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curves.jl
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curves.jl
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# arcs, circles, ellipses, curves, pie, sector, bezier
function circle(x::Real, y::Real, r::Real;
action=:none)
if action != :path
newpath()
end
Cairo.arc(get_current_cr(), x, y, r, 0, 2pi)
do_action(action)
return (Point(x, y) - (r, r), Point(x, y) + (r, r))
end
"""
circle(centerpoint::Point, r; action=:none)
circle(centerpoint::Point, r, action)
Make a circle of radius `r` centered at 'centerpoint', and add it to the current path.
`action` is one of the actions applied by `do_action`, defaulting to `:none`.
Returns a tuple of two points, the corners of a bounding box that encloses the circle.
You can also use `ellipse()` to draw circles and place them by their centerpoint.
"""
circle(centerpoint::Point, r::Real; action=:none) =
circle(centerpoint.x, centerpoint.y, r, action=action)
circle(pt::Point, r::Real, action::Symbol) = circle(pt, r, action = action::Symbol)
circle(x::Real, y::Real, r::Real, action::Symbol) = circle(Point(x, y), r, action = action::Symbol)
"""
circle(pt1::Point, pt2::Point; action=:none)
circle(pt1::Point, pt2::Point, action)
Make a circle that passes through two points that define the diameter, and add it to the current path.
"""
function circle(pt1::Point, pt2::Point;
action=:none)
center = midpoint(pt1, pt2)
radius = distance(pt1, pt2)/2
circle(center, radius, action=action)
end
circle(pt1::Point, pt2::Point, action::Symbol) = circle(pt1, pt2, action=action)
"""
circle(pt1::Point, pt2::Point, pt3::Point; action=:none)
circle(pt1::Point, pt2::Point, pt3::Point, action)
Make a circle that passes through three points, and add it to the current path.
"""
function circle(pt1::Point, pt2::Point, pt3::Point;
action=:none)
center = midpoint(pt1, pt2)
radius = distance(pt1, pt2)/2
circle(center3pts(pt1, pt2, pt3)..., action=action)
end
circle(pt1::Point, pt2::Point, pt3::Point, action::Symbol) =
circle(pt1, pt2, pt3; action=action)
"""
center3pts(a::Point, b::Point, c::Point)
Find the radius and center point for three points lying on a circle.
returns `(centerpoint, radius)` of a circle.
If there's no such circle, the function returns `(Point(0, 0), 0)`.
If two of the points are the same, use `circle(pt1, pt2)` instead.
"""
function center3pts(p1::Point, p2::Point, p3::Point)
norm2(p::Point) = (p.x)^2+(p.y)^2
α1 = norm2(p3-p2)*(norm2(p2-p1)+norm2(p1-p3)-norm2(p3-p2))
α2 = norm2(p1-p3)*(norm2(p3-p2)+norm2(p2-p1)-norm2(p1-p3))
α3 = norm2(p2-p1)*(norm2(p1-p3)+norm2(p3-p2)-norm2(p2-p1))
if α1+α2+α3 ≠ 0.0
c = (α1*p1+α2*p2+α3*p3)/(α1+α2+α3)
r = √(norm2(p1-c))
return c, r
else
@warn "center3pts(): There are no circles which pass through $p1, $p2 and $p3."
return (Point(0, 0), 0)
end
end
function ellipse(xc::Real, yc::Real, w::Real, h::Real;
action=:none)
x = xc - w/2
y = yc - h/2
# kappa = 4.0 * (sqrt(2.0) - 1.0) / 3.0
kappa = 0.5522847498307936
ox = (w / 2) * kappa # control point offset horizontal
oy = (h / 2) * kappa # control point offset vertical
xe = x + w # x-end
ye = y + h # y-end
xm = x + w / 2 # x-middle
ym = y + h / 2 # y-middle
move(x, ym)
curve(x, ym - oy, xm - ox, y, xm, y)
curve(xm + ox, y, xe, ym - oy, xe, ym)
curve(xe, ym + oy, xm + ox, ye, xm, ye)
curve(xm - ox, ye, x, ym + oy, x, ym)
do_action(action)
return (Point(xc, yc) - (w/2, h/2), Point(xc, yc) + (w/2, h/2))
end
ellipse(xc::Real, yc::Real, w::Real, h::Real, action::Symbol) =
ellipse(xc, yc, w, h, action=action)
"""
ellipse(centerpoint::Point, w, h; action=:none)
ellipse(centerpoint::Point, w, h; action)
Make an ellipse, centered at `centerpoint`, with width `w`, and height `h`, and add it to the current path.
