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Perspective View in a Sphere

Perspective View in a Sphere
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2020-05-06 14:25
Solid GeometryTransformationGeometric Construction
This example demonstrates a perspective view within a sphere. Key mathematical concepts include geometric transformations and rotation. Using GeoGebra commands such as Line, Circle, Segment, and Point, the perspective view within a sphere is constructed. Suitable for high school mathematics education, particularly in solid geometry and geometric transformations.

GeoGebra Commands & Parameters

  1. Line(A, zAxis) → f
  2. Circle(A, R, f) → c
  3. Segment(B, A) → g
  4. Point(g) → C
  5. CircumcircleArc(T, B_1, U) → e_2
  6. Polygon(I, K, L, F) → q1, i_3, k_3, l_3, f_3
  7. Sphere(A, R) → e
  8. PointIn(q1) → D
  9. PointIn(q1) → E
  10. Polygon(D, E, 4, q1) → poly1, i, j, k, l, G, H
  11. Line(E, f) → m
  12. Point(m) → J
  13. Prism(poly1, (-Distance[J, E])) → a, M, arêteDM, arêteEN, arêteGO, arêteHP, arêteMN, arêteNO, arêteOP, arêteMP, N, O, P, faceDENM, faceEGON, faceGHPO, faceDHPM, faceMNOP
  14. Midpoint(P, M) → Q
  15. Midpoint(H, D) → S
  16. Midpoint(H, P) → V
  17. Midpoint(M, D) → W
  18. Ray(P, C) → n
  19. Ray(Q, C) → p
  20. Ray(M, C) → q
  21. Ray(V, C) → r
  22. Ray(H, C) → s
  23. Ray(D, C) → t
  24. Ray(S, C) → b
  25. Ray(W, C) → d
  26. Ray(M, C) → f_1
  27. Intersect(n, e) → Z, A_1
  28. Intersect(p, e) → C_1, D_1
  29. Intersect(f_1, e) → E_1, F_1
  30. Intersect(r, e) → G_1, H_1
  31. Intersect(d, e) → I_1, J_1
  32. Intersect(s, e) → K_1, L_1
  33. Intersect(b, e) → M_1, N_1
  34. Intersect(t, e) → O_1, P_1
  35. CircumcircleArc(Z, C_1, E_1) → c_1
  36. CircumcircleArc(E_1, I_1, O_1) → d_1
  37. CircumcircleArc(Z, G_1, K_1) → e_1
  38. CircumcircleArc(O_1, M_1, K_1) → g_1
  39. Midpoint(J, O) → Q_1
  40. Midpoint(O, G) → R_1
  41. Midpoint(J, E) → S_1
  42. Ray(J, C) → h_1
  43. Ray(O, C) → i_1
  44. Ray(Q_1, C) → j_1
  45. Ray(G, C) → k_1
  46. Ray(E, C) → l_1
  47. Midpoint(E, G) → T_1
  48. Ray(R_1, C) → m_1
  49. Ray(S_1, C) → n_1
  50. Intersect(h_1, e) → U_1, V_1
  51. Intersect(j_1, e) → W_1, Z_1
  52. Intersect(n_1, e) → A_2, B_2
  53. Intersect(i_1, e) → C_2, D_2
  54. Intersect(l_1, e) → E_2, F_2
  55. Intersect(m_1, e) → G_2, H_2
  56. Intersect(k_1, e) → I_2, J_2
  57. Ray(T_1, C) → p_1
  58. Intersect(p_1, e) → K_2, L_2
  59. CircumcircleArc(C_2, W_1, U_1) → q_1
  60. CircumcircleArc(U_1, A_2, E_2) → r_1
  61. CircumcircleArc(C_2, G_2, I_2) → s_1
  62. CircumcircleArc(E_2, K_2, I_2) → t_1
  63. Midpoint(arêteOP) → M_2
  64. Midpoint(arêteMN) → N_2
  65. Midpoint(k) → O_2
  66. Midpoint(i) → P_2
  67. Ray(O_2, C) → a_1
  68. Ray(P_2, C) → b_1
  69. Ray(N_2, C) → f_2
  70. Ray(M_2, C) → g_2
  71. Intersect(f_2, e) → Q_2, R_2
  72. Intersect(g_2, e) → S_2, T_2
  73. Intersect(b_1, e) → U_2, V_2
  74. Intersect(a_1, e) → W_2, Z_2
  75. CircumcircleArc(U_1, Q_2, E_1) → c_2
  76. CircumcircleArc(O_1, U_2, E_2) → d_2
  77. CircumcircleArc(Z, S_2, C_2) → h_2
  78. CircumcircleArc(K_1, W_2, I_2) → k_2
  79. Segment(A, T_1) → i_2
  80. Segment(C, P) → j_2
  81. Segment(C, O) → l_2
  82. Segment(C, M) → m_2
  83. Segment(C, J) → n_2
  84. Segment(C, H) → p_2
  85. Segment(C, D) → q_2
  86. Segment(C, E) → s_2
  87. Segment(C, G) → r_2
  88. Segment(C, A) → t_2
  89. Sequence(CurveCartesian[x(A) + ((R * cos(k)) * cos(j)), y(A) + ((R * cos(k)) * sin(j)), z(A) + (R * sin(k)), j, ((-π)) / 2, π / 2], k, 0, π / 2, π / oo) → l2
  90. CircumcircleArc(T, B, U) → c_3
  91. Sequence(CurveCartesian[x(A) + ((R * cos(k)) * cos(j)), y(A) + ((R * cos(k)) * sin(j)), z(A) + (R * sin(k)), k, 0, π / 2], j, ((-π)) / 2, π / 2, (2 * π / (oo + 10))) → l1
  92. CircumcircleArc(U_1, Q_2, E_1) → d_3
  93. CircumcircleArc(O_1, U_2, E_2) → e_3
  94. CircumcircleArc(C_2, S_2, Z) → h_3
  95. CircumcircleArc(K_1, W_2, I_2) → g_3