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抛物线段：阿基米德洞察

抛物线段：阿基米德洞察
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2019-05-20 21:40
函数微积分几何作图
①本范例动态演示抛物线段的形成及其与切线、角度的关系；②涉及核心数学概念包括抛物线、切线、角度、直线、向量等；③作图要点：运用了Line, Segment, Slope, Intersect, Polygon, OrthogonalLine, Circle, Dilate, Vector, Translate, Point, Angle, Rotate, Midpoint等GeoGebra命令与对象；④适合高中及以上学生及数学爱好者用于理解抛物线性质及几何作图。

GeoGebra 命令及参数

  1. Line(A, B) → g
  2. Segment(A, B) → j
  3. Slope(g) → d
  4. Intersect(f', h, 1) → C
  5. Polygon(A, D, B) → t1, b_1, a_1, d_1
  6. OrthogonalLine(D, g) → l
  7. Intersect(g, l) → E
  8. Slope(b_1) → m
  9. Slope(a_1) → m_1
  10. Intersect(f', n) → F
  11. Intersect(f', p) → H
  12. Polygon(A, I, D) → t2, d_2, a_2, i_1
  13. Polygon(D, G, B) → t3, b_2, d_3, g_1
  14. Segment(A, D) → q
  15. Segment(D, B) → r
  16. Segment(B, G) → s
  17. Segment(G, D) → t
  18. Segment(A, I) → e
  19. Segment(I, D) → f_1
  20. Line(A, B) → h_1
  21. Circle(A, t) → c_1
  22. Intersect(c_1, j_1, 2) → J
  23. Circle(A, r) → e_1
  24. Circle(J, s) → p_1
  25. Intersect(e_1, p_1, 2) → K
  26. Dilate(I, If[0 < o ≤ 1, 1 - o, 0], J) → I'
  27. Polygon(A, I', D) → t4, d_4, a_3, i'
  28. Segment(A, I') → l_1
  29. Segment(I', D) → n_1
  30. Line(D, l_1) → q_1
  31. Dilate(D, If[3 < o ≤ 4, 4 - o, 0], K) → D'
  32. Polygon(A, I', D') → t5, d', a_4, i'_1
  33. Segment(I', D') → r_1
  34. Segment(D', A) → s_1
  35. Dilate(I', If[5 < o ≤ 6, 6 - o, 0], G) → I''
  36. Vector(I', I'') → u
  37. Translate(A, u) → A'
  38. Translate(I', u) → I''_1
  39. Translate(D', u) → D''
  40. Polygon(A', I''_1, D'') → t5', d'', a', i''_1
  41. Line(I', t) → t_1
  42. Point(t_1) → L
  43. Angle(L, I', A) → α
  44. Rotate(A', If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → A''
  45. Rotate(I''_1, If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → M
  46. Rotate(D'', If[5 < o ≤ 6, ((-(o - 5)) * α), (-α)], I''_1) → D'''
  47. Polygon(A'', M, D''') → t5'', d''', a'', m_2
  48. Midpoint(A, B) → N
  49. Midpoint(N, B) → O
  50. Midpoint(A, N) → P
  51. Midpoint(A, P) → Q
  52. Midpoint(P, N) → R
  53. Midpoint(N, O) → S
  54. Midpoint(O, B) → T
  55. Dilate(A, If[7 < o ≤ 8, 8 - o, 0], Q) → A'_1
  56. Segment(A, A'_1) → f_2
  57. Rotate(A, If[8 < o ≤ 9, ((o - 8) * π), π], Q) → A'_2
  58. Rotate(A'_1, If[8 < o ≤ 9, ((o - 8) * π), π], Q) → U
  59. Segment(A'_2, U) → g_2
  60. Rotate(A'_2, If[9 < o ≤ 10, ((o - 9) * π), π], A'_2) → V
  61. Rotate(U, If[9 < o ≤ 10, ((o - 9) * π), π], A'_2) → U'
  62. Segment(V, U') → h_2
  63. Rotate(V, If[10 < o ≤ 11, ((o - 10) * π), π], U') → V'
  64. Rotate(U', If[10 < o ≤ 11, ((o - 10) * π), π], U') → U''
  65. Segment(V', U'') → i_2
  66. Rotate(V', If[11 < o ≤ 12, ((o - 11) * π), π], V') → V''
  67. Rotate(U'', If[11 < o ≤ 12, ((o - 11) * π), π], V') → U'''
  68. Segment(V'', U''') → j_2
  69. Rotate(V'', If[12 < o ≤ 13, ((o - 12) * π), π], U''') → V'''
  70. Rotate(U''', If[12 < o ≤ 13, ((o - 12) * π), π], U''') → W
  71. Segment(V''', W) → k_2
  72. Rotate(V''', If[13 < o ≤ 14, ((o - 13) * π), π], V''') → Z
  73. Rotate(W, If[13 < o ≤ 14, ((o - 13) * π), π], V''') → W'
  74. Segment(Z, W') → l_2
  75. Rotate(Z, If[14 < o ≤ 15, ((o - 14) * π), π], W') → Z'
  76. Rotate(W', If[14 < o ≤ 15, ((o - 14) * π), π], W') → W''
  77. Segment(Z', W'') → n_2
  78. Circle(D, n_2) → c_2
  79. Intersect(c_2, k_1) → A_1, B_1
  80. Dilate(M, If[16 < o ≤ 17, 17 - o, 0], B_1) → M'
  81. Polygon(D, M', B) → t6, b_3, d_5, m'
  82. Segment(D, M') → p_2
  83. Segment(M', B) → q_2
  84. Midpoint(M', W') → C_1
  85. Line(A, B) → r_2
  86. Circle(A, (7 / 8 * Distance[A, B])) → e_2
  87. Intersect(e_2, r_2) → D_1, E_1
  88. Polygon(A, E_1, D) → t7, d_6, a_5, e_3
  89. Polygon(E_1, B, D) → t8, d_7, e_4, b_4
  90. Midpoint(M', E_1) → F_1
  91. Dilate(D, If[17 < o ≤ 18, 18 - o, 0], E_1) → D'_1
  92. Segment(D, D'_1) → s_2
  93. Rotate(D, If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → D'_2
  94. Rotate(M', If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → M''
  95. Rotate(B, If[19 < o ≤ 20, ((-(o - 19)) * π), (-π)], F_1) → B'
  96. Polygon(D'_2, M'', B') → t6', b', d'_2, m''