The Topology of Costa's Minimal Surface

(* Needs["costa1torx1geom`"];   pre-computed geometry for the shape targets *)
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(* first piece of torus is called "torus1" *)
(* Needs["costa1x1geom`"]; *)
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(* first piece of costa1 surface is called "costa1x1" *)
(* Needs["Objintrp`"]; -- Point-wise linear interpolation of Graphics3D::Points *)
polyintrp[poly1_Polygon,poly2_Polygon,degree_]:=Polygon[Map[((1-degree) #[[1]]+degree #[[2]])&,Transpose[{poly1[[1]],poly2[[1]]}],{1}]];

objintrp[graphics1_Graphics3D,graphics2_Graphics3D,degree_]:=Graphics3D[Map[polyintrp[#[[1]],#[[2]],degree]&,Transpose[{graphics1[[1]],graphics2[[1]]}],{1}]];

(* Needs["Orient`"]; *)
VecLen::usage="VecLen[vector] where vector is a list of {x, y, z}";

VecLen[{x_,y_,z_}]:=Sqrt[N[x^2+y^2+z^2]];

RotX::usage="RotX[angle] - returns the 3x3 versor to accomplish the rotation";

RotX[Xangle_]:=Return[{{1,0,0},{0,N[Cos[N[Xangle]]],N[Sin[N[Xangle]]]},{0,N[-Sin[N[Xangle]]],N[Cos[N[Xangle]]]}}];

RotY::usage="RotY[angle] - returns the 3x3 versor to accomplish the rotation";

RotY[Yangle_]:=Return[{{N[Cos[N[Yangle]]],0,N[Sin[N[Yangle]]]},{0,1,0},{N[-Sin[N[Yangle]]],0,N[Cos[N[Yangle]]]}}];

RotZ::usage="RotZ[angle] - returns 3x3 the versor to accomplish the rotation";

RotZ[Zangle_]:=Return[{{N[Cos[N[Zangle]]],N[Sin[N[Zangle]]],0},{N[-Sin[N[Zangle]]],N[Cos[N[Zangle]]],0},{0,0,1}}];

Rot4X::usage="Rot4X[angle] returns the 4x4 versor to accomplish the rotation";

Rot4X[Xangle_]:=Return[{{1,0,0,0},{0,N[Cos[N[Xangle]]],N[Sin[N[Xangle]]],0},{0,N[-Sin[N[Xangle]]],N[Cos[N[Xangle]]],0},{0,0,0,1}}];

Rot4Y::usage="Rot4Y[angle] returns the 4x4 versor to accomplish the rotation";

Rot4Y[Yangle_]:=Return[{{N[Cos[N[Yangle]]],0,N[Sin[N[Yangle]]],0},{0,1,0,0},{N[-Sin[N[Yangle]]],0,N[Cos[N[Yangle]]],0},{0,0,0,1}}];

Rot4Z::usage="Rot4Z[angle] returns the 4x4 versor to accomplish the rotation";

Rot4Z[Zangle_]:=Return[{{N[Cos[N[Zangle]]],N[Sin[N[Zangle]]],0,0},{N[-Sin[N[Zangle]]],N[Cos[N[Zangle]]],0,0},{0,0,1,0},{0,0,0,1}}];

ScaleMat[scale_List]:=Return[{{scale[[1]],0,0},{0,scale[[2]],0},{0,0,scale[[3]]}}];

Scale4[scale_List]:=Return[{{scale[[1]],0,0,0},{0,scale[[2]],0,0},{0,0,scale[[3]],0},{0,0,0,1}}];

translate[xlate_List]:=Return[{{1,0,0,0},{0,1,0,0},{0,0,1,0},{xlate[[1]],xlate[[2]],xlate[[3]],1}}];

