Astroid

This a summary of my demonstrations dealing with astroid.

Astroid extruded

In[]:=
ParametricPlot3D[{Cos[t]^3,0,Sin[t]^3},{t,0,2Pi}]
Out[]=
In[]:=
ParametricPlot3D[{Cos[t]^3,0,Sin[t]^3}+{0,s,0},{t,0,0.999×2Pi},{s,0,1},MeshNone,BoundaryStyleBlack]
Out[]=
In[]:=
ParametricPlot3D[{Cos[t]^3,0,Sin[t]^3}+{0,s,0},{t,0,0.999×2Pi},{s,0,2},Mesh20,BoundaryStyleBlack]
Out[]=

Astroid related to circles

In[]:=
a1=Rotate[ParametricPlot3D[{uCos[t]^3,uSin[t]^3,0},{t,-Pi,Pi},{u,0.4,1},BoundaryStyleDirective[Thick,Red],PlotPoints30,MeshNone][[1]],Pi/4,{0,0,1}];​​​​c1=ParametricPlot3D[{Cos[t],Sin[t],0},{t,0,2Pi}][[1]];​​c2=Table[Rotate[Translate[c1,{1.4241,0,0}],i2Pi/4,{0,0,1}],{i,4}];​​​​Graphics3D[{a1,c2},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100}]
Out[]=
In[]:=
a1=ParametricPlot3D[{uCos[t]^3,uSin[t]^3,0},{t,-Pi,Pi},{u,0,1},Mesh20,PlotPoints40,MeshShading{{Red,Yellow},{Green,Pink}}][[1]];​​c1=ParametricPlot3D[{Cos[t],Sin[t],0},{t,0,2Pi}][[1]];​​Graphics3D[{c1,a1},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,6}]
Out[]=

Astroid made from segments

In[]:=
Get["PolyhedronOperations`"];​​Manipulate[​​Module[{},​​ast=ParametricPlot3D[{Cos[x]^3,0,Sin[x]^3},{x,0,2Pi},PlotStyleTube[0.02]][[1]];​​​​tab1=Table[{Cos[x]^3,0,Sin[x]^3},{x,0,buildPi/2,Pi/2/floor}];​​tab2=RotationTransform[2Pi/segment,{0,0,1}][tab1];​​​​zone=poly[tab1,tab2];​​top={Table[Rotate[{zone},i2Pi/segment,{0,0,1}],{i,stripsegment}]};​​bottom={FaceForm[Yellow,Green],Rotate[top,Pi,{1,0,0}]};​​​​Graphics3D[{If[sast,ast,{}],If[sbottom,bottom,{}],Truncate[top,truncate]},BoxedFalse,SphericalRegionTrue,ViewAngle0.19,ViewPoint{4,3,1},ImageSize400,PlotRange1.1{{-1,1},{-1,1},{-1,1}}]],​​Grid[{​​{Control@{{floor,21,"number of floors"},1,30,1,Appearance"Labeled",ImageSizeSmall},​​Control@{{segment,40,"number of segments"},1,40,1,Appearance"Labeled",ImageSizeSmall}},​​{Control@{{truncate,0.5,"truncate"},0,0.5,0.01,Appearance"Labeled",ImageSizeSmall},​​Row[{"show: ",Control@{{sast,True,"astroid"},{True,False}},Spacer[10],​​Control@{{sbottom,True,"bottom"},{True,False}}}]},​​{Control@{{build,0.9,"build"},0,1,0.01,ImageSizeSmall,Appearance"Labeled"},​​Control@{{strip,0.9,"spread"},0,1,0.01,ControlPlacementLeft,ImageSizeSmall,Appearance"Labeled"}}​​},AlignmentLeft],​​TrackedSymbols{floor,segment,truncate,sast,sbottom,build,strip},​​Initialization(Get["PolyhedronOperations`"];​​f[{a_,b_,c_,d_}]:={a,b,d,c};​​poly[t1_,t2_]:=Polygon/@f/@Partition[Flatten[Transpose[{t1,t2}],1],4,2];),AlignmentCenter]

Bars of astroid

Astroid shaped spikes on dodecahedron

The bar of ellipsograph rolls on astroid

Astroid and its pedal curve

Astroid formed by rolling circles