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动画演示几何变换与函数图像

动画演示几何变换与函数图像
几何画板及绘图内容加载中，请稍等……
2017-04-21 01:38
向量函数三角函数解析几何几何作图
①本范例通过动画演示几何图形的旋转、平移与缩放变换，结合函数图像与三角函数关系；②涉及核心数学概念包括角度、向量、几何变换、三角函数、解析几何等；③构造过程中运用了 Rotate, Ray, Point, Segment, Midpoint, Polygon, Angle, Circle, Intersect, Line, CircleSector, Dilate, Vector, Translate 等GeoGebra命令与对象；④适合高中及以上数学学习者，用于理解几何变换与函数图像的关联，提升空间想象与逻辑推理能力。

GeoGebra 命令及参数

  1. Rotate(A, (a * π / 180), B) → A'
  2. Ray(B, A') → g
  3. Point(g) → C
  4. Rotate(A, If[0 < u ≤ 1, (u * π / 2), π / 2], C) → A'_1
  5. Rotate(C, If[0 < u ≤ 1, (u * π / 2), π / 2], C) → C'
  6. Segment(C', A'_1) → i'
  7. Midpoint(i') → V
  8. Polygon(C, A, 4) → poly1, j, k, l, m, D, E
  9. Angle(A, B, A') → α
  10. Segment(A, B) → f
  11. Segment(B, C) → h
  12. Segment(C, A) → i
  13. Polygon(B, C, 4) → poly2, n, p, q, r, F, G
  14. Angle(poly2) → β, γ, δ, ε
  15. Angle(poly1) → ζ, η, θ, ι
  16. Segment(C, E) → s
  17. Segment(E, D) → t
  18. Segment(D, A) → b
  19. Segment(B, G) → c
  20. Segment(G, F) → d
  21. Segment(F, C) → e
  22. Min(i, h) → o
  23. Circle(E, (0.15 * o)) → c_1
  24. Polygon(B, A, C) → poly3, c_2, b_1, a_1
  25. Intersect(c_1, s, 1) → H
  26. Intersect(c_1, t, 1) → I
  27. Line(B, E) → f_1
  28. Line(A, F) → g_1
  29. Line(G, D) → h_1
  30. Intersect(g_1, f_1) → J
  31. Circle(A, (0.15 * o)) → d_1
  32. Intersect(d_1, b, 1) → K
  33. Intersect(d_1, i, 1) → L
  34. CircleSector(A, K, L) → e_1
  35. Rotate(B, If[0 < u ≤ 1, (u * π / 2), π / 2], B) → B'
  36. Rotate(C, If[0 < u ≤ 1, (u * π / 2), π / 2], B) → C'_1
  37. Segment(B', C'_1) → h'
  38. Rotate(B', If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], G) → B''
  39. Rotate(C'_1, If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], G) → M
  40. Segment(B'', M) → h''
  41. Rotate(C', If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], E) → C''
  42. Rotate(A'_1, If[1 < u ≤ 2, ((u - 1) * π / 2), π / 2], E) → N
  43. Segment(C'', N) → i''
  44. Rotate(C'', If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], D) → C'''
  45. Rotate(N, If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], D) → N'
  46. Segment(C''', N') → i'''
  47. Rotate(B'', If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], F) → B'''
  48. Rotate(M, If[2 < u ≤ 3, ((u - 2) * π / 2), π / 2], F) → M'
  49. Segment(B''', M') → h'''
  50. Dilate(A, If[3.5 < u ≤ 4.5, 4.5 - u, 0], D) → A'_2
  51. Vector(A, A'_2) → v
  52. Translate(e_1, v) → k_1
  53. Dilate(A, If[3.5 < u ≤ 4.5, 4.5 - u, 0], C) → A'_3
  54. Vector(A, A'_3) → w
  55. Translate(e_1, w) → p_1
  56. Line(C, A) → i_1
  57. Point(i_1) → O
  58. Angle(O, C, F) → κ