Returns a tuple of two points, the corners of a bounding box that encloses the ellipse.
"""
ellipse(c::Point, w::Real, h::Real; action=:none) = ellipse(c.x, c.y, w, h, action=action)
ellipse(c::Point, w::Real, h::Real, action::Symbol) = ellipse(c, w, h, action=action)
"""
squircle(center::Point, hradius, vradius;
action=:none,
rt = 0.5, stepby = pi/40, vertices=false)
squircle(center::Point, hradius, vradius, action;
rt = 0.5, stepby = pi/40, vertices=false)
Make a squircle or superellipse (basically a rectangle with
rounded corners), and add it to the current path. Specify
the center position, horizontal radius (distance from center
to a side), and vertical radius (distance from center to top
or bottom):
The root (`rt`) option defaults to 0.5, and gives an intermediate shape. Values
less than 0.5 make the shape more rectangular. Values above make the shape more
round. The horizontal and vertical radii can be different.
"""
function squircle(center::Point, hradius::Real, vradius::Real;
action=:none,
rt = 0.5,
vertices=false,
stepby = pi/40,
reversepath=false)
points = Point[]
for theta in 0:stepby:2pi
xpos = center.x + ^(abs(cos(theta)), rt) * hradius * sign(cos(theta))
ypos = center.y + ^(abs(sin(theta)), rt) * vradius * sign(sin(theta))
push!(points, Point(xpos, ypos))
end
result = reversepath ? reverse(points) : points
if !vertices
poly(points, action, close=true, reversepath=reversepath)
end
return result
end
squircle(center::Point, hradius::Real, vradius::Real, action::Symbol;
rt = 0.5,
vertices=false,
stepby = pi/40,
reversepath=false) =
squircle(center, hradius, vradius;
action=action,
rt = rt,
vertices=false,
stepby = pi/40,
reversepath=false)
function arc(xc, yc, radius, angle1, angle2;
action=:none)
Cairo.arc(get_current_cr(), xc, yc, radius, angle1, angle2)
do_action(action)
end
arc(xc, yc, radius, angle1, angle2, the_action::Symbol) =
arc(xc, yc, radius, angle1, angle2, action=the_action)
"""
arc(centerpoint::Point, radius, angle1, angle2; action=:none)
arc(centerpoint::Point, radius, angle1, angle2, action)
Add an arc to the current path from `angle1` to `angle2` going clockwise, centered
at `centerpoint`.
Angles are defined relative to the x-axis, positive clockwise.
"""
arc(centerpoint::Point, radius, angle1, angle2; action=:none) =
arc(centerpoint.x, centerpoint.y, radius, angle1, angle2, action=action)
arc(centerpoint::Point, radius, angle1, angle2, action::Symbol) =
arc(centerpoint.x, centerpoint.y, radius, angle1, angle2, action=action)
function carc(xc, yc, radius, angle1, angle2;
action=:none)
Cairo.arc_negative(get_current_cr(), xc, yc, radius, angle1, angle2)
do_action(action)
end
"""
carc(centerpoint::Point, radius, angle1, angle2; action=:none)
carc(centerpoint::Point, radius, angle1, angle2, action)
Add an arc centered at `centerpoint` to the current path from `angle1` to
`angle2`, going counterclockwise.
Angles are defined relative to the x-axis, positive clockwise.
"""
carc(centerpoint::Point, radius, angle1, angle2; action=:none) =
carc(centerpoint.x, centerpoint.y, radius, angle1, angle2, action=action)
carc(centerpoint::Point, radius, angle1, angle2, action::Symbol) =
carc(centerpoint.x, centerpoint.y, radius, angle1, angle2, action=action)
carc(x::Real, y::Real, radius, angle1, angle2, action::Symbol=:none) =
carc(x, y, radius, angle1, angle2, action=action)
"""
arc2r(c1::Point, p2::Point, p3::Point; action=:none)
arc2r(c1::Point, p2::Point, p3::Point, action)
Add a circular arc centered at `c1` that starts at `p2` and ends at `p3`, going clockwise,
to the current path.
