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Homotopy: Roman Surface vs Boy Surface

Homotopy: Roman Surface vs Boy Surface
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2020-01-07 16:32
SequenceSolid GeometryTransformationParametric EquationSlider Animation
① This GeoGebra example dynamically demonstrates a homotopy between the Roman surface and the Boy surface, showing a continuous deformation between two classical immersed surfaces in 3D space. ② The core mathematical concepts include homotopy, surface parametrization, topological equivalence, and continuous deformations of immersed surfaces, with emphasis on the Roman and Boy surface constructions. ③ Construction highlights: The Surface command is used to create parametric surfaces, and the Sequence command generates a series of intermediate surface frames; a slider controls the deformation parameter. Auxiliary points (A1-A5, B1-B4, etc.) and curves (l1, l2) help define and constrain the geometry, while image objects (image2, image3, image4) enhance the visual presentation. ④ This is suitable for university-level mathematics students or advanced undergraduates studying topology and differential geometry, and serves as an excellent visualization tool for understanding abstract homotopy concepts.

GeoGebra Commands & Parameters

  1. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A1, A2) → B1
  2. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A2, A3) → B2
  3. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A3, A4) → B3
  4. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, π / 2, v, A4, A5) → B4
  5. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A1, A2) → C1
  6. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A2, A3) → C2
  7. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A3, A4) → C3
  8. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π / 2, (2 * π / 2), v, A4, A5) → C4
  9. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A1, A2) → D1
  10. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A1, A2) → E1
  11. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A2, A3) → E2
  12. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A3, A4) → E3
  13. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A2, A3) → D2
  14. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A3, A4) → D3
  15. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, π, (3 * π / 2), v, A4, A5) → D4
  16. Surface(x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, (3 * π / 2), (4 * π / 2), v, A4, A5) → E4
  17. Sequence(CurveCartesian[x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), u, 0, (2 * π)], v, 0, (2 * π), (2 * π / m)) → l1
  18. Sequence(CurveCartesian[x(O) + ((2 - a) * f(u, v)), y(O) + ((2 - a) * g(u, v)), z(O) + ((2 - a) * h(u, v)), v, 0, (2 * π)], u, 0, (2 * π), (2 * π / m)) → l2