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User:Sam Derbyshire/MuPAD

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Back to my user page, my talk page.

MuPad code

[edit]

Here is some code I used to generate some of my diagrams.

Be careful though, the code isn't exactly of high quality, it may not work as you'd expect it to.

General 2D function

[edit]
Airy Ai and Bi functions.
f := plot::Function2d(airyAi(x), x=-6..4, LineWidth=0.8, LineColor=RGB::Red):
g := plot::Function2d(airyBi(x), x=-6..4, LineWidth=0.8, LineColor=RGB::CornflowerBlue):
plot(f,g,
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 4, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-6..4, -2..8],
Height=100, Width=100, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

General 3D function

[edit]
A 3D plot with contour lines.
f := plot::Function3d(8*sin(x-cos(y))+(x^2+x*y),
                     Mesh=[30,30], Submesh=[1,1]):
plot(f,plot::Transform3d([0, 0, 0], [1, 0, 0, 0, 1, 0, 0, 0, 1], 
             plot::modify(f, ZContours = [Automatic,14],
                             LineWidth = 0.3,
                             LineColorType = Dichromatic,
                             LineColor = RGB::White.[0.2],
                             LineColor2 = RGB::White.[0.2],
                             XLinesVisible = FALSE,
                             YLinesVisible = FALSE,
                             Filled = FALSE)),
       plot::Transform3d([0, 0, -15], [1, 0, 0, 0, 1, 0, 0, 0, 0], 
             plot::modify(f, ZContours = [Automatic,14],
                             LineWidth = 0.5,
                             LineColorType = Dichromatic,
                             LineColor = RGB::Red.[0.9],
                             LineColor2 = RGB::CadmiumYellow.[0.9],
                             XLinesVisible = FALSE,
                             YLinesVisible = FALSE,
                             Filled = FALSE)),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, FillColor=RGB::Red, FillColor2=RGB::CadmiumYellow,
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
Height=100, Width=100,
ViewingBox=[-5..5,-5..5,-15..50],
//Scaling = Constrained,
AxesTitles = ["x", "y", "z"])

[[Category:]]

==Complex conformal plots==

A conformal plot with coloured grids.
m := 21:
M := 2:
f := z -> 1/z:
hsv1 := z -> RGB::fromHSV([(arg(z))*180/PI, 0.5+abs(z)/(2*M*sqrt(2)), 0.25 + abs(z)/(4/3*M*sqrt(2)), 0.6]):
hsv2 := z -> RGB::fromHSV([(arg(z))*180/PI, 0.5+abs(z)/(2*M*sqrt(2)), 0.25 + abs(z)/(4/3*M*sqrt(2)), 1]):
p := arctan(Im(z)/Re(z)):
q := abs(z)/(M*sqrt(2)):
plot(plot::Conformal(z, z=-M*(1+I)..M*(1+I), Mesh=[m,m],LineColorType=Functional, 
LineColorFunction=(hsv1)),
plot::Conformal(f(z), z=-M*(1+I)..M*(1+I), Mesh=[m,m], LineColorType=Functional, AdaptiveMesh=2, Submesh=[3,3],
LineColorFunction=(hsv2)), Axes=Boxed, GridVisible = TRUE, SubgridVisible = TRUE,GridLineWidth = 0.1, SubgridLineWidth = 0.1,
 GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.15], SubgridLineColor = RGB::SlateGreyDark.[0.1], ViewingBox = [-5..5, -5..5])

Contour plots

[edit]
A contour plot (it is in fact a 3D plot viewed from above).
f := sin(PI*x)*sin(PI*y):

  conts := 21:
  xlimit := 1:
  ylimit := 1:
  submeshlevel := 10:

a:=1:
zmin := -1:
zmax := 1:
range:=zmax-zmin:
plotmin := -1:
plotmax := 1:
projectionlevel:=plotmin:
colour := x -> RGB::fromHSV([3*180/PI*x,1,1]):
colfunc := (x,y,z) -> colour(floor((conts-1)/2*z)/((conts-1)/2)):

funcplot := plot::Function3d(f(x,y),
                             x = -xlimit..xlimit,
                             y = -ylimit..ylimit,
                             Mesh = [21, 21],
                             Submesh = [submeshlevel,submeshlevel],
                             LineColor = RGB::Black.[0.4],
                             LineWidth = 0.15,
                             FillColorFunction = colfunc,
                             AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
                             ViewingBoxZRange = -1..1
                            ):

contours := plot::modify(funcplot,
                         ZContours = [Automatic, conts],
                         LineWidth = 0.4,
                         LineColor = RGB::Gray20.[0.9],
                         XLinesVisible = FALSE,
                         YLinesVisible = FALSE,
                         Filled = FALSE
                        ):
                                                     