  59. Rotate(p_1, If[4.5 < u ≤ 5.5, ((u - 4.5) * (π / 2 + κ)), π / 2 + κ], C) → q_1
  60. Rotate(k_1, If[4.5 < u ≤ 5.5, ((u - 4.5) * π / 2), π / 2], D) → r_1
  61. Dilate(D, If[5.5 < u ≤ 6.5, 6.5 - u, 0], E) → D'
  62. Dilate(C, If[5.5 < u ≤ 6.5, 6.5 - u, 0], F) → C'_2
  63. Vector(C, C'_2) → u_1
  64. Vector(D, D') → v_1
  65. Translate(r_1, v_1) → s_1
  66. Translate(q_1, u_1) → t_1
  67. Rotate(s_1, If[6.5 < u ≤ 7.5, ((u - 6.5) * π / 2), π / 2], E) → d_2
  68. Rotate(t_1, If[6.5 < u ≤ 7.5, ((u - 6.5) * π / 2), π / 2], F) → e_2
  69. Dilate(F, If[7.5 < u ≤ 8.5, 8.5 - u, 0], G) → F'
  70. Dilate(E, If[7.5 < u ≤ 8.5, 8.5 - u, 0], V) → E'
  71. Vector(E, E') → w_1
  72. Vector(F, F') → j_1
  73. Translate(d_2, w_1) → f_2
  74. Translate(e_2, j_1) → g_2
  75. Rotate(g_2, If[8.5 < u ≤ 9.5, ((u - 8.5) * π / 2), π / 2], G) → h_2
  76. Circle(G, 2) → p_2
  77. Circle(D, 2) → q_2
  78. Circle(F, 2) → r_2
  79. Intersect(r_2, g_1, 2) → P
  80. Intersect(p_2, h_1, 1) → Q
  81. Intersect(d_1, g_1, 1) → R
  82. Intersect(q_2, h_1, 2) → S
  83. Ray(D, S) → l_1
  84. Ray(G, Q) → m_1
  85. Ray(F, P) → n_1
  86. Ray(A, R) → i_2
  87. Circle(B, 2) → s_2
  88. Intersect(s_2, f_1, 1) → T
  89. Intersect(c_1, f_1, 2) → U
  90. Ray(B, T) → j_2
  91. Ray(E, U) → l_2
  92. Rotate(f_2, If[8.5 < u ≤ 9.5, ((u - 8.5) * π / 2), π / 2], V) → k_2
  93. Dilate(V, If[9.5 < u ≤ 10.5, 10.5 - u, 0], C) → V'
  94. Vector(V, V') → u_2
  95. Translate(k_2, u_2) → t_2
  96. Dilate(B, If[11 < u ≤ 12, 12 - u, 0], E) → B'_1
  97. Segment(B, B'_1) → m_2
  98. Dilate(A, If[12 < u ≤ 13, 13 - u, 0], J) → A'_4
  99. Segment(A, A'_4) → n_2
  100. Dilate(J, If[13 < u ≤ 14, 14 - u, 0], F) → J'
  101. Segment(J, J') → a_2
  102. Dilate(G, If[14 < u ≤ 15, 15 - u, 0], J) → G'
  103. Segment(G, G') → b_2
  104. Circle(J, (0.15 * o)) → c_3
  105. Intersect(c_3, g_1, 1) → W
  106. Intersect(c_3, h_1, 2) → Z
  107. Sector(c_3, W, Z) → d_3
  108. Dilate(JJ, If[15 < u ≤ 16, 16 - u, 0], D) → JJ'
  109. Segment(JJ, JJ') → f_3
  110. Line(G, n_2) → g_3
  111. Point(g_3) → A_1
  112. Line(G, j_2) → h_3
  113. Angle(F, G, A_1) → λ
  114. Rotate(h_2, If[16 < u ≤ 17, ((u - 16) * λ), λ], G) → e_3
  115. Dilate(G, If[17 < u ≤ 18, 18 - u, 0], JJ) → G'_1
  116. Vector(G, G'_1) → v_2
  117. Translate(e_3, v_2) → k_3
  118. Angle(A, JJ, D) → μ
  119. Rotate(W, If[18 < u ≤ 19, ((u - 18) * π / 4), π / 4], JJ) → W'
  120. CircleSector(JJ, W, W') → p_3
  121. Rotate(p_3, If[19.5 < u ≤ 20.5, ((u - 19.5) * π / 4), π / 4], JJ) → q_3
  122. Circle(B, (0.15 * o)) → r_3
  123. Intersect(r_3, f, 1) → B_1
  124. Intersect(r_3, h, 1) → C_1
  125. CircleSector(B, B_1, C_1) → s_3