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Homotopy: Mug and Donut Topological Equivalence

Homotopy: Mug and Donut Topological Equivalence
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2020-01-07 01:42
FunctionSolid GeometryTransformationParametric EquationSlider Animation
① This GeoGebra example demonstrates the famous topological concept of homotopy by showing a continuous deformation from a coffee mug into a donut (torus), illustrating that they are topologically equivalent. ② Core mathematical concepts include homotopy, topological equivalence, continuous deformation, parametric surfaces, and spatial curves in 3D. ③ Construction highlights: uses GeoGebra commands such as CurveCartesian to define space curves, OrthogonalLine for auxiliary orthogonal lines, and Surface to generate dynamic surfaces; key objects include parameters m, r, R, e, c, change, A, B, i, C, j, l, n, b, d, g, h, a, f, o, s, fact, p, and multiple images (image1, image2, image3) to control the deformation process via sliders or variables. ④ Suitable for university-level topology, geometry, or mathematical visualization courses, as well as science outreach, helping learners intuitively grasp abstract homotopy concepts and continuous deformation ideas.

GeoGebra Commands & Parameters

  1. CurveCartesian(A + (t * (B - A)), t, 0, 1) → i
  2. OrthogonalLine(C, j) → l
  3. Surface(i, (2 * π), l) → n
  4. Surface((cos(t) * (R + ((r + change) * sin(s)))), (sin(t) * (R + ((r + change) * sin(s)))), ((r + change) * cos(s)), t, π / 2, (3 * π / 2), s, 0, (2 * π)) → b
  5. Surface(((2 * sin(change)) * b(t, s)) + (cos(change) * n(((-t)) / π + 3 / 2, s + π / 2)), t, (1.325 * π / 2), (2.65 * π / 2), s, 0, (2 * π)) → d
  6. CurveCartesian(d((2.65 * π / 2), t), t, 0, (2 * π)) → g
  7. CurveCartesian(d(t, π), t, π / 2, (3 * π / 2)) → h
  8. Surface((k * g(t)) + ((1 - k) * g(t + π)), k, 0, 1, t, 0, π) → a
  9. CurveCartesian(d((1.325 * π / 2), t), t, 0, (2 * π)) → f
  10. Surface((cos(t) * (R + ((r + change) * sin(s)))), (sin(t) * (R + ((r + change) * sin(s)))), ((r + change) * cos(s)), t, ((-π)) / 2, π / 2, s, 0, (2 * π)) → o
  11. Surface((cos(t) * (R + (3 * change) + ((r + change) * sin(s)))), (sin(t) * (R + (3 * change) + ((r + change) * sin(s)))), ((r + change) * cos(s)), t, π / 2, (3 * π / 2), s, 0, (2 * π)) → s
  12. Surface((cos(t) * (R + (fact * change) + ((r + fact) * sin(s)))), (sin(t) * (R + (fact * change) + ((r + fact) * sin(s)))), ((r + fact) * cos(s)), t, π / 2, (3 * π / 2), s, 0, (2 * π)) → p

AI Drawing Prompt Reference

Draw the continuous deformation from a coffee mug to a torus to illustrate topological equivalence. Start by drawing a space curve, then generate auxiliary orthogonal lines and dynamic surfaces. Next, construct a torus surface with adjustable parameters, and finally demonstrate the transformation from the coffee mug to the torus through continuous deformation.
The prompt above was compiled from the original drawing commands and is for reference only.