                             
ploteverything := plot::Canvas(funcplot, contours,
                               Width = 200, 
                               Height = 200,
                               TicksLabelFont=["Cambria",10,RGB::SlateGreyDark],
                               plot::AmbientLight(1), Lighting=None, OrthogonalProjection = TRUE, plot::Camera([0, 0, 4*plotmax], [0, 0, (plotmin+plotmax)/2],PI/6)
                              ): 
                       
plot(ploteverything);

Thanks to Inductiveload for helping out here.

Riemann Sphere

[edit]

Simple projection of a curve (and coloured grid, cartesian/polar) back onto the Riemann sphere, can be done other way round.

Nephroid and coloured cartesian grid projected onto the sphere.
r := 1:
V := 5:
A := (X,Y) -> [X/(1+0.25*X^2+0.25*Y^2),Y/(1+0.25*X^2+0.25*Y^2),(-1+0.25*X^2+0.25*Y^2)/(1+0.25*X^2+0.25*Y^2)+1]:
Am := (x,y,z) -> [2*x/(2-z), 2*y/(2-z)]:
hsv1 := (t,x,y,z) -> RGB::fromHSV([(arg(x+I*y))*180/PI, 0.5+abs(x+I*y)/(2*V*sqrt(2)), 0.25 + abs(x+I*y)/(4/3*V*sqrt(2)), 0.9]):
hsv2 := (t,x,y,z) -> RGB::fromHSV([(arg(Am(x,y,z)[1]+I*Am(x,y,z)[2]))*180/PI,
     0.5+abs(Am(x,y,z)[1]+I*Am(x,y,z)[2])/(2*V*sqrt(2)), 0.25 + abs(Am(x,y,z)[1]+I*Am(x,y,z)[2])/(4/3*V*sqrt(2)), 0.9]):
fgx := (i,j) -> (i,t,0)://fgx := (i,j) -> (i*cos(t),i*sin(t),0)://
fgy := (i,j) -> (t,j,0)://fgy := (i,j) -> (cos(j*2*PI/V)*t,sin(j*2*PI/V)*t,0)://
P := plot::Surface([u, v,0], u = -V .. V, v = -V .. V, 
UMesh=2*V+1, VMesh=2*V+1, Submesh=[1,1], FillColor=[1,1,1,1], FillColor2=[1,1,1,1], ULinesVisible=FALSE, VLinesVisible=FALSE):
S := plot::Surface([cos(u)*sin(v), sin(u)*sin(v),cos(v)+1], u = 0 .. 2*PI, v = 0 .. PI,
     Submesh=[1,1], FillColor=[1,1,1,1], FillColor2=[1,1,1,1], ULinesVisible=FALSE, VLinesVisible=FALSE):
cx := t -> 1*(3*cos(t) - cos(3*t)):
cy := t -> 1*(3*sin(t) - sin(3*t)):
C := plot::Curve3d([cx(t),cy(t),0],t=0..2*PI, LineColor=RGB::Black.[0.9],LineWidth=0.8, Mesh=200):
D := plot::Curve3d(A(cx(t),cy(t)),t=0..2*PI,LineColor=RGB::Black.[0.9],LineWidth=0.7, Mesh=100):
Gx := i -> plot::Curve3d([fgx(i,j)],t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv1)):
Gy := j -> plot::Curve3d([fgy(i,j)],t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv1)):
Gsx := i -> plot::Curve3d(A(fgx(i,j)),t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv2)):
Gsy := j -> plot::Curve3d(A(fgy(i,j)),t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv2)):
Grid := plot::Group3d(Gx(k-V-1)$ k = 1..2*V+1 ,Gy(k-V-1) $ k = 1..2*V+1):
SGrid := plot::Group3d(Gsx(k-V-1)$ k = 1..2*V+1 ,Gsy(k-V-1) $ k = 1..2*V+1):
plot(P,S,C,D, Grid, SGrid,
     ViewingBox=[-V..V,-V..V,-3..3], Height = 250, Width=250, Scaling=Constrained, AxesVisible=FALSE)

Projection of the grid and the curve onto the Riemann Sphere, Euclidean motion of it ("rotation" matrix + translation) and projection back down.