`c1`-`p2` really determines the radius. If `p3` doesn't lie on the circular path,
it will be used only as an indication of the arc's length, rather than its position.
"""
function arc2r(c1::Point, p2::Point, p3::Point;
action=:none)
r = distance(c1, p2)
startangle = atan(p2.y - c1.y, p2.x - c1.x)
endangle = atan(p3.y - c1.y, p3.x - c1.x)
if endangle < startangle
endangle = mod2pi(endangle + 2pi)
end
arc(c1, r, startangle, endangle, action=action)
end
arc2r(c1::Point, p2::Point, p3::Point, action::Symbol) = arc2r(c1, p2, p3, action = action)
"""
carc2r(c1::Point, p2::Point, p3::Point; action=:none)
Add a circular arc centered at `c1` that starts at `p2` and ends at `p3`,
going counterclockwise, to the current path.
`c1`-`p2` really determines the radius. If `p3` doesn't lie on the circular
path, it will be used only as an indication of the arc's length, rather than its position.
"""
function carc2r(c1::Point, p2::Point, p3::Point;
action=:none)
r = distance(c1, p2)
startangle = atan(p2.y - c1.y, p2.x - c1.x)
endangle = atan(p3.y - c1.y, p3.x - c1.x)
if startangle < endangle
startangle = mod2pi(startangle + 2pi)
end
carc(c1, r, startangle, endangle, action=action)
end
carc2r(c1::Point, p2::Point, p3::Point, action::Symbol) = carc2r(c1, p2, p3, action = action)
"""
isarcclockwise(c::Point, A::Point, B::Point)
Return `true` if an arc centered at `c` going from `A` to `B` is clockwise.
If `c`, `A`, and `B` are collinear, then a hemispherical arc could be
either clockwise or not.
"""
function isarcclockwise(c::Point, A::Point, B::Point)
a = A - c
b = B - c
return crossproduct(a, b) > 0
end
"""
sector(centerpoint::Point, innerradius, outerradius, startangle, endangle;
action=:none)
Make an annular sector centered at `centerpoint`, and add it to the current path.
TODO - return something more useful than a Boolean
"""
function sector(centerpoint::Point, innerradius::Real, outerradius::Real,
startangle::Real, endangle::Real;
action=:none)
(innerradius > outerradius) && throw(DomainError(outerradius, "outer radius must be larger than inner radius $(innerradius)"))
gsave()
translate(centerpoint)
newpath()
move(innerradius * cos(startangle), innerradius * sin(startangle))
line(outerradius * cos(startangle), outerradius * sin(startangle))
arc(0, 0, outerradius, startangle, endangle, action=:none)
line(innerradius * cos(endangle), innerradius * sin(endangle))
carc(0, 0, innerradius, endangle, startangle, action=:none)
closepath()
grestore()
do_action(action)
end
sector(centerpoint::Point, innerradius::Real, outerradius::Real,
startangle::Real, endangle::Real, action::Symbol) =
sector(centerpoint, innerradius, outerradius, startangle, endangle, action=action)
"""
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real;
action=:none)
Make an annular sector centered at the origin, and add it to the current path.
"""
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real;
action=:none) =
sector(O, innerradius, outerradius, startangle, endangle, action=action)
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real, action::Symbol) =
sector(innerradius, outerradius, startangle, endangle, action=action)
"""
sector(centerpoint::Point, innerradius, outerradius,
startangle, endangle, cornerradius;
action:none)
Make an annular sector with rounded corners, basically a bent sausage shape,
centered at `centerpoint`, and add it to the current path.
TODO: The results aren't 100% accurate at the moment. There are small
discontinuities where the curves join.
TODO - return something more useful than a Boolean
The cornerradius is reduced from the supplied value if neceesary to prevent overshoots.
