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Inflating a Tetrahedron onto a Sphere

Inflating a Tetrahedron onto a Sphere
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2017-05-29 15:53
Solid GeometryTransformationGeometric Construction
This example demonstrates how to inflate a tetrahedron onto a sphere. Key mathematical concepts include solid geometry and geometric construction. Using GeoGebra commands such as Intersect, Rotate, Circle, etc., the intersection points, rotations, and circles are constructed. Suitable for high school or university solid geometry courses.

GeoGebra Commands & Parameters

  1. Intersect(zAxis, xOyPlane) → A
  2. Sphere(A, 1) → a
  3. Intersect(zAxis, a) → B, C
  4. Line(A, B) → i
  5. Rotate(C, 109.5°, xAxis) → C'
  6. Rotate(C', 120°, zAxis) → C''
  7. Circle(C, C', C'') → c_1
  8. Center(c_1) → N
  9. Ray(A, N) → f_1
  10. Intersect(f_1, a) → P, O
  11. OrthogonalPlane(O, f_1) → v
  12. Rotate(C'', 120°, zAxis) → C'''
  13. CircleArc(A, C, C') → c
  14. CircleArc(A, C, C'') → d
  15. CircleArc(A, C', C'') → e
  16. CircleArc(A, C, C''') → f
  17. CircleArc(A, C', C''') → g
  18. CircleArc(A, C'', C''') → h
  19. Circle(B, 1, i) → k
  20. Intersect(j, k) → D, E
  21. Line(D, C') → l
  22. Intersect(l, i) → F
  23. Line(F, C'') → n
  24. Intersect(n, k) → G,
  25. Ray(B, G) → j_1
  26. Line(F, C''') → m
  27. Intersect(m, k) → H,
  28. Point(g) → I
  29. Plane(I, C, A) → o
  30. Intersect(o, k) → J, K
  31. Line(J, I) → p
  32. Intersect(p, i) → L
  33. Locus(M, I) → loc1
  34. Rotate(I, 120°, zAxis) → I'
  35. Rotate(I', 120°, zAxis) → I''
  36. Rotate(M, 120°, zAxis) → M'
  37. Rotate(M', 120°, zAxis) → M''
  38. Locus(M', I) → loc2
  39. Locus(M'', I) → loc3
  40. Segment(C', D) → q
  41. Segment(C''', H) → r
  42. Segment(C'', G) → s
  43. Circle(O, 1, f_1) → d_1
  44. Rotate(d_1, (-(120°)), i) → e_1
  45. Rotate(e_1, (-(120°)), i) → g_1
  46. Ray(A, C') → h_1
  47. Rotate(C', 120°, h_1) → C''_1
  48. Circle(P, 1, f_1) → k_1
  49. Rotate(k_1, (-(120°)), i) → p_1
  50. Rotate(p_1, (-(120°)), i) → q_1
  51. Center(q_1) → Q
  52. Center(p_1) → R
  53. OrthogonalPlane(P, f_1) → u
  54. Ray(B, D) → i_1
  55. Intersect(i_1, u) → S
  56. Ray(B, H) → l_1
  57. Intersect(j_1, v) → U
  58. Intersect(j_1, u) → T
  59. Plane(A, T, U) → w
  60. Mirror(M', w) → M''_1
  61. Locus(M''_1, I) → loc4
  62. Mirror(D, w) → D'
  63. Mirror(G, w) → G'
  64. Polygon(C', C'', C''') → poly1, c''', c', c''
  65. Pyramid(poly1, C) → a_1, faceCC'C'', faceCC''C''', faceCC'C''', edgeCC', edgeCC'', edgeCC'''
  66. Circle(C, C''', C') → r_1
  67. Midpoint(edgeCC') → V
  68. Ray(V, A) → m_1
  69. Midpoint(edgeCC') → W
  70. Intersect(e, c, 2) → Z
  71. CircleArc(Ce, C, Z) → s_1
  72. Rotate(s_1, 120°, zAxis) → t_1
  73. Rotate(t_1, 120°, zAxis) → c_2
  74. Mirror(Ce, A) → Ce'
  75. CircleArc(Ce', C'', C''') → d_2
  76. Rotate(d_2, 120°, zAxis) → e_2
  77. Rotate(e_2, 120°, zAxis) → f_2

AI Drawing Prompt Reference

Draw a tetrahedron and inflate it onto a sphere. Start by drawing a sphere a with radius 1 at point A where the z-axis intersects the xOy plane. Then, find the intersection points B and C of sphere a with the z-axis, and draw line i connecting A and B. Rotate point C 109.5 degrees around the x-axis to get C', and then rotate C' 120 degrees around the z-axis to get C''. Draw circles with centers C, C', and C'' and passing through each other. Find the center N of circle c1, and draw ray f1 from A to N, intersecting sphere a at points P and O. Draw a plane v perpendicular to f1 at O, and rotate C'' 120 degrees around the z-axis to get C'''. Draw arcs from A to C', C, C'', and C''', and arcs from B to C, C', C'', and C'''. Draw a circle k from B to C, find the intersection points D and E of k and line j from B to C', draw line l from D to C', intersecting k at F, draw line n from F to C'', intersecting k at G and H. Draw a ray j1 from G to B, a line m from F to C''', intersecting k at H, a ray h1 from C' to A, and rotate C' 120 degrees around h1 to get C''', and draw a circle k1 from P to A.
The prompt above was compiled from the original drawing commands and is for reference only.