Nephroid and coloured cartesian grid projected onto the sphere, which is then moved (translated and rotated in this case), and then the curves are projected back onto the plane.
K := [-2,0,3]:
R := [K[1],K[2],K[3]+1]:
tra := [K[1],K[2],K[3]]:
V := 6:
T :=  matrix([[0.707,0,0.707],[0,1,0],[-0.707,0,0.707]]):
p1 := plot::Point3d(K, PointSize = 2, PointColor=RGB::Black):
p2 := plot::Point3d(R, PointSize = 2, PointColor=RGB::Black):
Ab := (X,Y) -> [2*X/(1+X^2+Y^2),2*Y/(1+X^2+Y^2),(-1+X^2+Y^2)/(1+X^2+Y^2)]:
Af := (x,y,z) -> [x/(1.01-z), y/(1.01-z)]:
Ab1 := (X,Y) -> Ab(X,Y):
Af1 := (x,y,z) -> Af(x,y,z):
Ab2 := (X,Y) -> zip(Ab((X-K[1])/R[3],(Y-K[2])/R[3]),[K[1],K[2],K[3]], _plus):
Af2 := (x,y,z) -> zip(Af(R[3]*(x-K[1]),R[3]*(y-K[2]),z-K[3]),[K[1],K[2]], _plus):
hsv := (t,x,y,z) -> RGB::fromHSV([(arg(x+I*y))*180/PI, 0.5+abs(x+I*y)/(2*V*sqrt(2)), 0.25 + abs(x+I*y)/(4/3*V*sqrt(2)), 0.3]):
hsv1 := (t,x,y,z) -> RGB::fromHSV([(arg(x+I*y))*180/PI, 0.5+abs(x+I*y)/(2*V*sqrt(2)), 0.25 + abs(x+I*y)/(4/3*V*sqrt(2)), 0.9]):
hsv2 := (t,x,y,z) -> hsv1(t,Af1(x,y,z)[1],Af1(x,y,z)[2],z):
hsv3 := (t,x,y,z) -> hsv2(t,(T^(-1)*(matrix([Ab2(x,y)[1],Ab2(x,y)[2],Ab2(x,y)[3]])-matrix(tra)))[1],
     (T^(-1)*(matrix([Ab2(x,y)[1],Ab2(x,y)[2],Ab2(x,y)[3]])-matrix(tra)))[2],
     (T^(-1)*(matrix([Ab2(x,y)[1],Ab2(x,y)[2],Ab2(x,y)[3]])-matrix(tra)))[3]):
fgx := (i,j) -> (i*cos(t),i*sin(t),0)://fgx := (i,j) -> (i,t,0)://
fgy := (i,j) -> (cos(j*2*PI/V)*t,sin(j*2*PI/V)*t,0)://fgy := (i,j) -> (t,j,0)://
P := plot::Surface([u, v,0], u = -V .. V, v = -V .. V, 
UMesh=2*V+1, VMesh=2*V+1, Submesh=[1,1], FillColor=[1,1,1,1], FillColor2=[1,1,1,1], ULinesVisible=FALSE, VLinesVisible=FALSE):
S1 := plot::Surface([cos(u)*sin(v), sin(u)*sin(v),cos(v)], u = 0 .. 2*PI, v = 0 .. PI,
     Submesh=[1,1], FillColor=[1,1,1,1], FillColor2=[1,1,1,1], ULinesVisible=FALSE, VLinesVisible=FALSE):
S2 := plot::Surface([cos(u)*sin(v)+K[1], sin(u)*sin(v)+K[2],cos(v)+K[3]], u = 0 .. 2*PI, v = 0 .. PI,
     Submesh=[1,1], FillColor=[1,1,1,1], FillColor2=[1,1,1,1], ULinesVisible=FALSE, VLinesVisible=FALSE):
cx := t -> 2.25*cos(t)-0.7*cos(13*t):