GimbGen::usage="GimbGen[x_axis (, y_axis)] - returns a rotation matrix
to orient a vector into a rotated coordinate system defined by x_axis, y_axis
and x_axis X y_axis.  y_axis is optional";

xvert::usage="xvert[point,matrix] gives the result of applying the
4D matrix to the 3D point (i.e., Drop[Append[point,1].matrix,-1]).";

xvect::usage="xvect[vector,matrix] gives the result of applying the
4D matrix to the 3D vector (i.e., Drop[vector.Drop[matrix,-1],-1]).";

Objtran::usage="Objtran[Graphics3D, transforms], where transforms
is a list of 4 X 4 matrices, returns a translate/rotate/scaled Graphics3D";

GimbGen[x_]:=Block[{y,z},y=If[x[[1]]==0&&x[[2]]==0,Cross[x,{1,0,0}],Cross[x,{0,0,1}]];
z=Cross[x,y];
UnitVector[3,#]&/@{x,y,z}];

GimbGen[x_,y_]:=UnitVector[3,#]&/@{x,y,Cross[x,y]};

xvert[{x_?NumberQ,y_?NumberQ,z_?NumberQ},xform_]:=Drop[{x,y,z,1}.xform,-1];

xvect[{x_?NumberQ,y_?NumberQ,z_?NumberQ},xform_]:=Drop[{x,y,z}.Drop[xform,-1],-1];

Objtran[polygon_,xform_]:=Map[xvert[#,xform]&,polygon,{-2}];

Objtran[polygon_,translate[{0,0,0}]]:=polygon;
xvert[{x_?NumberQ,y_?NumberQ,z_?NumberQ},xform_]:=Drop[{x,y,z,1}.xform,-1];

xvect[{x_?NumberQ,y_?NumberQ,z_?NumberQ},xform_]:=Drop[{x,y,z}.Drop[xform,-1],-1];

Scale4[scale_List]:=Return[{{scale[[1]],0,0,0},{0,scale[[2]],0,0},{0,0,scale[[3]],0},{0,0,0,1}}];

Objtran[polygon_,xform_]:=Map[xvert[#,xform]&,polygon,{-2}];

(* Needs["Revel`"];  -- Reverse Graphics3D::Polygons *)
reverts[vertlist_Polygon]:=Polygon[Reverse[vertlist[[1]]]];
Revel[polyList_]:=polyList/.Polygon[x_]:>reverts[Polygon[x]];

torus2 = Revel[Objtran[torus1, Scale4[{-1, 1, 1}]]];
torus3 = Revel[Objtran[torus1, Scale4[{1, -1, 1}]]];
torus4 = Revel[Objtran[torus2, Scale4[{1, -1, 1}]]];
costa1x1a = Objtran[costa1x1, Scale4[{0.5, 0.5, 0.5}]];
costa1x2 = Revel[Objtran[costa1x1, Scale4[{-0.5, 0.5, 0.5}]]];
costa1x3 = Revel[Objtran[costa1x1, Scale4[{0.5, -0.5, 0.5}]]];
costa1x4 = Revel[Objtran[costa1x2, Scale4[{1, -1, 1}]]];
torus = Show[Flatten[{torus1, torus2,
	 torus3, torus4}]];
costa1 = Show[Flatten[{costa1x1a, costa1x2,
	 costa1x3, costa1x4}]];

Manipulate[Show[objintrp[torus, costa1, N[morph]], ViewPoint->{1.0, -3.0, 1.0}, Boxed->False,
	PlotRange->All, Background->Black,ImageSize->{450,450}],{{morph, 0.0, "morph"}, 0, 1, Appearance->"Labeled"},SaveDefinitions->True]

CAPTION

Costa's minimal surface is described as a torus with three points removed. How is this done? The points are literally removed from (ℝ^3) and flung to infinity. Here is a Demonstration of specifically where on the torus the points are removed and how the torus turns itself inside-out when the resulting ends of the surface are flung to infinity.

THIS NOTEBOOK IS THE SOURCE CODE FROM

"The Topology of Costa's Minimal Surface" from the Wolfram Demonstrations Project
 http://demonstrations.wolfram.com/TheTopologyOfCostasMinimalSurface/

Contributed by: Stewart Dickson

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