xMinNF, xMaxNF, yMinNF, yMaxNF)
ptsSuchThat?(singPts, foo) =>
error "singular pts in region of sketch"
-- compute critical points
htanPts := rsolve([p, dpdx], [x, y], EPSILON)
vtanPts := rsolve([p, dpdy], [x, y], EPSILON)
critPts := append(htanPts, vtanPts)
-- if there are critical points on the boundary, then enlarge
-- the region, but be sure that the new region does not contain
-- any singular points
hInc : RN := (1/20) * (xMax - xMin)
vInc : RN := (1/20) * (yMax - yMin)
foo : L NF -> B := s +-> onVertSeg?(s, yMinNF, yMaxNF, xMinNF)
if ptsSuchThat?(critPts, foo) then
xMin := newX(critPts, singPts, yMinNF, yMaxNF, xMinNF, xMin, -hInc)
xMinNF := RNtoNF xMin
foo : L NF -> B := s +-> onVertSeg?(s, yMinNF, yMaxNF, xMaxNF)
if ptsSuchThat?(critPts, foo) then
xMax := newX(critPts, singPts, yMinNF, yMaxNF, xMaxNF, xMax, hInc)
xMaxNF := RNtoNF xMax
foo : L NF -> B := s +-> onHorzSeg?(s, xMinNF, xMaxNF, yMinNF)
if ptsSuchThat?(critPts, foo) then
yMin := newY(critPts, singPts, xMinNF, xMaxNF, yMinNF, yMin, -vInc)
yMinNF := RNtoNF yMin
foo : L NF -> B := s +-> onHorzSeg?(s, xMinNF, xMaxNF, yMaxNF)
if ptsSuchThat?(critPts, foo) then
yMax := newY(critPts, singPts, xMinNF, xMaxNF, yMaxNF, yMax, vInc)
yMaxNF := RNtoNF yMax
htans := listPtsInRect(htanPts, xMinNF, xMaxNF, yMinNF, yMaxNF)
vtans := listPtsInRect(vtanPts, xMinNF, xMaxNF, yMinNF, yMaxNF)
crits := append(htans, vtans)
-- conversions to DoubleFloats
xMinSF := RNtoSF xMin; xMaxSF := RNtoSF xMax
yMinSF := RNtoSF yMin; yMaxSF := RNtoSF yMax
corners := makeCorners(xMinSF, xMaxSF, yMinSF, yMaxSF)
pSF := coerceCoefsToSFs p
dpdxSF := coerceCoefsToSFs dpdx
dpdySF := coerceCoefsToSFs dpdy
delta := min((xMaxSF - xMinSF)/25, (yMaxSF - yMinSF)/25)
err := min(delta/100, PLOTERR/100)
bound : PI := 10
-- compute points on the boundary
pRN := coerceCoefsToRNs(p)
lf : L Pt := listPtsOnVertBdry(pRN, x, xMin, yMinNF, yMaxNF)
rt : L Pt := listPtsOnVertBdry(pRN, x, xMax, yMinNF, yMaxNF)
bt : L Pt := listPtsOnHorizBdry(pRN, y, yMin, xMinNF, xMaxNF)
tp : L Pt := listPtsOnHorizBdry(pRN, y, yMax, xMinNF, xMaxNF)
bdPts : BoundaryPts := [lf, rt, bt, tp]
bran := traceBranches(pSF, dpdxSF, dpdySF, x, y, corners, delta, err, _
bound, crits, bdPts)
[p, x, y, xMin, xMax, yMin, yMax, bdPts, htans, vtans, bran]
refine(plot, stepFraction) ==
p := plot.poly; x := plot.xVar; y := plot.yVar
dpdx := differentiate(p, x); dpdy := differentiate(p, y)
pSF := coerceCoefsToSFs p
dpdxSF := coerceCoefsToSFs dpdx
dpdySF := coerceCoefsToSFs dpdy
xMin := plot.minXVal; xMax := plot.maxXVal
yMin := plot.minYVal; yMax := plot.maxYVal
xMinSF := RNtoSF xMin; xMaxSF := RNtoSF xMax
yMinSF := RNtoSF yMin; yMaxSF := RNtoSF yMax
corners := makeCorners(xMinSF, xMaxSF, yMinSF, yMaxSF)
pSF := coerceCoefsToSFs p
dpdxSF := coerceCoefsToSFs dpdx
dpdySF := coerceCoefsToSFs dpdy
delta :=
stepFraction * min((xMaxSF - xMinSF)/25, (yMaxSF - yMinSF)/25)
err := min(delta/100, PLOTERR/100)
bound : PI := 10
crits := append(plot.hTanPts, plot.vTanPts)
bdPts := plot.bdryPts
bran := traceBranches(pSF, dpdxSF, dpdySF, x, y, corners, delta, err, _
bound, crits, bdPts)
htans := plot.hTanPts; vtans := plot.vTanPts
[p, x, y, xMin, xMax, yMin, yMax, bdPts, htans, vtans, bran]
traceBranches(pSF, dpdxSF, dpdySF, x, y, corners, delta, err, bound, _
crits, bdPts) ==
-- for boundary points, trace curve from boundary to boundary
-- add the branch to the list of branches
-- update list of boundary points by deleting first and last
-- points on this branch
-- update list of critical points by deleting any critical
-- points which were plotted
lf := bdPts.left; rt := bdPts.right
tp := bdPts.top ; bt := bdPts.bottom
bdry := append(append(lf, rt), append(bt, tp))
bran : L L Pt := []
while not(empty?(bdry)) repeat
pt := first bdry
p0 := dummyFirstPt(pt, dpdxSF, dpdySF, x, y, lf, rt, bt, tp)
segInfo := listPtsOnSegment(pSF, dpdxSF, dpdySF, x, y, p0, pt, _
corners, delta, err, bound, crits, bdry)
bran := cons(first segInfo, bran)
crits := second segInfo
bdry := third segInfo
-- trace loops beginning and ending with critical points
-- add the branch to the list of branches
-- update list of critical points by deleting any critical
-- points which were plotted
while not(empty?(crits)) repeat
pt := first crits
segInfo := listPtsOnLoop(pSF, dpdxSF, dpdySF, x, y, pt, _
corners, delta, err, bound, crits, bdry)
bran := cons(first segInfo, bran)
crits := second segInfo
bran
dummyFirstPt(p1, dpdxSF, dpdySF, x, y, lf, rt, bt, tp) ==
-- The function 'computeNextPt' requires 2 points, p0 and p1.