cy := t -> 2.25*sin(t)-0.7*sin(13*t):
C1 := plot::Curve3d([cx(t),cy(t),0],t=0..2*PI, LineColor=RGB::Black.[0.3],LineWidth=0.8, Mesh=400):
CS := plot::Curve3d(Ab1(cx(t),cy(t)),t=0..2*PI,LineColor=RGB::Black.[0.9],LineWidth=0.7, Mesh=500):
CS2 := plot::Transform3d(tra,T,CS):
C2 := plot::Curve3d(Af2((T*matrix([Ab1(cx(t),cy(t))[1],Ab1(cx(t),cy(t))[2],Ab1(cx(t),cy(t))[3]])+matrix(tra))[1],
     (T*matrix([Ab1(cx(t),cy(t))[1],Ab1(cx(t),cy(t))[2],Ab1(cx(t),cy(t))[3]])+matrix(tra))[2],
     (T*matrix([Ab1(cx(t),cy(t))[1],Ab1(cx(t),cy(t))[2],Ab1(cx(t),cy(t))[3]])+matrix(tra))[3]).[0],
     t=0..2*PI, LineColor=RGB::Black.[0.9],LineWidth=0.8, Mesh=600):
Gx := i -> plot::Curve3d([fgx(i,j)],t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv)):
Gy := j -> plot::Curve3d([fgy(i,j)],t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv)):
Gsx := i -> plot::Curve3d(Ab1(fgx(i,j)),t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv2)):
Gsy := j -> plot::Curve3d(Ab1(fgy(i,j)),t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv2)):
G2x := i -> plot::Curve3d(Af2((T*matrix(Ab1(fgx(i,j)))+matrix(tra))[1],(T*matrix(Ab1(fgx(i,j)))+matrix(tra))[2],(T*matrix(Ab1(fgx(i,j)))+matrix(tra))[3]).[0],
     t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv3)):
G2y := j -> plot::Curve3d(Af2((T*matrix(Ab1(fgy(i,j)))+matrix(tra))[1],(T*matrix(Ab1(fgy(i,j)))+matrix(tra))[2],(T*matrix(Ab1(fgy(i,j)))+matrix(tra))[3]).[0],
     t=-V..V,LineWidth=0.4, LineColor=RGB::Black.[0.9], LineColor=RGB::Black.[0.9], LineColorType=Functional, LineColorFunction=(hsv3)):
Grid := plot::Group3d(Gx(k-V-1)$ k = 1..2*V+1 ,Gy(k-V-1) $ k = 1..2*V+1):
SGrid := plot::Group3d(Gsx(k-V-1)$ k = 1..2*V+1 ,Gsy(k-V-1) $ k = 1..2*V+1):
SGrid2 := plot::Group3d(plot::Transform3d(tra,T,Gsx(k-V-1))$ k = 1..2*V+1 ,plot::Transform3d(tra,T,Gsy(k-V-1)) $ k = 1..2*V+1):
Grid2 := plot::Group3d(G2x(k-V-1)$ k = 1..2*V+1 ,G2y(k-V-1) $ k = 1..2*V+1):
plot(P,S2,C1,CS2,C2, Grid, Grid2, SGrid2, p1,p2, S1,SGrid,CS,
     ViewingBox=[-V-3..V+3,-V-3..V+3,-4..4], Height = 250, Width=250, Scaling=Constrained, AxesVisible=FALSE)

