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圆锥截面演示 by Bernat

圆锥截面演示 by Bernat
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2019-01-24 14:05
函数解析几何立体几何几何作图
①本范例演示圆锥与不同几何元素（直线、平面）相交形成的各种截面图形；②涉及核心数学概念包括圆锥、截面、解析几何、向量运算；③构造过程中运用了Line（直线）、Surface（曲面）、Intersect（相交）、OrthogonalPlane（正交平面）等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者学习圆锥几何性质与截面变换。

GeoGebra 命令及参数

  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