-- When computing the second point on a branch which starts
-- on the boundary, we use the boundary point as p1 and the
-- 'dummy' point returned by this function as p0.
x1 := xCoord p1; y1 := yCoord p1
zero := 0$SF; one := 1$SF
px := ground(eval(dpdxSF, [x, y], [x1, y1]))
py := ground(eval(dpdySF, [x, y], [x1, y1]))
if px * py < zero then -- positive slope at p1
member?(p1, lf) or member?(p1, bt) =>
makePt(x1 - one, y1 - one)
makePt(x1 + one, y1 + one)
else
member?(p1, lf) or member?(p1, tp) =>
makePt(x1 - one, y1 + one)
makePt(x1 + one, y1 - one)
listPtsOnSegment(pSF, dpdxSF, dpdySF, x, y, p0, p1, corners, _
delta, err, bound, crits, bdry) ==
-- p1 is a boundary point; p0 is a 'dummy' point
bdry := remove(p1, bdry)
pointList : L Pt := [p1]
ptInfo := computeNextPt(pSF, dpdxSF, dpdySF, x, y, p0, p1, corners, _
delta, err, bound, crits, bdry)
p2 := ptInfo.newPt
ptInfo.type = BDRY =>
bdry := remove(p2, bdry)
pointList := cons(p2, pointList)
[pointList, crits, bdry]
if ptInfo.type = CRIT then crits := remove(p2, crits)
pointList := cons(p2, pointList)
repeat
pt0 := second pointList; pt1 := first pointList
ptInfo := computeNextPt(pSF, dpdxSF, dpdySF, x, y, pt0, pt1, corners, _
delta, err, bound, crits, bdry)
p2 := ptInfo.newPt
ptInfo.type = BDRY =>
bdry := remove(p2, bdry)
pointList := cons(p2, pointList)
return [pointList, crits, bdry]
if ptInfo.type = CRIT then crits := remove(p2, crits)
pointList := cons(p2, pointList)
--!! delete next line (compiler bug)
[pointList, crits, bdry]
listPtsOnLoop(pSF, dpdxSF, dpdySF, x, y, p1, corners, _
delta, err, bound, crits, bdry) ==
x1 := xCoord p1; y1 := yCoord p1
p0 := makePt(x1 - 1$SF, y1 - 1$SF)
pointList : L Pt := [p1]
ptInfo := computeNextPt(pSF, dpdxSF, dpdySF, x, y, p0, p1, corners, _
delta, err, bound, crits, bdry)
p2 := ptInfo.newPt
ptInfo.type = BDRY =>
error "boundary reached while on loop"
if ptInfo.type = CRIT then
p1 = p2 =>
error "first and second points on loop are identical"
crits := remove(p2, crits)
pointList := cons(p2, pointList)
repeat
pt0 := second pointList; pt1 := first pointList
ptInfo := computeNextPt(pSF, dpdxSF, dpdySF, x, y, pt0, pt1, corners, _
delta, err, bound, crits, bdry)
p2 := ptInfo.newPt
ptInfo.type = BDRY =>
error "boundary reached while on loop"
if ptInfo.type = CRIT then
crits := remove(p2, crits)
p1 = p2 =>
pointList := cons(p2, pointList)
return [pointList, crits, bdry]
pointList := cons(p2, pointList)
--!! delete next line (compiler bug)
[pointList, crits, bdry]
computeNextPt(pSF, dpdxSF, dpdySF, x, y, p0, p1, corners, _
delta, err, bound, crits, bdry) ==
-- p0=(x0, y0) and p1=(x1, y1) are the last two points on the curve.
-- The function computes the next point on the curve.
-- The function determines if the next point is a critical point
-- or a boundary point.