Epitrochoids and Hypotrochoids

[edit]
Here an hypotrochoid.
r := 2.1:
R := 3:
s := -1: // 1 for epi, -1 for hypo
displacement := 0:
angle := 0:
period := 14*PI:
range := R + 2*r*heaviside(s)+displacement*heaviside(displacement):
centerx := a -> (R+s*r)*cos(+a):
centery := a -> (R+s*r)*sin(+a):
center2x := a -> cos(angle)*centerx(a) - sin(angle)*centery(a):
center2y := a -> sin(angle)*centerx(a) + cos(angle)*centery(a):
pointx := t -> center2x(t)+(r+displacement)*cos(s*(R/r+s*1)*t):
pointy := t -> center2y(t)+(r+displacement)*sin(s*(R/r+s*1)*t):
point2x := t -> cos(-angle)*pointx(t) - sin(-angle)*pointy(t):
point2y := t -> sin(-angle)*pointx(t) + cos(-angle)*pointy(t):

plot(
plot::Circle2d(R, LineColor=RGB::Blue, LineWidth=0.6),
plot::Circle2d(r, [centerx(a),centery(a)], a=0..period, LineColor=RGB::Black, LineWidth=0.6),
plot::Curve2d([point2x(t),point2y(t)], t=0..a, a=0..period, LineWidth=0.6, LineColor=RGB::Red, Mesh=500),
plot::Line2d([centerx(a),centery(a)],[point2x(a),point2y(a)], a=0..period, LineColor=RGB::Black),
plot::Point2d([centerx(a),centery(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
plot::Point2d([point2x(a),point2y(a)], a=0..period, PointSize=2, PointColor=RGB::Red),

GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=100, TimeEnd=5,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Pedal curve

[edit]
Pedal curve of an ellipse.
fx := t -> 2*cos(t):
fy := t -> sin(t):
f := t -> [fx(t),fy(t)]:
range :=2.5:
px := 0:
py := 0:
h := 0.1:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fx(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
ix := a ->((slope(a)*(-fx(a))+fy(a))-(-(1/slope(a))*(-px)+py))/(-slope(a)-1/slope(a)):
iy := a -> slope(a)*(ix(a)-fx(a))+fy(a):
period := 2*PI:
plot(
plot::Curve2d(f(t),t=0..2*PI, LineColor=RGB::Black, LineWidth=0.6, Mesh=200),
plot::Function2d(slope(a)*(x-fx(a))+fy(a),x=-range..range, a=0..period, LineColor=RGB::Blue, LineWidth=0.5),
plot::Function2d(-(1/slope(a))*(x-px)+py,x=-range..range, a=0..period, LineColor=RGB::SapGreen, LineWidth=0.5),
plot::Curve2d([ix(t),iy(t)],t=0..period, LineWidth=0.6, LineColorFunction = ((u, x, y, a) -> [1, 0, 0,1]), Mesh=400, Submesh=3, AdaptiveMesh=2),
plot::Point2d([px,py], PointSize=2, PointColor=RGB::Black),
plot::Point2d([fx(a),fy(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
plot::Point2d([ix(a),iy(a)], a=0..period, PointSize=2, PointColor=RGB::Red),
plot::Polygon2d([[ix(a)+h*cos(angle(a)), iy(a)+h*cos(angle(a))*slope(a)],
                 [ix(a)+h*cos(angle(a))+h*sin(angle(a)), iy(a)+h*cos(angle(a))*slope(a)-h*sin(angle(a))/slope(a)],
                 [ix(a)+h*sin(angle(a)), iy(a)-h*sin(angle(a))/slope(a)]], 
                 a=0..period, LineColor=RGB::Black, LineWidth=0.5),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=400, TimeEnd=10,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Orthotomic curve

[edit]
Orthotomic curve of an ellipse.
fx := t -> 2*cos(t):
fy := t -> sin(t):
f := t -> [fx(t),fy(t)]:
range :=4.5:
px := 0:
py := 1:
h := 0.1:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fx(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
ix := a ->((slope(a)*(-fx(a))+fy(a))-(-(1/slope(a))*(-px)+py))/(-slope(a)-1/slope(a)):
iy := a -> slope(a)*(ix(a)-fx(a))+fy(a):
period := 2*PI:
plot(
plot::Curve2d(f(t),t=0..2*PI, LineColor=RGB::Black, LineWidth=0.6, Mesh=200),
plot::Function2d(slope(a)*(x-fx(a))+fy(a),x=-range..range, a=0..period, LineColor=RGB::Blue, LineWidth=0.5),
plot::Function2d(-(1/slope(a))*(x-px)+py,x=-range..range, a=0..period, LineColor=RGB::SapGreen, LineWidth=0.5),
plot::Curve2d([px+2*(ix(t)-px),py+2*(iy(t)-py)],t=0..period, LineWidth=0.6, LineColorFunction = ((u, x, y, a) -> [1, 0, 0,1]), Mesh=400, Submesh=3, AdaptiveMesh=2),
plot::Point2d([px,py], PointSize=2, PointColor=RGB::Black),
plot::Point2d([fx(a),fy(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
plot::Point2d([ix(a),iy(a)], a=0..period, PointSize=2, PointColor=RGB::Blue),
plot::Point2d([px+2*(ix(a)-px),py+2*(iy(a)-py)], a=0..period, PointSize=2, PointColor=RGB::Red),
plot::Polygon2d([[ix(a)+h*cos(angle(a)), iy(a)+h*cos(angle(a))*slope(a)],
                 [ix(a)+h*cos(angle(a))+h*sin(angle(a)), iy(a)+h*cos(angle(a))*slope(a)-h*sin(angle(a))/slope(a)],
                 [ix(a)+h*sin(angle(a)), iy(a)-h*sin(angle(a))/slope(a)]], 
                 a=0..period, LineColor=RGB::Black, LineWidth=0.5),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=400, TimeEnd=10,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Contrapedal curve

