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Clifford Torus Stereographic Projection

Clifford Torus Stereographic Projection
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2021-07-20 23:08
Coordinate GeometryTransformationGeometric Construction
This example demonstrates the stereographic projection of a Clifford Torus from 4D to 3D. Key mathematical concepts include coordinate geometry and transformation. Using GeoGebra commands such as If, Text, Vector, Polygon, etc., the construction process of the torus is shown. Suitable for high school students and math enthusiasts.

GeoGebra Commands & Parameters

  1. If(χStereoPrj, StereoPrj(x, y), If[χParallelPrj, (x * 6), 0]) → Prj
  2. If(χPunctured, (R * π / 8), 0) → punc
  3. Text("x^2+z^2=y^2+w^2=1 ↔ x^2+w^2=y^2+z^2=1; Rotate: ϕ=arg(z,w); Stereo-projected from w=R", (0, 0), false, true) → txtExplain
  4. Vector(OW - (1, 0), OW + (1.8, 0)) → vW
  5. Polygon(OW + (sZW * (Wzxy(0, 0), Zw(0, 0))), OW + (sZW * (Wzxy(0, π), Zw(0, π))), 4) → TZW
  6. Vector(OW - (0, 1), OW + (0, 1.5)) → vZ
  7. Angle(PR + (-1, 1), PR, PR + (-1, -1)) → αWR
  8. Sequence(Vector[PR + (0, z), PR + (-0.1, z)], z, -1, 1, 0.1) → ParallelPrjRays, {(-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0), (-0.1, 0)}
  9. Vector(OW + (sZW * (Wzxy(0, 0), Zw(0, 0))), OW + (sZW * (Wzxy(π, 0), Zw(π, 0)))) → vZWU
  10. Vector(OW + (sZW * (Wzxy(0, π), Zw(0, π))), OW + (sZW * (Wzxy(π, π), Zw(π, π)))) → vZWUπ
  11. Vector(OW + (sZW * (Wzxy(0, 0), Zw(0, 0))), OW + (sZW * (Wzxy(0, π), Zw(0, π)))) → vZWV
  12. Vector(OW + (sZW * (Wzxy(π, 0), Zw(π, 0))), OW + (sZW * (Wzxy(π, π), Zw(π, π)))) → vZWVπ
  13. Angle(OW + (sZW * (Wzxy(0, π), Zw(0, π))), OW + (sZW * (Wzxy(0, 0), Zw(0, 0))), OW + (sZW * (Wzxy(π, 0), Zw(π, 0)))) → αPuncture
  14. Vector(PUV00, PUV00 + (0, 2, 0)) → vV00
  15. Vector(PUV0π, PUV0π + (0, -2, 0)) → vV0π
  16. Vector(PUVπ0, PUVπ0 + (0, 2, 0)) → vVπ0
  17. Vector(PUVππ, PUVππ + (0, -2, 0)) → vVππ
  18. Vector(PUV0π, PUV0π + (2, 0, 0)) → vU0π
  19. Vector(PUVπ0, PUVπ0 + (-2, 0, 0)) → vUπ0
  20. Vector(PUV00, PUV00 + (2, 0, 0)) → vU00
  21. Vector(PUVππ, PUVππ + (-2, 0, 0)) → vUππ
  22. Surface(PrjX(u, v), PrjY(u, v), PrjZ(u, v), u, punc, (2 * π) - punc, v, punc, (2 * π) - punc) → Torus
  23. Surface(PrjX(u, v), PrjY(u, v), PrjZ(u, v), u, 0, (2 * π), v, (-0.005 * π), (0.005 * π)) → TorusU
  24. CurveCartesian(PrjX(u, π), PrjY(u, π), PrjZ(u, π), u, 0, (2 * π)) → TorusUπ
  25. CurveCartesian(PrjX(π, v), PrjY(π, v), PrjZ(π, v), v, 0, (2 * π)) → TorusVπ
  26. Surface(PrjX(u, v), PrjY(u, v), PrjZ(u, v), u, (-0.005 * π), (0.005 * π), v, 0, (2 * π)) → TorusV
  27. CurveCartesian(PrjX(u, punc), PrjY(u, punc), PrjZ(u, punc), u, (-punc), punc) → TorusPuncU
  28. CurveCartesian(PrjX(u, (-punc)), PrjY(u, (-punc)), PrjZ(u, (-punc)), u, (-punc), punc) → TorusPuncUn
  29. CurveCartesian(PrjX(punc, v), PrjY(punc, v), PrjZ(punc, v), v, (-punc), punc) → TorusPuncV
  30. CurveCartesian(PrjX((-punc), v), PrjY((-punc), v), PrjZ((-punc), v), v, (-punc), punc) → TorusPuncVn
  31. Surface(PrjX(u, v), PrjY(u, v), PrjZ(u, v), u, punc, (2 * π) - punc, v, (-punc), punc) → TorusPatchU
  32. Surface(PrjX(u, v), PrjY(u, v), PrjZ(u, v), u, (-punc), punc, v, punc, (2 * π) - punc) → TorusPatchV

AI Drawing Prompt Reference

Draw a stereographic projection of a Clifford Torus. The figure should show a three-dimensional torus shape composed of multiple torus patches joined together. The patches should be connected through circular holes, forming a complete torus structure. The projection should reflect the mapping from four-dimensional space to three-dimensional space, with annotations of coordinate relationships and rotation angles. The overall figure should clearly display the geometric features of the torus and its projected appearance, suitable for math enthusiasts.
The prompt above was compiled from the original drawing commands and is for reference only.