"""
function sector(centerpoint::Point, innerradius::Real, outerradius::Real, startangle::Real,
endangle::Real, cornerradius::Real;
action=:none)
(innerradius > outerradius) && throw(DomainError(outerradius, "outer radius must be larger than inner radius $(innerradius)"))
gsave()
translate(centerpoint)
# some work is done using polar coords to calculate the points
# attempts to prevent pathological cases
cornerradius = min(cornerradius, abs(outerradius-innerradius)/2)
if endangle < startangle
endangle = mod2pi(endangle + 2pi)
end
# TODO reduce given corner radius to prevent messes when spanning angle too small
# 4 is a magic number
while abs(endangle - startangle) < 4.0(atan(cornerradius, innerradius))
cornerradius *= 0.75
end
# first inner corner
alpha1 = asin(cornerradius/(innerradius+cornerradius))
p1p2center = (innerradius + cornerradius, startangle + alpha1)
p1 = (innerradius, startangle + alpha1)
p2 = (innerradius + cornerradius, startangle)
# first outer
alpha2 = asin(cornerradius/(outerradius-cornerradius))
p3p4center = (outerradius - cornerradius, startangle + alpha2)
p3 = (outerradius - cornerradius, startangle)
p4 = (outerradius, startangle + alpha2)
# last outer
p5p6center = (outerradius - cornerradius, endangle - alpha2)
p5 = (outerradius, endangle - alpha2)
p6 = (outerradius - cornerradius, endangle)
# last inner
p7p8center = (innerradius + cornerradius, endangle - alpha1)
p7 = (innerradius + cornerradius, endangle)
p8 = (innerradius, endangle - alpha1)
# make path
move(@polar(p1))
newpath()
# inner corner
arc(@polar(p1p2center), cornerradius, slope(@polar(p1p2center), @polar(p1)),
slope(@polar(p1p2center), @polar(p2)), action=:none)
line(@polar(p3))
# outer corner
arc(@polar(p3p4center), cornerradius, slope(@polar(p3p4center), @polar(p3)),
slope(@polar(p3p4center), @polar(p4)), action=:none)
# outside arc
arc(O, outerradius, slope(O, @polar(p4)), slope(O, @polar(p5)), action=:none)
# last outside corner
arc(@polar(p5p6center), cornerradius, slope(@polar(p5p6center), @polar(p5)),
slope(@polar(p5p6center), @polar(p6)), action=:none)
line(@polar(p7))
# last inner corner
arc(@polar(p7p8center), cornerradius, slope(@polar(p7p8center), @polar(p7)),
slope(@polar(p7p8center), @polar(p8)), action=:none)
s1, s2 = slope(O, @polar(p8)), slope(O, @polar(p1))
if s1 < s2
s2 = mod2pi(s2 + 2pi)
end
carc(O, innerradius, s1, s2, action=:none)
closepath()
do_action(action)
grestore()
end
sector(centerpoint::Point, innerradius::Real, outerradius::Real, startangle::Real, endangle::Real,
cornerradius::Real, action::Symbol) = sector(centerpoint,
innerradius, outerradius, startangle, endangle,
cornerradius, action=action)
"""
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real,
cornerradius::Real, action)
Make an annular sector with rounded corners, centered at the
current origin, and add it to the current path.
"""
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real,
cornerradius::Real, action::Symbol) =
sector(O, innerradius, outerradius, startangle, endangle, cornerradius, action=action)
sector(innerradius::Real, outerradius::Real, startangle::Real, endangle::Real, cornerradius::Real; action=:stroke) =
sector(O, innerradius, outerradius, startangle, endangle, cornerradius, action=action)
"""
pie(x, y, radius, startangle, endangle; action=:none)
pie(centerpoint, radius, startangle, endangle; action=:none)
Make a pie shape centered at `x`/`y`. Angles start at the positive x-axis and
are measured clockwise, and add it to the current path.
TODO - return something more useful than a Boolean
"""
function pie(x::Real, y::Real, radius::Real, startangle::Real, endangle::Real;
action=:none)
gsave()
translate(x, y)
newpath()
move(0, 0)
line(radius * cos(startangle), radius * sin(startangle))
arc(0, 0, radius, startangle, endangle, action=:none)
closepath()
grestore()
do_action(action)
end
pie(x::Real, y::Real, radius::Real, startangle::Real, endangle::Real, action::Symbol) =
pie(x, y, radius, startangle, endangle, action=action)
pie(centerpoint::Point, radius::Real, startangle::Real, endangle::Real; action=:none) =
pie(centerpoint.x, centerpoint.y, radius, startangle, endangle, action=action)
"""
pie(centerpoint::Point, radius::Real, startangle::Real, endangle::Real, action::Symbol)
"""
pie(centerpoint::Point, radius::Real, startangle::Real, endangle::Real, action::Symbol) =
pie(centerpoint.x, centerpoint.y, radius, startangle, endangle, action=action)
"""
pie(radius, startangle, endangle;
action=:none)
Make a pie shape centered at the origin, and add it to the current path.