[edit]
Contrapedal curve of an ellipse.
fx := t -> 2*cos(t):
fy := t -> sin(t):
f := t -> [fx(t),fy(t)]:
range :=2.5:
px := 0:
py := 0:
h := 0.1:
slope := a -> piecewise([diff(fy(a),a)=0,-10^10],[diff(fy(a),a) <> 0,-diff(fx(a),a)/(diff(fy(a),a))]):
angle := a -> arctan(slope(a)):
ix := a ->((slope(a)*(-fx(a))+fy(a))-(-(1/slope(a))*(-px)+py))/(-slope(a)-1/slope(a)):
iy := a -> slope(a)*(ix(a)-fx(a))+fy(a):
period := 2*PI:
plot(
plot::Curve2d(f(t),t=0..2*PI, LineColor=RGB::Black, LineWidth=0.6, Mesh=200),
plot::Function2d(slope(a)*(x-fx(a))+fy(a),x=-range..range, a=0..period, LineColor=RGB::Blue, LineWidth=0.5),
plot::Function2d(-(1/slope(a))*(x-px)+py,x=-range..range, a=0..period, LineColor=RGB::SapGreen, LineWidth=0.5),
plot::Curve2d([ix(t),iy(t)],t=0..period, LineWidth=0.6, LineColorFunction = ((u, x, y, a) -> [1, 0, 0,1]), Mesh=400, Submesh=3, AdaptiveMesh=2),
plot::Point2d([fx(a),fy(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
plot::Point2d([px,py], PointSize=2, PointColor=RGB::Black),
plot::Point2d([ix(a),iy(a)], a=0..period, PointSize=2, PointColor=RGB::Red),
plot::Polygon2d([[ix(a)+h*cos(angle(a)), iy(a)+h*cos(angle(a))*slope(a)],
                 [ix(a)+h*cos(angle(a))+h*sin(angle(a)), iy(a)+h*cos(angle(a))*slope(a)-h*sin(angle(a))/slope(a)],
                 [ix(a)+h*sin(angle(a)), iy(a)-h*sin(angle(a))/slope(a)]], 
                 a=0..period, LineColor=RGB::Black, LineWidth=0.5),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=400, TimeEnd=10,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Evolute

[edit]
Evolute of an ellipse.
fx := t -> 2*cos(t):
fy := t -> sin(t):
f := t -> [fx(t),fy(t)]:
range :=5:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fx(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
period := 2*PI:
lines_number := 50:
timend := 5:

  lines := [FAIL $ lines_number+1]:
  for i from 0 to lines_number do
    theta := i*2*PI/(lines_number):
    lines[i+1] := plot::Function2d((-1/slope(t)*(x-fx(t))+fy(t))|(t=theta), 
                   VisibleAfter = timend*i/(lines_number+1),
                   Color = RGB::Blue
                     ):
end_for:

cx :=t -> fx(t)+diff(fy(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fx(t),t,t)*diff(fy(t),t)-diff(fy(t),t,t)*diff(fx(t),t)):
cy :=t -> fy(t)+diff(fx(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fy(t),t,t)*diff(fx(t),t)-diff(fx(t),t,t)*diff(fy(t),t)): 
plot(
plot::Curve2d(f(t),t=0..2*PI, LineColor=RGB::Black, LineWidth=0.6, Mesh=200),
plot::Group2d(op(lines)),
plot::Curve2d([cx(t),cy(t)],t=0..a,a=0..2*PI, LineColor=RGB::Blue, LineWidth=0.6, Mesh=200),
plot::Point2d([cx(a),cy(a)],a=0..2*PI, PointSize=2,Color=RGB::Black),
plot::Point2d([fx(a),fy(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=400, TimeEnd=timend,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

