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Elliptic Plane: Half Sphere to Möbius Strip

Elliptic Plane: Half Sphere to Möbius Strip
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2020-05-18 17:38
TrigonometrySolid GeometryTransformationGeometric Construction
This example demonstrates the construction of a Möbius strip from a specific region on a hemisphere, visually illustrating the geometric properties of the elliptic plane. Key mathematical concepts include spherical geometry, geometric transformations, and topology basics. The construction utilizes GeoGebra commands and objects such as CircumcircleArc, Circle, Line, Sphere, Intersect, Polygon, Semicircle, Segment, Plane, IntersectPath, Point, CurveCartesian, Surface, Angle, Rotate, Mirror to dynamically generate the figure. Suitable for high school students and math enthusiasts studying spatial geometry and topological transformations.

GeoGebra Commands & Parameters

  1. CircumcircleArc((0, 0, -1), (0, 1, 0), (0, 0, 1)) → d
  2. Circle(P, π, xOyPlane) → c
  3. Line(P, yAxis) → f
  4. Sphere(P, π / 2) → g
  5. Intersect(f, g) → G, H
  6. Sphere(G, Width / 2) → j
  7. Line(H, zAxis) → i
  8. Line(G, zAxis) → h
  9. Intersect(h, j) → I, J
  10. Line(I, f) → n
  11. Intersect(n, i) → K
  12. Line(J, f) → k
  13. Intersect(i, k) → L
  14. Polygon(I, K, L, J) → Rectangle, i_1, k_4, l_1, j_1
  15. Intersect(xOyPlane, Rectangle) → S, U
  16. Line(S, U) → l
  17. Intersect(zAxis, xAxis) → F
  18. Semicircle(O, N, xOyPlane) → EquatorLineSphere
  19. Sphere(F, P) → e
  20. Segment(G, H) → q
  21. Line(N, q) → f_1
  22. Plane(f_1, zAxis) → p_1
  23. IntersectPath(p_1, e) → r
  24. Point(d) → M
  25. Plane(M, xAxis) → q_1
  26. IntersectPath(q_1, e) → FullLine
  27. Intersect(FullLine, r, 2) → Q
  28. CircumcircleArc(Q, P, M) → LineSphere_1
  29. Point(LineSphere_1) → S_S
  30. Plane(S_S, zAxis) → r_1
  31. Intersect(r_1, EquatorLineSphere) → R, T
  32. CurveCartesian(cos(t), sin(t), 0, t, 0, (2 * π)) → EquatorLineMobius
  33. Point(EquatorLineMobius) → B
  34. Point(EquatorLineMobius) → C
  35. Point(EquatorLineMobius) → D
  36. Point(EquatorLineMobius) → E
  37. Surface((cos(u) * cos(v)), (cos(u) * sin(v)), sin(u), u, 0, (2 * π), v, 0, π) → Sphere
  38. CurveCartesian(((1 + (Distance * cos(t / 2))) * cos(t)), ((1 + (Distance * cos(t / 2))) * sin(t)), (Distance * sin(t / 2)), t, 0, (4 * π)) → Equidistant
  39. Point(EquatorLineSphere) → A
  40. Angle(N, F, A, xOyPlane) → α
  41. Circle(P, α, xOyPlane) → p
  42. Circle(P, (2 * π) - α, xOyPlane) → p2
  43. Surface(1 + (v * cos(u)), u, (v * sin(u)), u, ((-π)) / 2, π / 2, v, ((-Width)) / 2, Width / 2) → Twisted
  44. Surface(sin(u), (cos(u) * cos(v)), (cos(u) * sin(v)), u, 0, π, v, 0, π) → HalfSphere
  45. Rotate(N, (2 * α), F, xOyPlane) → A_m
