Math 22, Homework 12,  by  Oliver

Problem 12.1  Create your own Parametrized Curve  (space example)

r={Cos[t],Sin[3t],Cos[t]}+{Cos[17t],Sin[17t],0}/10;
S=Graphics3D[{Red,Tube[Table[r,{t,-Pi/2,Pi/2,0.01}],0.04]},ViewPoint->{-2.0522,-0.854289,2.5511},ViewVertical->{-0.541,-0.22644,2.032},Boxed->False,Axes->False]

Problem 12.2  Create your own Parametrized Surface

r={(3+Cos[3s])Cos[t],(3+Cos[3s])Sin[2t],4Sin[s]};ParametricPlot3D[r,{s,0,2Pi},{t,0,2Pi},Mesh->False,PlotPoints->40,PlotStyle->{Directive->Specularity["Red",100],Yellow},Boxed->False,Axes->False]

an example with texture

picture=Import["https://www.ibiblio.org/wm/paint/auth/pollock/pollock.key.jpg"];
r=(Sin[2 t]+Cos[3 s]) {Cos[t] Sin[s],Sin[t] Sin[s],Cos[s]};
ParametricPlot3D[r,{s,0,Pi},{t,0,2 Pi},Lighting->"Neutral",Mesh->False,PlotPoints->40,TextureCoordinateFunction->({#5,#4}&),PlotStyle->{Directive->Specularity["White",100],Texture[picture]},Boxed->False,Axes->False,ViewPoint->{-3.0825,-1.26182,-0.5639}]

Problem 12.3 Create your own level surface

ContourPlot3D[x^4 y^2+y^4 z^2+z^4 x^2==0.03,{x,-2,2},{y,-2,2},{z,-2,2},PlotPoints->30,MeshStyle->{Yellow},ContourStyle->{Directive->Specularity["Yellow",100],Orange},Boxed->False,Axes->False]

Problem 12.4 Create your own coordinate system 3D example

f={x+Sin[y z]/10,y+Sin[x z]/3,z+Sin[x y]/10}; rad=0.02;
X=Table[{Red,Tube[Table[f,{x,-2,2,0.1}],rad]},{y,-2,2,1},{z,-2,2,1}];
Y=Table[{Green,Tube[Table[f,{y,-2,2,0.1}],rad]},{x,-2,2,1},{z,-2,2,1}];
Z=Table[{Blue,Tube[Table[f,{z,-2,2,0.1}],rad]},{x,-2,2,1},{y,-2,2,1}];
Graphics3D[{X,Y,Z},Boxed->False]
DistortionFactor=Det[{D[f,x],D[f,y],D[f,z]}] /. x->0.5 /. y->0.7/. z->0.8

2D example  Spherical  coordinates reference

Clear[r,s,t];f={r Sin[s] Cos[t],r Sin[s] Sin[t],r Cos[s]}; rad=0.01;
X=Table[{Red,Tube[Table[f,{r,0,1,0.1}],rad]},{s,0,Pi,Pi/3},{t,0,2Pi,Pi/3}];
Y=Table[{Green,Tube[Table[f,{t,0,2Pi,0.1}],rad]},{r,0,1,0.2},{s,0,Pi,Pi/3}];
Z=Table[{Blue,Tube[Table[f,{s,0,Pi,0.1}],rad]},{r,0,1,0.2},{t,0,2Pi,Pi/3}];
Graphics3D[{X,Y,Z},Boxed->False]
DistortionFactor=Simplify[Det[{D[f,r],D[f,s],D[f,t]}] ]

2D example

f={x+Sin[4y ]/10,y+Sin[4x]/10};rad=0.01;
X=Table[{Red,Thickness->rad,Line[Table[f,{x,-2,2,0.1}]]},{y,-2,2,1}];
Y=Table[{Green,Thickness->rad,Line[Table[f,{y,-2,2,0.1}]]},{x,-2,2,1}];
Graphics[{X,Y}]
DistortionFactor=Det[{D[f,x],D[f,y]}] /. x->0.5/. y->0.6

2D example Polar coordinates reference

Clear[r,t];f={r Cos[t],r Sin[t]};rad=0.01;
X=Table[{Red,Thickness->rad,Line[Table[f,{r,0,1,0.1}]]},{t,0,2Pi,Pi/6}];
Y=Table[{Green,Thickness->rad,Line[Table[f,{t,0,2Pi,0.1}]]},{r,0,1,0.2}];
Graphics[{X,Y}]
DistortionFactor=Simplify[Det[{D[f,r],D[f,t]}] ]