With parallel curves

[edit]
fx := t -> 2*cos(t):
fy := t -> sin(t):
f := t -> [fx(t),fy(t)]:
g := (t,a) -> [fx(t) + a*diff(fy(t),t)/(sqrt((diff(fx(t),t))^2+(diff(fy(t),t)^2))),fy(t) - a*diff(fx(t),t)/(sqrt((diff(fx(t),t))^2+(diff(fy(t),t)^2)))]: 
range :=5:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fx(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
period := 2*PI:
lines_number := 21:
timend := 5:

  lines := [FAIL $ lines_number+1]:
  for i from 0 to lines_number do
    theta := i*2*PI/(lines_number):
    lines[i+1] := plot::Function2d((-1/slope(t)*(x-fx(t))+fy(t))|(t=theta), 
                   VisibleAfter = timend*i/(lines_number+1),
                   Color = RGB::Blue,
                   TimeEnd = 5
                     ):
end_for:

cx :=t -> fx(t)+diff(fy(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fx(t),t,t)*diff(fy(t),t)-diff(fy(t),t,t)*diff(fx(t),t)):
cy :=t -> fy(t)+diff(fx(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fy(t),t,t)*diff(fx(t),t)-diff(fx(t),t,t)*diff(fy(t),t)): 
plot(
plot::Curve2d(f(t),t=0..2*PI, LineColor=RGB::Red, LineWidth=0.6, Mesh=200),
plot::Group2d(op(lines)),
plot::Curve2d(g(t,a),t=0..2*PI,a=0..-5, LineColor=RGB::Black, LineWidth=0.6, Mesh=200, TimeBegin=5, TimeEnd=10, VisibleBeforeBegin = FALSE),
//plot::Curve2d([cx(t),cy(t)],t=0..a,a=0..2*PI, LineColor=RGB::Blue, LineWidth=0.6, Mesh=200),
//plot::Point2d([cx(a),cy(a)],a=0..2*PI, PointSize=2,Color=RGB::Black),
plot::Point2d([fx(a),fy(a)], a=0..period, PointSize=2, PointColor=RGB::Black),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-range..range, -range..range], Frames=400, TimeEnd=timend,
Height=200, Width=200, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Involute

[edit]

Arclength parametrization case

[edit]

Disclaimer : Works only with arclenth parametrization !

Involute of a Catenary
fx := t -> arcsinh(t):
fy := t -> sqrt(t^2+1):
f := t -> [fx(t),fy(t)]:
range :=4:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fy(a),a)=0,10^-10],[diff(fx(a),a) <> 0 and diff(fy(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
period := 2*PI:
lines_number := 50:
timend := 5:
cx := t -> fx(t)-t*diff(fx(t),t):
cy := t -> fy(t)-t*diff(fy(t),t): 
plot(
plot::Curve2d(f(t),t=-6..6, LineColor=RGB::Blue, LineWidth=0.6, Mesh=200),
plot::Point2d([cx(a),cy(a)], a=-5..5, PointSize=2, PointColor=RGB::Black, VisibleFromTo=0..timend),
plot::Line2d([fx(a),fy(a)],[cx(a),cy(a)], a=-5..5, LineWidth=0.4, LineColor=RGB::Black, VisibleFromTo=0..timend),
plot::Curve2d([cx(t),cy(t)], t=-30..30, LineWidth=0.6, LineColor=[1,0.8,0.8], Mesh=200),
plot::Curve2d([cx(t),cy(t)], t=-5..a,a=-5..5, LineWidth=0.6, LineColor=RGB::Red, Mesh=200),
plot::Curve2d([cx(t),cy(t)], t=-30..30, LineWidth=0.6, LineColor=RGB::Red, VisibleAfter=timend, Mesh=200),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-3..3, -1..5], Frames=400, TimeEnd=timend,
Height=160, Width=120, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Variant, all parametrizations

[edit]

To be used against an evolute : first equations are those of the evolute.