"""
pie(radius::Real, startangle::Real, endangle::Real; action=:none) =
pie(O, radius, startangle, endangle, action=action)
pie(radius::Real, startangle::Real, endangle::Real, action::Symbol) =
pie(O, radius, startangle, endangle, action=action)
"""
curve(x1, y1, x2, y2, x3, y3)
curve(p1, p2, p3)
Add a Bézier curve to the current path..
The spline starts at the current position, finishing at `x3/y3` (`p3`),
following two control points `x1/y1` (`p1`) and `x2/y2` (`p2`).
"""
curve(x1, y1, x2, y2, x3, y3) = Cairo.curve_to(get_current_cr(), x1, y1, x2, y2, x3, y3)
curve(pt1, pt2, pt3) = curve(pt1.x, pt1.y, pt2.x, pt2.y, pt3.x, pt3.y)
"""
circlepath(center::Point, radius;
action=:none,
reversepath=false,
kappa = 0.5522847498307936)
circlepath(center::Point, radius, action;
reversepath=false,
kappa = 0.5522847498307936)
Make a circle using Bézier curves, and add it to the current
path.
One benefit of using this rather than `circle()` is
that you can use the `reversepath` option to draw the circle
clockwise rather than `circle`'s counterclockwise.
The magic value, `kappa`, is `4.0 * (sqrt(2.0) - 1.0) / 3.0`.
Return two points, the corners of a bounding box.
"""
function circlepath(center::Point, radius;
action=:none,
reversepath=false,
kappa = 0.5522847498307936)
function northtoeast(center::Point, radius, kappa)
curve(center.x + (radius * kappa), center.y + radius, center.x + radius,
center.y + (radius * kappa), center.x + radius, center.y )
end
function easttosouth(center::Point, radius, kappa)
curve(center.x + radius, center.y - (radius * kappa), center.x + (radius * kappa),
center.y - radius, center.x, center.y - radius)
end
function southtowest(center::Point, radius, kappa)
curve(center.x - (radius * kappa), center.y - radius, center.x - radius,
center.y - (radius * kappa), center.x - radius, center.y)
end
function westtonorth(center::Point, radius, kappa)
curve(center.x - radius, center.y + (radius * kappa), center.x - (radius * kappa),
center.y + radius, center.x, center.y + radius)
end
function northtowest(center::Point, radius, kappa)
curve(center.x - (radius * kappa), center.y + radius, center.x - radius,
center.y + (radius * kappa), center.x - radius, center.y )
end
function westtosouth(center::Point, radius, kappa)
curve(center.x - radius, center.y - (radius * kappa), center.x - (radius * kappa),
center.y - radius, center.x, center.y - radius )
end
function southtoeast(center::Point, radius, kappa)
curve(center.x + (radius * kappa), center.y - radius, center.x + radius,
center.y - (radius * kappa), center.x + radius, center.y)
end
function easttonorth(center::Point, radius, kappa)
curve(center.x + radius, center.y + (radius * kappa), center.x + (radius * kappa),
center.y + radius, center.x, center.y + radius )
end
move(center.x, center.y + radius)
if !reversepath
northtoeast(center, radius, kappa)
easttosouth(center, radius, kappa)
southtowest(center, radius, kappa)
westtonorth(center, radius, kappa)
else
northtowest(center, radius, kappa)
westtosouth(center, radius, kappa)
southtoeast(center, radius, kappa)
easttonorth(center, radius, kappa)
end
do_action(action)
return (center - (radius, radius), center + (radius, radius))
end
circlepath(center::Point, radius, action::Symbol; reversepath=false, kappa = 0.5522847498307936) = circlepath(center, radius;
action=action,
reversepath=reversepath,
kappa = kappa)
"""
ellipse(focus1::Point, focus2::Point, k;
action=:none,
stepvalue=pi/100,
vertices=false,
reversepath=false)
Build a polygon approximation to an ellipse, given two
points and a distance, `k`, which is the sum of the
distances to the focii of any points on the ellipse (or the
shortest length of string required to go from one focus to
the perimeter and on to the other focus), and add it to the
current path.
