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3D Opposition Loops

3D Opposition Loops
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2015-06-10 03:20
Coordinate GeometrySolid GeometryGeometric Construction
This example demonstrates opposition loops in 3D space. Key mathematical concepts include ellipses, cubes, and linear functions. Using GeoGebra commands such as Point, Ellipse, Intersect, Rotate, etc., complex 3D geometric shapes can be constructed. Suitable for high school and above students or teachers for teaching and research.

GeoGebra Commands & Parameters

  1. Point(xAxis) → C
  2. Ellipse(A, B, C) → c
  3. Intersect(xAxis, yAxis) → D
  4. Circle(D, C) → d
  5. Rotate(C, ε, A) → C'_2
  6. Angle(C, A, C'_2) → ζ
  7. Line(D, C'_2) → f
  8. Vector(D, A) → u
  9. Translate(f, u) → f'
  10. Circle(A, ((2 * ne) * h)) → g
  11. Intersect(g, f') → F, G
  12. Midpoint(A, G) → H
  13. Circle(H, h) → k
  14. Intersect(k, f') → I, J
  15. Ellipse(A, G, I) → p
  16. Point(c) → K
  17. Point(c) → L
  18. Point(c) → M
  19. Point(c) → N
  20. Point(c) → O
  21. Point(c) → P
  22. Point(c) → Q
  23. Point(c) → R
  24. Point(d) → S
  25. Point(d) → T
  26. Point(d) → U
  27. Point(d) → V
  28. Point(d) → W
  29. Line(A, W) → i
  30. Point(i) → Z
  31. Ray(A, Z) → j
  32. OrthogonalLine(A, j) → l
  33. Intersect(d, j) → C_1,
  34. Circle(H, h) → q
  35. Rotate(Z, η, A) → Z'
  36. Angle(Z, A, Z') → θ
  37. Line(A, Z') → m
  38. Intersect(d, xAxis, 2) → I_1
  39. Rotate(I_1, α + (((0.0167 * 180) * ° / 3.141) * sin(α)) + ((0.5 * (((0.0167 * 180) * ° / 3.141))^(2)) * sin((2 * α))), D) → E
  40. Angle(I_1, D, E) → β
  41. Rotate(J, λ + (((ne * 180) * ° / 3.141) * sin(λ)) + ((0.5 * (((ne * 180) * ° / 3.141))^(2)) * sin((2 * λ))), H) → J'
  42. Angle(J, H, J') → γ
  43. Ray(H, J') → b
  44. Intersect(p, b) → D_1,
  45. Segment(E, D_1) → n
  46. Ray(A, E) → r
  47. Rotate(p, δ, m) → p'
  48. Rotate(D_1, δ, m) → R_1
  49. Rotate(I, δ, m) → I'
  50. Rotate(J, δ, m) → J'_1
  51. Rotate(H, δ, m) → H'
  52. Rotate(G, δ, m) → G'
  53. Line(J'_1, I') → s
  54. Segment(E, R_1) → t
  55. Vector(E, D) → w
  56. Translate(E, w) → E'
  57. Translate(R_1, w) → S_1
  58. Segment(E', S_1) → t'
  59. Intersect(d, xAxis, 2) → A_1
  60. Intersect(d, xAxis, 1) → B_1
  61. Translate(E, w) → E'_1
  62. Translate(D_1, w) → E_1
  63. Segment(E'_1, E_1) → n'
  64. Sphere(D, h) → c_1
  65. Ray(D, S_1) → d_1
  66. Intersect(d_1, c_1) → F_1, G_1
  67. Textfield(v) → Textfeld1
  68. Plane(N, O, P) → l_1
  69. Rotate(l_1, -23.4*°, j) → m_1
  70. IntersectPath(m_1, c_1) → p_1
  71. Textfield(h) → Textfeld2