Involute of an epitrochoid.
fx := t -> (3+1)*(cos(t)-cos((3+1)*t)/(3+1)):
fy := t -> (3+1)*(sin(t)-sin((3+1)*t)/(3+1)):
f := t -> [fx(t),fy(t)]:
range :=4:
slope := a -> piecewise([diff(fx(a),a)=0,10^10],[diff(fy(a),a)=0,10^-10],[diff(fx(a),a) <> 0 and diff(fy(a),a) <> 0,diff(fy(a),a)/(diff(fx(a),a))]):
angle := a -> arctan(slope(a)):
period := 2*PI:
lines_number := 100:
timend := 10:
cx :=t -> fx(t)+diff(fy(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fx(t),t,t)*diff(fy(t),t)-diff(fy(t),t,t)*diff(fx(t),t)):
cy :=t -> fy(t)+diff(fx(t),t)*((diff(fx(t),t))^2+(diff(fy(t),t))^2)/(diff(fy(t),t,t)*diff(fx(t),t)-diff(fx(t),t,t)*diff(fy(t),t)): 
plot(
plot::Curve2d(f(t),t=-PI..a,a=-PI..PI, LineColor=RGB::Red, LineWidth=0.6, Mesh=200),
//plot::Curve2d([-0.6*fx(t),-0.6*fy(t)],t=-2*PI..2*PI, LineColor=RGB::Purple, LineWidth=0.6, Mesh=200),
plot::Curve2d([cx(t),cy(t)],t=-2*PI..2*PI, LineColor=RGB::Blue, LineWidth=0.6, Mesh=200),
plot::Point2d([fx(a),fy(a)], a=-PI..PI, PointSize=2, PointColor=RGB::Black),
//plot::Point2d([cx(a),cy(a)], a=-PI..PI, PointSize=2, PointColor=RGB::Black),
plot::Line2d([cx(a),cy(a)],[fx(a),fy(a)], a=-PI..PI, LineColor=RGB::Black, LineWidth=0.4),
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1],
TicksNumber = Low, TicksBetween = 1, TicksLabelFont=["Cambria",10,RGB::SlateGreyDark], AxesTitleFont=["Calibri",12,RGB::SlateGreyDark],
ViewingBox = [-6..6, -6..6], Frames=400, TimeEnd=timend,
Height=160, Width=120, Scaling=Constrained,
AxesTitles = ["x", "y"]
)

Two coupled nonlinear differential equations

[edit]

Here Lotka-Volterra equations with decreasing returns.

Numerical resolution of a differential equation.
[r0,r1,r2,r3,r4,r5] := [10, 4.00000, 0.00000,-0.50000,-0.05000, 0.00000]: 
[h0,h1,h2,h3,h4,h5] := [10,-1.00000, 0.00000, 0.30000, 0.00000, 0.00000]:
fr := (yr,yh) ->             r1*yr + r2*yh + r3*yr*yh + r4*yr^2 + r5*yh^2:
fh := (yr,yh) ->             h1*yh + h2*yr + h3*yh*yr + h4*yh^2 + h5*yr^2:
Y := [yr,yh]:
Y0 := [r0,h0]:
Z[0]:=Y0:
f := (t,Y) -> [fr(Y[1],Y[2]),fh(Y[1],Y[2])]:

N := 800:

for i from 0 to N do  
 t[i] := i/20
end_for:

for i from 1 to N do  
  Z[i] := numeric::odesolve(f, t[i-1]..t[i], Z[i-1])
end_for:



plot(plot::Listplot([ [ t[i],Z[i][1] ] $ i = 0..N ], LineWidth=0.8, LineColor=RGB::SapGreen),
plot::Listplot([ [ t[i],Z[i][2] ] $ i = 0..N ], LineWidth=0.8, LineColor=RGB::Red), 
PointsVisible=FALSE, AxesTitles = ["Time", "Population"],
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1]):

plot(plot::Listplot([ [ Z[i][1],Z[i][2] ] $ i = 0..N ], LineWidth=0.6,LineColor=RGB::Blue), 
PointsVisible=FALSE, AxesTitles = ["Prey", "Predator"],
GridVisible = TRUE, SubgridVisible = TRUE, GridLineWidth = 0.2, SubgridLineWidth = 0.1, 
GridInFront = FALSE, GridLineColor = RGB::SlateGreyDark.[0.2], SubgridLineColor = RGB::SlateGreyDark.[0.1]):
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