"""
function ellipse(focus1::Point, focus2::Point, k;
action=:none,
stepvalue=pi/100,
vertices=false,
reversepath=false)
a = k/2 # a = ellipse's major axis, the widest part
cpoint = midpoint(focus1, focus2)
dc = distance(focus1, cpoint)
b = sqrt(abs(a^2 - dc^2)) # minor axis, hopefuly not 0
phi = slope(focus1, focus2) # angle between the major axis and the x-axis
points = Point[]
drawing = false
for t in 0:stepvalue:2pi
xt = cpoint.x + a * cos(t) * cos(phi) - b * sin(t) * sin(phi)
yt = cpoint.y + a * cos(t) * sin(phi) + b * sin(t) * cos(phi)
push!(points, Point(xt, yt))
end
vertices ? points : poly(points, action, close=true, reversepath=reversepath)
end
ellipse(f1::Point, f2::Point, k, action::Symbol;
stepvalue=pi/100,
vertices=false,
reversepath=false) =
ellipse(f1, f2, k,
action=action,
stepvalue=stepvalue,
vertices=vertices,
reversepath=reversepath)
"""
ellipse(focus1::Point, focus2::Point, pt::Point;
action=:none,
stepvalue=pi/100,
vertices=false,
reversepath=false)
Build a polygon approximation to an ellipse, given two points and a point somewhere on the
ellipse.
"""
function ellipse(focus1::Point, focus2::Point, pt::Point;
action=:none,
stepvalue=pi/100,
vertices=false,
reversepath=false)
k = distance(focus1, pt) + distance(focus2, pt)
ellipse(focus1, focus2, k, action=action, stepvalue=stepvalue,
vertices=vertices, reversepath=reversepath)
end
ellipse(focus1::Point, focus2::Point, pt::Point, action::Symbol;
stepvalue=pi/100,
vertices=false,
reversepath=false) = ellipse(focus1, focus2, pt;
action=action,
stepvalue=stepvalue,
vertices=vertices,
reversepath=reversepath)
"""
hypotrochoid(R, r, d;
action=:none,
stepby=0.01,
period=0.0,
vertices=false)
hypotrochoid(R, r, d, action;
stepby=0.01,
period=0.0,
vertices=false)
Make a hypotrochoid with short line segments, and add it to
the current path. (Like a Spirograph.) The curve is traced
by a point attached to a circle of radius `r` rolling around
the inside of a fixed circle of radius `R`, where the point
is a distance `d` from the center of the interior circle.
Things get interesting if you supply non-integral values.
Special cases include the hypocycloid, if `d` = `r`, and an
ellipse, if `R` = `2r`.
`stepby`, the angular step value, controls the amount of
detail, ie the smoothness of the polygon,
If `period` is not supplied, or 0, the lowest period is
calculated for you.
The function can return a polygon (a list of points), or
draw the points directly using the supplied `action`. If the
points are drawn, the function returns a tuple showing how
many points were drawn and what the period was (as a
multiple of `pi`).
"""
function hypotrochoid(R, r, d;
action=:none,
close = true,
stepby = 0.01,
period = 0.0,
vertices = false)
function nextposition(t)
x = (R - r) * cos(t) + (d * cos(((R - r)/r) * t))
y = (R - r) * sin(t) - (d * sin(((R - r)/r) * t))
return Point(x, y)
end
# try to calculate the period exactly
if isapprox(period, 0)
period = 2pi * (r/gcd(convert(Int, floor(R)), convert(Int, floor(r))))
end
counter=1
points=Point[]
for t = 0:stepby:period
push!(points, nextposition(t))
end
# don't repeat end point if it's more or less the same as the start point
isapprox(points[1], points[end]) && pop!(points)
vertices ? points : (poly(points, action, close=close); (length(points), period/pi))
end
hypotrochoid(R, r, d, action::Symbol;
close = true,
stepby = 0.01,
period = 0.0,
vertices = false) = hypotrochoid(R, r, d;
action= action,
close = close,
stepby = stepby,
period = period,
vertices = vertices)
"""
epitrochoid(R, r, d;
action=:none,
stepby=0.01,
period=0,
vertices=false)
epitrochoid(R, r, d, action;
stepby=0.01,
period=0,
vertices=false)
Make a epitrochoid with short line segments, and add it to
the current path. (Like a Spirograph.) The curve is traced
by a point attached to a circle of radius `r` rolling around
the outside of a fixed circle of radius `R`, where the point
is a distance `d` from the center of the circle. Things get
interesting if you supply non-integral values.
