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Plane Intersecting a Cone

Plane Intersecting a Cone
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2017-08-03 05:51
TrigonometryCoordinate GeometrySolid GeometryCalculusGeometric Construction
This example demonstrates the geometric cross-section shapes formed by the intersection of a plane and a cone. It involves core mathematical concepts such as conic sections (ellipses, parabolas, hyperbolas), plane geometry, quadratic equations, matrices, and differential calculus. The construction utilizes GeoGebra commands and objects like Vector, OrthogonalLine, Translate, Mirror, Polygon, Midpoint, Conic, Segment, Line, Point, Intersect, and AngularBisector. It is suitable for high school and above students to understand the connection between spatial geometry and analytical geometry, as well as the application of parametric equations.

GeoGebra Commands & Parameters

  1. Vector(B_1, X) → u
  2. Vector(B_1, Z) → w
  3. OrthogonalLine(Vues, yAxis) → a
  4. OrthogonalLine(Vues, xAxis) → b
  5. Vector(B_1, Y) → v
  6. Translate(O, Vector[(R * u)]) → O'_1
  7. Translate(O, Vector[(R * v)]) → O'_2
  8. Mirror(O'_2, O) → O''
  9. Mirror(O'_1, O) → A
  10. Polygon(A, O'_2, O'_1, O'') → poly1, a_1, o', o'_1, o''
  11. Midpoint(a_1) → B
  12. Midpoint(o'') → C
  13. Midpoint(o'_1) → D
  14. Midpoint(o') → E
  15. Conic(E, B, C, D, E) → d
  16. Midpoint(O, O'_2) → F
  17. Midpoint(F, B) → G
  18. Midpoint(O, G) → H
  19. Midpoint(O, G) → I
  20. Midpoint(G, I) → J
  21. Mirror(J, G) → J'
  22. Conic(E, J', B, C, D) → e
  23. Translate(O, Vector[(h * w)]) → S
  24. Segment(S, O) → f
  25. Line(O'_2, f) → i
  26. Line(A, f) → j
  27. Line(O'', f) → l
  28. Point(j) → K
  29. Point(i) → L
  30. Point(l) → M
  31. Segment(K, L) → m
  32. Segment(K, M) → n
  33. Line(L, n) → p
  34. Line(M, m) → q
  35. Intersect(p, q) → N
  36. Polygon(K, L, N, M) → poly2, k_1, l_1, n_1, m_1
  37. Midpoint(k_1) → P
  38. Midpoint(l_1) → Q
  39. Midpoint(m_1) → S_2
  40. Midpoint(n_1) → T
  41. Line(S, E) → r
  42. Line(B, S) → s
  43. Line(S, D) → t
  44. Line(S, C) → b_1
  45. Line(S_2, Q) → c_1
  46. Line(P, T) → d_1
  47. Intersect(c_1, r) → U
  48. Intersect(c_1, b_1) → V
  49. Intersect(s, d_1) → W
  50. Intersect(t, d_1) → A_1
  51. Segment(S, D) → e_1
  52. Segment(S, C) → f_1
  53. Segment(S, B) → g_1
  54. Segment(S, E) → h_1
  55. Line(V, m_1) → i_1
  56. Line(W, k_1) → j_1
  57. Line(U, l_1) → p_1
  58. Line(A_1, n_1) → q_1
  59. Intersect(p_1, j_1) → C_1
  60. Intersect(i_1, j_1) → D_1
  61. Intersect(q_1, i_1) → E_1
  62. Intersect(q_1, p_1) → F_1
  63. Polygon(D_1, C_1, F_1, E_1) → poly3, d_2, c_2, f_2, e_2
  64. Midpoint(C_1, E_1) → G_1
  65. Mirror(V, G_1) → V'
  66. Conic(U, V', W, V, A_1) → r_1
  67. Segment(O'_1, A) → s_1
  68. Segment(O'_2, O'') → t_1
  69. Intersect(e, s_1) → H_1, I_1
  70. Intersect(e, t_1) → J_1, K_1
  71. Segment(S, I_1) → a_2
  72. Segment(S, J_1) → b_2
  73. Segment(S, H_1) → g_2
  74. Segment(S, K_1) → h_2
  75. AngularBisector(O'_2, O, B) → i_2
  76. AngularBisector(A, O, B) → j_2
  77. AngularBisector(C, O, A) → k_2
  78. AngularBisector(O'', O, C) → l_2
  79. Midpoint(B, A) → L_1
  80. Midpoint(O'_2, B) → M_1
  81. Midpoint(O'_2, E) → N_1
  82. Midpoint(A, C) → O_1
  83. Line(O, L_1) → m_2
  84. Line(N_1, O) → n_2
  85. Line(M_1, O) → p_2
  86. Line(O_1, O) → q_2
  87. Intersect(e, p_2) → P_1, Q_1
  88. Intersect(e, m_2) → R_1, S_1
  89. Intersect(e, q_2) → T_1, U_1
  90. Intersect(e, n_2) → V_1, W_1
  91. Segment(Q_1, S) → r_2
  92. Segment(W_1, S) → s_2
  93. Segment(T_1, S) → t_2
  94. Segment(S_1, S) → a_3
  95. Segment(R_1, S) → b_3
  96. Segment(U_1, S) → c_3
  97. Segment(S, V_1) → d_3
  98. Segment(P_1, S) → e_3
  99. Intersect(e, r_1) → Z_1, A_2, B_2, C_2
  100. Conic(E, J', B, C, D) → f_3
  101. Conic(U, V', W, V, A_1) → g_3
  102. Polygon(K, L, N, M) → poly4, k_3, l_3, n_3, m_3
  103. Polygon(A, O'_2, O'_1, O'') → poly5, a_4, o'_2, o'_3, o''_1

AI Drawing Prompt Reference

Draw a geometric illustration showing the cross-section shapes formed by the intersection of a cone and a plane, including ellipses, parabolas, or hyperbolas. Start by constructing the basic shapes of the cone and the plane, then create key points using vectors, reflections, and translations. Draw polygons and conic sections using these points. Finally, add auxiliary lines and intersection points to complete a full illustration showcasing the shapes and geometric relationships of the cross-sections.
The prompt above was compiled from the original drawing commands and is for reference only.