-- The function returns a record of the form
-- Record(newPt: Pt, type: String).
-- If the new point is a boundary point, then 'type' is
-- "boundary point" and 'newPt' is a boundary point to be
-- deleted from the list of boundary points yet to be plotted.
-- Similarly, if the new point is a critical point, then 'type' is
-- "critical point" and 'newPt' is a critical point to be
-- deleted from the list of critical points yet to be plotted.
-- If the new point is neither a critical point nor a boundary
-- point, then 'type' is "nothing in particular".
xMinSF := getXMin corners; xMaxSF := getXMax corners
yMinSF := getYMin corners; yMaxSF := getYMax corners
x0 := xCoord p0; y0 := yCoord p0
x1 := xCoord p1; y1 := yCoord p1
px := ground(eval(dpdxSF, [x, y], [x1, y1]))
py := ground(eval(dpdySF, [x, y], [x1, y1]))
-- let m be the slope of the tangent line at p1
-- if |m| < 1, we will increment the x-coordinate by delta
-- (indicated by 'incVar = x'), find an approximate
-- y-coordinate using the tangent line, then find the actual
-- y-coordinate using a Newton iteration
if abs(py) > abs(px) then
incVar0 := incVar := x
deltaX := (if x1 > x0 then delta else -delta)
x2Approx := x1 + deltaX
y2Approx := y1 + (-px/py)*deltaX
-- if |m| >= 1, we interchange the roles of the x- and y-
-- coordinates
else
incVar0 := incVar := y
deltaY := (if y1 > y0 then delta else -delta)
x2Approx := x1 + (-py/px)*deltaY
y2Approx := y1 + deltaY
lookingFor := NADA
-- See if (x2Approx, y2Approx) is out of bounds.
-- If so, find where the line segment connecting (x1, y1) and
-- (x2Approx, y2Approx) intersects the boundary and use this
-- point as (x2Approx, y2Approx).
-- If the resulting point is on the left or right boundary,
-- we will now consider x as the 'incremented variable' and we
-- will compute the y-coordinate using a Newton iteration.
-- Similarly, if the point is on the top or bottom boundary,
-- we will consider y as the 'incremented variable' and we
-- will compute the x-coordinate using a Newton iteration.
if x2Approx >= xMaxSF then
incVar := x
lookingFor := BDRY
x2Approx := xMaxSF
y2Approx := y1 + (-px/py)*(x2Approx - x1)
else
if x2Approx <= xMinSF then
incVar := x
lookingFor := BDRY
x2Approx := xMinSF
y2Approx := y1 + (-px/py)*(x2Approx - x1)
if y2Approx >= yMaxSF then
incVar := y
lookingFor := BDRY
y2Approx := yMaxSF
x2Approx := x1 + (-py/px)*(y2Approx - y1)
else
if y2Approx <= yMinSF then
incVar := y
lookingFor := BDRY
y2Approx := yMinSF
x2Approx := x1 + (-py/px)*(y2Approx - y1)
-- set xLo = min(x1, x2Approx), xHi = max(x1, x2Approx)
-- set yLo = min(y1, y2Approx), yHi = max(y1, y2Approx)
if x1 < x2Approx then
xLo := x1
xHi := x2Approx
else
xLo := x2Approx
xHi := x1
if y1 < y2Approx then
yLo := y1
yHi := y2Approx
else
yLo := y2Approx
yHi := y1
-- check for critical points (x*, y*) with x* between
-- x1 and x2Approx or y* between y1 and y2Approx
-- store values of x2Approx and y2Approx
x2Approxx := x2Approx
y2Approxx := y2Approx
-- xPointList will contain all critical points (x*, y*)
-- with x* between x1 and x2Approx
xPointList : L Pt := []
-- yPointList will contain all critical points (x*, y*)
-- with y* between y1 and y2Approx
yPointList : L Pt := []
for pt in crits repeat
xx := xCoord pt; yy := yCoord pt
-- if x1 = x2Approx, then p1 is a point with horizontal
-- tangent line
-- in this case, we don't want critical points with
-- x-coordinate x1
if xx = x2Approx and not (xx = x1) then
if min(abs(yy-yLo), abs(yy-yHi)) < delta then
xPointList := cons(pt, xPointList)
if ((xLo < xx) and (xx < xHi)) then
if min(abs(yy-yLo), abs(yy-yHi)) < delta then
xPointList := cons(pt, [])
x2Approx := xx
if xx < x1 then xLo := xx else xHi := xx
-- if y1 = y2Approx, then p1 is a point with vertical
-- tangent line
-- in this case, we don't want critical points with
-- y-coordinate y1
if yy = y2Approx and not (yy = y1) then
yPointList := cons(pt, yPointList)
if ((yLo < yy) and (yy < yHi)) then
if min(abs(xx-xLo), abs(xx-xHi)) < delta then
yPointList := cons(pt, [])
y2Approx := yy
if yy < y1 then yLo := yy else yHi := yy
-- points in both xPointList and yPointList
if not(empty?(xPointList)) and not(empty?(yPointList)) then
xPointList = yPointList =>
-- this implies that the lists have only one point
incVar := incVar0