`stepby`, the angular step value, controls the amount of
detail, ie the smoothness of the polygon.
If `period` is not supplied, or 0, the lowest period is
calculated for you.
The function can return a polygon (a list of points), or
draw the points directly using the supplied `action`. If the
points are drawn, the function returns a tuple showing how
many points were drawn and what the period was (as a
multiple of `pi`).
"""
function epitrochoid(R, r, d;
action = :none,
close = true,
stepby = 0.01,
period = 0,
vertices = false)
function nextposition(t)
x = (R + r) * cos(t) - (d * cos(((R - r)/r) * t))
y = (R + r) * sin(t) - (d * sin(((R - r)/r) * t))
return Point(x, y)
end
# try to calculate the period exactly
if isapprox(period, 0)
period = 2pi * (r/gcd(convert(Int, floor(R)), convert(Int, floor(r))))
end
counter=1
points=Point[]
for t = 0:stepby:period
push!(points, nextposition(t))
end
# don't repeat end point if it's more or less the same as the start point
isapprox(points[1], points[end]) && pop!(points)
vertices ? points : (poly(points, action, close=close); (length(points), period/pi))
end
epitrochoid(R, r, d, action::Symbol;
close = true,
stepby = 0.01,
period = 0.0,
vertices = false) = epitrochoid(R, r, d;
action= action,
close = close,
stepby = stepby,
period = period,
vertices = vertices)
"""
spiral(a, b;
action = :none,
stepby = 0.01,
period = 4pi,
vertices = false,
log =false)
spiral(a, b, action;
stepby = 0.01,
period = 4pi,
vertices = false,
log =false)
Make a spiral, and add it to the current path. The two
primary parameters `a` and `b` determine the start radius,
and the tightness.
For linear spirals (`log=false`), `b` values are:
lituus: -2
hyperbolic spiral: -1
Archimedes' spiral: 1
Fermat's spiral: 2
For logarithmic spirals (`log=true`):
golden spiral: b = ln(phi)/ (pi/2) (about 0.30)
Values of `b` around 0.1 produce tighter, staircase-like spirals.
"""
function spiral(a, b;
action=:none,
stepby = 0.01,
period = 4pi,
vertices = false,
log=false)
function nextpositionlog(t)
ebt = exp(t * b)
if abs(ebt) < 10^8 # arbitrary cutoff to avoid NaNs and Infs
x = a * ebt * cos(t)
y = a * ebt * sin(t)
return Point(x, y)
else
return nothing
end
end
function nextpositionlin(t)
tpn = t ^ (1/b)
if tpn < 10^8
x = a * tpn * cos(t)
y = a * tpn * sin(t)
return Point(x, y)
else
return nothing
end
end
log ? nextpos=nextpositionlog : nextpos = nextpositionlin
points = Point[]
for t = 0:stepby:period
pt = nextpos(t)
if isa(nextpos(t), Point)
push!(points, pt)
end
end
if vertices == false
poly(points, action) # no close by default :)
end
return points
end
spiral(a, b, action::Symbol;
stepby = 0.01,
period = 4pi,
vertices = false,
log=false) = spiral(a, b;
action=action,
stepby = stepby,
period = period,
vertices = vertices,
log=log)
"""
intersection2circles(pt1, r1, pt2, r2)
Find the area of intersection between two circles, the first centered at `pt1` with radius
`r1`, the second centered at `pt2` with radius `r2`.
If one circle is entirely within another, that circle's area is returned.
