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Perspective Ellipse Demo

Perspective Ellipse Demo
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2018-05-08 00:07
Coordinate GeometryParametric EquationGeometric Construction
This example demonstrates the formation and properties of ellipses under perspective projection. It involves core mathematical concepts such as ellipses, perspective geometry, geometric mean, sine function, parametric curves, rhombuses, and probability statistics. The construction utilizes GeoGebra commands and objects like Intersect (intersection), Line (line), OrthogonalLine (perpendicular line), Point (point), Circle (circle), Segment (segment), Midpoint (midpoint), and Conic (conic). Suitable for high school and above learners to understand the integration of spatial geometry and analytical geometry.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, yAxis) → A
  2. Line(A, B) → a
  3. OrthogonalLine(B, a) → b
  4. Intersect(b, xAxis) → C
  5. Line(A, C) → c
  6. OrthogonalLine(B, c) → d
  7. Point(d) → D
  8. Line(A, D) → e
  9. Line(D, C) → f
  10. Circle(C, B) → g
  11. Circle(A, B) → h
  12. Intersect(g, c, 1) → E
  13. Intersect(h, c, 2) → F
  14. Point(d) → G
  15. OrthogonalLine(D, d) → i
  16. Circle(D, G) → k
  17. Intersect(k, i, 2) → H
  18. Intersect(k, i, 1) → I
  19. Line(I, E) → j
  20. Line(H, F) → l
  21. Intersect(e, l) → J
  22. Intersect(f, j) → K
  23. Line(J, d) → m
  24. Line(K, d) → n
  25. Line(A, G) → p
  26. Line(G, C) → q
  27. Intersect(m, p) → L
  28. Intersect(n, q) → M
  29. Line(M, A) → r
  30. Line(L, C) → s
  31. Intersect(r, s) → N
  32. Line(K, A) → t
  33. Line(J, C) → a_1
  34. Intersect(t, a_1) → O
  35. Line(D, L) → b_1
  36. Line(J, G) → c_1
  37. Intersect(b_1, c_1) → P
  38. Line(K, N) → d_1
  39. Line(O, M) → e_1
  40. Intersect(d_1, e_1) → Q
  41. Line(P, A) → f_1
  42. Intersect(m, f_1) → R
  43. Intersect(d, f_1) → S
  44. Line(P, m) → g_1
  45. Line(Q, d) → h_1
  46. Intersect(t, h_1) → T
  47. Intersect(e, g_1) → U
  48. Intersect(p, g_1) → V
  49. Intersect(r, h_1) → W
  50. Line(O, N) → i_1
  51. Line(Q, A) → j_1
  52. Intersect(i_1, j_1) → Z
  53. Intersect(n, j_1) → A_1
  54. Segment(L, J) → k_1
  55. Segment(J, D) → l_1
  56. Segment(G, L) → m_1
  57. Segment(N, L) → n_1
  58. Segment(M, G) → p_1
  59. Segment(K, M) → q_1
  60. Segment(D, K) → r_1
  61. Segment(G, D) → s_1
  62. Segment(M, N) → t_1
  63. Segment(O, N) → a_2
  64. Segment(J, O) → b_2
  65. Segment(K, O) → c_2
  66. Segment(P, Q) → d_2
  67. Midpoint(S, V) → B_1
  68. Midpoint(R, U) → C_1
  69. Line(G, B_1) → e_2
  70. Line(J, C_1) → f_2
  71. Intersect(e_2, f_2) → D_1
  72. Point(l_1) → E_1
  73. Line(P, E_1) → g_2
  74. Line(U, S) → h_2
  75. Intersect(g_2, h_2) → F_1
  76. Line(V, F_1) → i_2
  77. Line(R, E_1) → j_2
  78. Intersect(i_2, j_2) → G_1
  79. Conic(R, U, G_1, S, V) → k_2
  80. Circle(J, D_1) → p_2
  81. Line(G, D_1) → l_2
  82. Line(J, D_1) → m_2
  83. Intersect(p_2, m_2, 2) → H_1
  84. Intersect(p_2, l_2, 2) → I_1
  85. Line(V, S) → n_2
  86. Line(R, U) → q_2
  87. Line(D_1, n_2) → r_2
  88. Line(D_1, q_2) → s_2
  89. Intersect(p_2, r_2, 2) → J_1
  90. Intersect(p_2, s_2, 2) → K_1
  91. Line(H_1, K_1) → t_2
  92. Line(J_1, I_1) → a_3
  93. Intersect(t_2, a_3) → L_1
  94. Line(L_1, J) → b_3
  95. Intersect(p_2, b_3, 2) → M_1
  96. Line(M_1, D_1) → c_3
  97. OrthogonalLine(D_1, c_3) → d_3
  98. Intersect(k_2, d_3, 1) → N_1
  99. Intersect(k_2, c_3, 2) → O_1
  100. Intersect(k_2, c_3, 1) → P_1
  101. Intersect(k_2, d_3, 2) → Q_1
  102. Segment(N_1, Q_1) → e_3
  103. Segment(O_1, P_1) → f_3
  104. Segment(D_1, C) → g_3
  105. OrthogonalLine(P, d_2) → h_3

AI Drawing Prompt Reference

Draw a demonstration of an ellipse under perspective projection, including two perpendicular line segments that intersect the coordinate axes at four points. Construct two more perpendicular line segments based on these points and draw two circles. Find the intersections of these circles with the line segments, then draw two line segments using these intersection points. Locate their intersection point. Finally, draw two more line segments using these intersection points and find their intersection point to form a closed ellipse.
The prompt above was compiled from the original drawing commands and is for reference only.