Lean In
A similar idea to Fall Out , one dimension up. Now I have a 2-sphere in the 3-sphere which I map to 3-space by extending the line containing a point until it intersects the w=1 hyperplane, then inverting in the unit sphere and reflecting through the origin.
The underlying object is just the standard equatorial 2-sphere (x,y,z,0) with x^2+y^2+z^2=1, which then gets rotated by \pi/4 in the plane spanned by \cos s (0,0,1,0) + \sin s (1,0,0,0) and (0,0,0,1) as s varies from 0 to 2\pi.
Here’s the code:
inversion[p_] := p/Norm[p]^2;
With[{n = 50, m = 31, d = .01,
cols = Darker[RGBColor[#]] & /@ {"#43DDE6", "#FC5185", "#364F6B"}},
Manipulate[
Graphics3D[
{Sphere[#, .02] & /@
Flatten[
Table[
i * inversion[#1[[1 ;; -2]]/#1[[-1]]] &[
RotationMatrix[Pi/4, {Cos[s] {0, 0, 1, 0} + Sin[s] {1, 0, 0, 0}, {0, 0, 0, 1}}]
.{Sqrt[1 - z^2] Cos[θ], Sqrt[1 - z^2] Sin[θ], z, 0}],
{i, {-1, 1}}, {θ, 0., 2 Pi - 2 Pi/n,
2 Pi/n}, {z, -1., 1, 2/m}], 2]},
PlotRange -> 1.2, Boxed -> False, ViewPoint -> {0, 5, 0},
ImageSize -> 540, Background -> cols[[-1]],
Lighting -> {{"Point", cols[[1]], 1/2 {Sin[s], 0, Cos[s]}},
{"Point", cols[[2]], 1/2 {-Sin[s], 0, -Cos[s]}}, {"Ambient", cols[[-1]], {0, 5, 0}}}],
{s, 0, 2 Pi}]
]
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