"""
function intersection2circles(pt1, rad1, pt2, rad2)
# via casey and jùlio on slack
# squared radii
rr1, rr2 = rad1 * rad1, rad2 * rad2
d = distance(pt1, pt2)
# trivial cases
d ≥ rad1 + rad2 && return 0.0
d ≤ abs(rad2 - rad1) && return π * min(rr1, rr2)
# First center point to the middle line
a_distancecenterfirst = (rr1 - rr2 + (d^2)) / (2d)
# Second centre point to the middle line
b_distancecentersecond = d - a_distancecenterfirst
# Half of the middle line
h_height = sqrt(rr1 - a_distancecenterfirst^2)
# central angle for the first circle
alpha = mod2pi(atan(h_height, a_distancecenterfirst) * 2.0 + 2π)
# Central angle for the second circle
beta = mod2pi(atan(h_height, b_distancecentersecond) * 2.0 + 2π)
# Area of the first circular segment
A1 = rr1 / 2.0 * (alpha - sin(alpha))
# Area of the second circular segment
A2 = rr2 / 2.0 * (beta - sin(beta))
return A1 + A2
end
"""
intersectioncirclecircle(cp1, r1, cp2, r2)
Find the two points where two circles intersect, if they do. The first circle is centered
at `cp1` with radius `r1`, and the second is centered at `cp1` with radius `r1`.
Returns
(flag, ip1, ip2)
where `flag` is a Boolean `true` if the circles intersect at the points `ip1` and `ip2`. If
the circles don't intersect at all, or one is completely inside the other, `flag` is `false`
and the points are both Point(0, 0).
Use `intersection2circles()` to find the area of two overlapping circles.
In the pure world of maths, it must be possible that two circles 'kissing' only have a
single intersection point. At present, this unromantic function reports that two kissing
circles have no intersection points.
"""
function intersectioncirclecircle(cp1, r1, cp2, r2)
r1squared = r1^2
r2squared = r2^2
d = distance(cp1, cp2)
if d > (r2 + r1) # circles do not overlap
return (false, O, O)
elseif (d <= abs(r1 - r2)) && (r1 >= r2)
# second circle is completely inside first circle
return (false, O, O)
elseif (d <= abs(r1 - r2)) && (r1 < r2)
# first circle is completely inside second circle
return (false, O, O)
end
a = (r1squared - r2squared + d^2) / 2d
h = sqrt(r1squared - a^2)
p0 = cp1 + a * (cp2 - cp1)/d
p3 = Point(
p0.x + h * (cp2.y - cp1.y )/d,
p0.y - h * (cp2.x - cp1.x )/d)
p4 = Point(
p0.x - h * (cp2.y - cp1.y )/d,
p0.y + h * (cp2.x - cp1.x )/d)
return (true, p3, p4)
end
"""
circlepointtangent(through::Point, radius, targetcenter::Point, targetradius)
Find the centers of up to two circles of radius `radius` that pass through point
`through` and are tangential to a circle that has radius `targetradius` and
center `targetcenter`.
This function returns a tuple:
* (0, O, O) - no circles exist
* (1, pt1, O) - 1 circle exists, centered at pt1
* (2, pt1, pt2) - 2 circles exist, with centers at pt1 and pt2
(The O are just dummy points so that three values are always returned.)
"""
function circlepointtangent(through::Point, radius, targetcenter::Point, targetradius)
distx = targetcenter.x - through.x
disty = targetcenter.y - through.y
dsq = distance(through, targetcenter)^2
if isless(dsq, 10e-6) # coincident
return (0, O, O)
else
sqinv=0.5/dsq
s = dsq - ((2radius + targetradius) * targetradius)
root = 4(radius^2) * dsq - s^2
s *= sqinv
if isless(dsq, 0.0) # no center possible
return (0, O, O)
else
if isless(root, 10e-6) # only one circle possible
x = through.x + distx * s
y = through.y + disty * s
if isless(abs(distance(through, Point(x, y)) - radius), 10e-6)
return (1, Point(x, y), O)
else
return (0, O, O)
end
else # two circles are possible
root = sqrt(root) * sqinv
xconst = through.x + distx * s
yconst = through.y + disty * s
xvar = disty * root
yvar = distx * root
return (2, Point(xconst - xvar, yconst + yvar), Point(xconst + xvar, yconst - yvar))
end
end
end
end
"""
circletangent2circles(radius, circle1center::Point, circle1radius, circle2center::Point, circle2radius)
Find the centers of up to two circles of radius `radius` that are tangent to the