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Conic Sections Demo by Bernat

Conic Sections Demo by Bernat
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2019-01-24 14:05
FunctionCoordinate GeometrySolid GeometryGeometric Construction
This example demonstrates various conic sections formed by intersections between a cone and different geometric elements like lines and planes. It involves core concepts such as cones, sections, coordinate geometry, and vector operations. GeoGebra commands like Line, Surface, Intersect, and OrthogonalPlane are used in construction. Suitable for high school students and math enthusiasts exploring conic geometry and section transformations.

GeoGebra Commands & Parameters

  1. Line(A, B) → f
  2. OrthogonalPlane(A, f) → a
  3. OrthogonalVector(a) → u
  4. PointIn(a) → D
  5. Surface(((m * u) * cos(v)), ((m * u) * sin(v)), u, u, -10, 10, v, 0, (2 * π)) → con
  6. Intersect(ff, gg, 0, (2 * π)) → G, K
  7. CurveCartesian(con(ff(t), t), t, 0, x(G)) → b_1
  8. Mirror(G, q) → G'
  9. CurveCartesian(con(ff(t), t), t, x(G'), (2 * π)) → b_2
  10. CurveCartesian(con(10, t), t, 0, (2 * π)) → n
  11. Line(C, J) → g
  12. CurveCartesian(con(10, t), t, 0, x(G)) → n_1
  13. Mirror(G, g) → F
  14. CurveCartesian(con(ff(t), t), t, 0, (2 * π)) → b
  15. InfiniteCone((0, 0, 0), Vector[(0, 0, 1)], α) → c
  16. Line((0, 0, 0), (0, 0, 1)) → h
  17. Circle(A, 1, Vector[(0, 1, 0)]) → d
  18. If(β ≥ π / 2, H, I) → P_b
  19. IntersectPath(a, c) → k_1
  20. CurveCartesian(con(z(P_b), t), t, 0, (2 * π)) → j
  21. Surface(con((k * ff(t)) + ((1 - k) * z(P_b)), t), k, 0, 1, t, 0, (2 * π)) → i
  22. Surface((k * b(t)) + ((1 - k) * b(t + π)), k, 0, 1, t, 0, π) → l
  23. Circle(zAxis, P_b) → p
  24. Surface(((m * u) * cos(v)), ((m * u) * sin(v)), u, u, 0, z(P_b), v, 0, (2 * π)) → con2
  25. Intersect(ff, gg, (-92.56337323042047, 9.999999999999849)) → E
  26. If(z(P_b) < 10, x(G), (2 * π)) → lim
  27. Surface(con((k * ff(t)) + ((1 - k) * 10), t), k, 0, 1, t, 0, x(G)) → i_1
  28. Surface(con((k * z(P_b)) + ((1 - k) * 10), t), k, 0, 1, t, 0, (2 * π)) → t
  29. Midpoint(P_1, P_2) → P_3
  30. CurveCartesian(P_1 + (k * (P_3 - P_1)), k, 0, 1) → n_3
  31. CurveCartesian(P_3 + (k * (P_2 - P_3)), k, 0, 1) → n_4
  32. Surface((k * n_1(t)) + ((1 - k) * n_3((aux_2 * t) + 1)), k, 0, 1, t, 0, x(G)) → v
  33. Surface((k * n(t)) + ((1 - k) * n(t + π)), k, 0, 1, t, 0, π) → w
  34. CurveCartesian(con(10, t), t, x(G), x(G')) → n_5
  35. CurveCartesian(P_1 + (k * (P_2 - P_1)), k, 0, 1) → n_6
  36. Circle(zAxis, I) → p'
  37. Cone(p', (-z(I))) → con3, c_1, d_1
  38. Line(P_4, P_3) → f_1
  39. CurveCartesian(con(z(I), t), t, 0, x(G)) → j'_1
  40. CurveCartesian(con(z(I), t), t, x(F), (2 * π)) → j'_2
  41. Surface(((m * u) * cos(v)), ((m * u) * sin(v)), u, u, 0, z(I), v, 0, (2 * π)) → con4
  42. Line(P_3, P_4) → f'_1
  43. CurveCartesian(con(z(I), t), t, x(G), x(G')) → j'_3
  44. Surface((k * b_1(t)) + ((1 - k) * j'_1(t)), k, 0, 1, t, 0, x(G)) → v_2
  45. Translate(i_1, Vector[(k * e)]) → t_1
  46. Translate(n_1, Vector[(k * e)]) → ct_3
  47. Mirror(n_1, g) → n_2
  48. Translate(n_2, Vector[(k * e)]) → ct_4
  49. Translate(n_3, Vector[(k * e)]) → ct_5
  50. Translate(b_1, Vector[(k * e)]) → ct_1
  51. Translate(n_4, Vector[(k * e)]) → ct_6
  52. Translate(t, Vector[(k * e)]) → t'_2
  53. Translate(i, Vector[(k * e)]) → t'_3
  54. Translate(w, Vector[(k * e)]) → t'_4
  55. Translate(n, Vector[(k * e)]) → ct'_1
  56. Translate(b, Vector[(k * e)]) → ct'_2
  57. Translate(l, Vector[(k * e)]) → t'_5
  58. Surface(con((k * ff(t)) + ((1 - k) * 10), t), k, 0, 1, t, x(G'), (2 * π)) → i_2
  59. Surface((k * n_5(t)) + ((1 - k) * n_6((aux_3 * t) + aux_4)), k, 0, 1, t, x(G), x(G')) → v_1
  60. Surface((k * n_5(t)) + ((1 - k) * j'_3(t)), k, 0, 1, t, x(G), x(G')) → v_4
  61. Surface((k * b_1(t)) + ((1 - k) * b_2((aux_1 * t) + (2 * π))), k, 0, 1, t, 0, x(G)) → l_1
  62. Surface((k * b_2(t)) + ((1 - k) * j'_2(t)), k, 0, 1, t, x(G'), (2 * π)) → v_3
  63. Translate(b_2, Vector[(k * e)]) → ct_2
  64. CurveCartesian(con(z(I), t), t, 0, (2 * π)) → j'
  65. Surface((k * b(t)) + ((1 - k) * j'(t)), k, 0, 1, t, 0, (2 * π)) → o
  66. Translate(l_1, Vector[(k * e)]) → t_2
  67. Mirror(v, f_1) → v'
  68. Translate(v, Vector[(k * e)]) → t_3
  69. Translate(v', Vector[(k * e)]) → t_4
  70. Translate(i_2, Vector[(k * e)]) → t_5
  71. IntersectPath(a, c) → e_1
  72. Focus(e_1) → L, M
  73. Axes(e_1) → g_1, h_1