  46. Angle(N, F, A_m, xOyPlane) → δ
  47. Angle(R, F, S_S) → γ
  48. Angle(Q, F, S_S, q_1) → β
  49. Angle(q_1, xOyPlane) → ε
  50. CurveCartesian(sin(u), (cos(u) * cos(ε)), (cos(u) * sin(ε)), u, 0, π) → LineSphere
  51. Surface(1 - (v * sin((Twisting * u))), u - π / 2, (v * cos((Twisting * u))), u, 0, π, v, ((-Width)) / 2, Width / 2) → Twist
  52. Surface(((-cos(u)) * sin(v)), (cos(u) * cos(v)), sin(u), u, (-atan(Width / 2)) - π, atan(Width / 2) - π, v, 0, π) → SphericalStrip
  53. Surface(((1 - Flattening) * ((-cos(u)) * sin(v))) + Flattening, (((1 - Flattening) * cos(u)) * cos(v)) + (Flattening * (v - π / 2)), ((1 - Flattening) * sin(u)) + (Flattening * tan((-u) + π)), u, (-atand(Width / 2)) - π, atand(Width / 2) - π, v, 0, π) → Flatten
  54. Surface(1 + (v * cos(u)), ((1 + Prolonging) * u), (v * sin(u)), u, ((-π)) / 2, π / 2, v, ((-Width)) / 2, Width / 2) → ProlongTwisted
  55. Surface(1 + (v * cos(u / 2)), u, (v * sin(u / 2)), u, (-π), π, v, ((-Width)) / 2, Width / 2) → Prolonged
  56. Surface(((1 + (v * cos(u / 2))) * cos(u)), ((1 + (v * cos(u / 2))) * sin(u)), (v * sin(u / 2)), v, ((-Width)) / 2, Width / 2, u, 0, (2 * π)) → MobiusStrip
  57. Surface(((1 + (v * cos(u / 2))) * cos((Glueing * u))), ((1 - Glueing) * u) + ((Glueing * (1 + (v * cos(u / 2)))) * sin((Glueing * u))), (v * sin(u / 2)), u, (-π), π, v, ((-Width)) / 2, Width / 2) → Glue
  58. CurveCartesian(1, u - π / 2, (-tan(asin((cos(u) * sin(ε))))), u, 0, π) → LineRectangle
  59. CurveCartesian(1 - (tan(asin((cos(u - π / 2) * sin(ε)))) * cos(u)), u, ((-tan(asin((cos(u - π / 2) * sin(ε))))) * sin(u)), u, ((-π)) / 2, π / 2) → LineTwisted
  60. CurveCartesian(1 - (tan(asin((cos(u / 2 - π / 2) * sin(ε)))) * cos(u / 2)), u, ((-tan(asin((cos(u / 2 - π / 2) * sin(ε))))) * sin(u / 2)), u, (-π), π) → LineProlonged
  61. CurveCartesian(((1 - (tan(asin((cos(u / 2 - π / 2) * sin(ε)))) * cos(u / 2))) * cos(u)), ((1 - (tan(asin((cos(u / 2 - π / 2) * sin(ε)))) * cos(u / 2))) * sin(u)), ((-tan(asin((cos(u / 2 - π / 2) * sin(ε))))) * sin(u / 2)), u, (-π), π) → LineMobius
  62. IntersectPath(xOyPlane, Rectangle) → EquatorLineRectangle
  63. Intersect(xOyPlane, p2) → V,
  64. Intersect(l, c) → W, Z
  65. Segment(W, Z) → EquatorLineProlonged
  66. Circle(F, 1, xOyPlane) → Equator
  67. Mirror(S_S, F) → S'
  68. Mirror(A, F) → A'

AI Drawing Prompt Reference

Draw a hemisphere with a semicircle on its surface as the equator. Start from a specific region on the equator and transform it through rotations and reflections to create a Möbius strip. The final image should illustrate the geometric transformation from the hemisphere to the Möbius strip, highlighting spherical geometry, geometric construction, and basic topological concepts.
The prompt above was compiled from the original drawing commands and is for reference only.