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动画175：多几何元素动态构造演示

动画175：多几何元素动态构造演示
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2017-01-23 11:02
多边形向量变换几何作图滑块动画
所属专题 Congruence (CO) Cluster Congruence (Vol 2) CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 CCSS High School: Geometry (Congruence) Volume 2 Gail LeBlancCCSS High School: Geometry (Congruence) Volume 2 Area, Surface Area, Volume, Cross Section, 3D
① 本范例展示了一个包含旋转、平移、缩放等多种变换的复合几何动画，通过滑块控制角度等参数，动态呈现点、线、圆、多边形、向量等元素之间的位置关系变化。② 核心数学概念涉及平面几何中的旋转与平移变换、向量运算、平行四边形的性质、直线与圆的相交、中点与垂线等，同时包含复数与立体几何（圆柱）的初步元素，以及利息计算等跨学科应用背景。③ 作图要点：从构造摘要可见，使用了Rotate（旋转）、Angle（角度）、Ray（射线）、Segment（线段）、Intersect（交点）、Midpoint（中点）、Line（直线）、OrthogonalLine（垂线）、Polygon（多边形）、Point（点）、Circle（圆）、Dilate（缩放）、Min（最小值）、CircleSector（扇形）、Vector（向量）、Translate（平移）等GeoGebra命令，并创建了众多对象如A、B、α、β、A'、γ、f、B'、δ、g、h、C、i、j、D、E、l、m、F、n、poly1、c、d、e、G、p、poly2、b、d_1、e_1、a、q、r、H、t、I、k、s、c_1、f_1，通过滑块驱动角度参数实现动态变化。④ 适合学段：适用于高中及以上学生，用于几何变换、向量与复数综合教学，也可作为数学建模或动态演示的参考案例。

GeoGebra 命令及参数

  1. Rotate(A, α, B) → A'
  2. Angle(A, B, A') → γ
  3. Ray(B, A') → f
  4. Rotate(B, (-β), A) → B'
  5. Angle(B', A, B) → δ
  6. Ray(A, B') → g
  7. Segment(B, A) → h
  8. Intersect(f, g) → C
  9. Segment(B, C) → i
  10. Segment(A, C) → j
  11. Midpoint(i) → D
  12. Midpoint(j) → E
  13. Line(B, A) → l
  14. OrthogonalLine(C, l) → m
  15. Intersect(l, m) → F
  16. Segment(C, F) → n
  17. Polygon(D, E, C) → poly1, c, d, e
  18. Midpoint(n) → G
  19. Line(C, l) → p
  20. Polygon(B, D, E, A) → poly2, b, d_1, e_1, a
  21. Segment(D, C) → q
  22. Segment(B, A) → r
  23. Point(p) → H
  24. OrthogonalLine(H, p) → t
  25. Intersect(l, t) → I
  26. Segment(H, I) → s
  27. Circle(I, 3) → c_1
  28. Circle(H, 3) → f_1
  29. Intersect(f_1, p, 1) → J
  30. Intersect(f_1, p, 2) → K
  31. Intersect(c_1, l, 2) → L
  32. Intersect(c_1, s, 1) → M
  33. Intersect(c_1, l, 1) → N
  34. Angle(J, H, I) → ε
  35. Angle(M, I, N) → ζ
  36. Dilate(H, If[0 < k ≤ 1, 1 - k, 0], I) → H'
  37. Segment(H, H') → g_1
  38. Angle(C, F, B) → η
  39. Dilate(C, If[1 < k ≤ 2, 2 - k, 0], G) → C'
  40. Segment(C, C') → h_1
  41. Dilate(G, If[2 < k ≤ 3, 3 - k, 0], F) → G'
  42. Segment(G, G') → i_1
  43. Rotate(C, If[3 < k ≤ 4, ((-(k - 3)) * π), (-π)], G) → C'_1
  44. Rotate(C', If[3 < k ≤ 4, ((-(k - 3)) * π), (-π)], G) → C''
  45. Segment(C'_1, C'') → j_1
  46. Dilate(G, If[4 < k ≤ 5, 5 - k, 0], D) → G'_1
  47. Dilate(G, If[4 < k ≤ 5, 5 - k, 0], E) → G'_2
  48. Segment(G, G'_1) → k_1
  49. Segment(G, G'_2) → l_1
  50. Segment(B, F) → m_1
  51. Segment(B, D) → n_1
  52. Segment(D, G) → p_1
  53. Segment(D, C) → q_1
  54. Min({m_1, n_1, p_1, q_1}) → o
  55. Circle(B, (0.6 * o)) → r_1
  56. Intersect(r_1, l) → O, P
  57. Segment(B, C) → s_1
  58. Intersect(r_1, s_1, 1) → Q
  59. CircleSector(B, P, Q) → t_1
  60. Rotate(D, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → D'
  61. Rotate(E, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → E'
  62. Rotate(C, If[6 < k ≤ 7, ((-(k - 6)) * π), (-π)], E) → C'_2
  63. Polygon(D', E', C'_2) → poly1', c'_2, d', e'
  64. Segment(D, C) → a_1
  65. Segment(D, B) → b_1
  66. Segment(C, E) → f_2
  67. Segment(E, A) → g_2
  68. Segment(E, C'_2) → h_2
  69. Segment(C'_2, D') → i_2
  70. Segment(D', E) → j_2
  71. Dilate(D, If[7 < k ≤ 8, 8 - k, 0], C) → D'_1
  72. Segment(D, D'_1) → k_2
  73. Dilate(B, If[8 < k ≤ 9, 9 - k, 0], D) → B'_1
  74. Vector(B, B'_1) → u
  75. Translate(t_1, u) → c_2
  76. Dilate(B, If[9 < k ≤ 10, 10 - k, 0], A) → B'_2
  77. Vector(B, B'_2) → v
  78. Translate(t_1, v) → d_2
  79. Circle(A, Radius[r_1]) → e_2
  80. Intersect(e_2, l, 1) → R
  81. Intersect(e_2, g_2, 1) → S
  82. Segment(B, A) → l_2
  83. Segment(A, C) → m_2
  84. Min({l_2, m_2}) → w
  85. Circle(A, (0.25 * w)) → p_2
  86. Intersect(g_2, m_2) → T
  87. Intersect(g_2, m_2) → U
  88. Intersect(p_2, m_2) → V,
  89. Intersect(p_2, l_2) → W,
  90. CircleSector(A, V, W) → q_2