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动态几何图形演示

动态几何图形演示
几何画板及绘图内容加载中，请稍等……
2018-07-26 21:27
向量解析几何变换几何作图
①本范例动态演示了由 Antonio Gutierrez 在 Twitter 上提出的一个几何问题，通过可调节的参数展示图形变换过程；②涉及的核心数学概念包括几何变换（旋转、缩放）、解析几何、向量与轨迹；③作图要点提炼自 GeoGebra 命令：Rotate（旋转）、Dilate（缩放）、Polygon（多边形）、Midpoint（中点）、CircleArc（圆弧）、Point（点）、Segment（线段）、Intersect（交点）、Circle（圆）、Vector（向量）、Line（直线）、Angle（角度）、Mirror（镜像）、CircleSector（圆扇形）、Ray（射线）、OrthogonalLine（垂线）、Translate（平移）；④适合高中及以上学生及数学爱好者用于理解几何变换与动态几何的原理，可作为教学演示或探究工具。

GeoGebra 命令及参数

  1. Rotate(A, If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A'_1
  2. Dilate(A, If[0 < e ≤ 1, 1 - e, 0], B_2) → A'
  3. Rotate(A', If[1 < e ≤ 2, ((-(e - 1)) * π / 2), -1.5707963267948966], B_2) → A''
  4. Polygon(A, B_2, 4) → poly1, a, b, c, d, C, D
  5. Midpoint(A, C) → V
  6. CircleArc(V, C, D) → g_2
  7. Point(g_2) → B
  8. Segment(C, B) → j_1
  9. Segment(A, B) → n
  10. Rotate(A'', If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → A'''
  11. Rotate(A'_1, If[2 < e ≤ 3, ((-(e - 2)) * π / 2), -1.5707963267948966], C) → R
  12. Segment(R, A''') → f''
  13. Intersect(n, f'') → Z
  14. Segment(B, Z) → l_1
  15. Segment(A, A') → f
  16. Segment(A'_1, A'') → f'
  17. Segment(A, B_2) → g
  18. Segment(B_2, C) → h
  19. Segment(C, D) → i
  20. Segment(D, A) → j
  21. Circle(B_2, (0.15 * f)) → p
  22. Circle(C, Radius[p]) → k
  23. Circle(A, (0.1 * j)) → q
  24. Circle(D, (0.1 * j)) → r
  25. Segment(A, C) → l
  26. Segment(D, B_2) → m
  27. Intersect(r, j, 1) → F
  28. Intersect(r, m, 1) → G
  29. Intersect(r, i, 1) → H
  30. Intersect(k, i, 1) → I
  31. Intersect(k, l, 1) → J
  32. Intersect(k, h, 1) → K
  33. Intersect(p, h, 1) → L
  34. Intersect(p, m, 1) → M
  35. Intersect(p, g, 1) → N
  36. Intersect(q, g, 1) → O
  37. Intersect(q, l, 1) → P
  38. Intersect(q, j, 1) → Q
  39. Rotate(R, If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → R'
  40. Rotate(A''', If[3 < e ≤ 4, ((-(e - 3)) * π / 2), -1.5707963267948966], D) → S
  41. Segment(R', S) → f'''
  42. Dilate(B_2, If[4 < e ≤ 5, 5 - e, 0], C) → B'
  43. Vector(B_2, B') → u
  44. Dilate(C, If[6 < e ≤ 7, 7 - e, 0], D) → C'
  45. Vector(C, C') → v
  46. Dilate(D, If[8 < e ≤ 9, 9 - e, 0], A) → D'
  47. Vector(D, D') → w
  48. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], C) → B'_1
  49. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], A) → B'_2
  50. Dilate(B_2, If[18 < e ≤ 19, 19 - e, 0], D) → B'_3
  51. Vector(B_2, B'_1) → v_1
  52. Vector(B_2, B'_3) → w_1
  53. Vector(B_2, B'_2) → i_1
  54. Circle(A, n - j_1) → h_2
  55. Dilate(B, If[10 < e ≤ 11, 11 - e, 0], C) → W'
  56. Dilate(B_2, If[13 < e ≤ 14, 14 - e, 0], B) → B'_4
  57. Vector(B_2, B'_4) → u_1
  58. Line(B, f') → a_1
  59. Point(a_1) → U
  60. Angle(U, B, C) → α
  61. Rotate(B, If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W'_2
  62. Rotate(W', If[14 < e ≤ 15, ((-(e - 14)) * π / 2), ((-π)) / 2], B) → W''
  63. Line(A, B_2) → h_1
  64. Angle(B, A, D) → β
  65. Dilate(D, If[19 < e ≤ 20, 20 - e, 0], B) → D'_1
  66. Mirror(C, D) → C'_1
  67. Rotate(C, ((-π)) / 3, B_2) → C'_2
  68. Segment(B_2, C'_2) → f_1
  69. Segment(C, C'_2) → g_1
  70. Intersect(k, g_1, 1) → E
  71. Intersect(k, h) → T,
  72. CircleSector(C, T, E) → d_1
  73. Rotate(D, 15°, C) → D'_2
  74. Angle(D, C, D'_2) → γ
  75. Ray(C, D'_2) → k_1
  76. Rotate(C, (-(15°)), D) → C'_3
  77. Angle(C'_3, D, C) → δ
  78. Ray(D, C'_3) → m_1
  79. Intersect(k_1, m_1) → W
  80. Rotate(B_2, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → A_1
  81. Rotate(C, If[10 < e ≤ 11, (((-π)) / 3 * (e - 10)), ((-π)) / 3], B_2) → C'_4
  82. Rotate(B_2, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → B_1
  83. Rotate(C, If[11 < e ≤ 12, (π / 3 * (e - 11)), π / 3], C) → C'_5
  84. Rotate(E, If[12 < e ≤ 13, (((-π)) / 6 * (e - 12)), ((-π)) / 6], C) → E'
  85. Polygon(C, W, D) → poly2, d_2, c_2, w_2
  86. Polygon(A, W, D) → poly3, d_3, a_2, w_3
  87. Polygon(B_2, W, C) → poly4, c_3, b_2, w_4
  88. Polygon(A, B_2, W) → poly5, w_5, a_3, b_1
  89. Rotate(T, 15°, C) → T'
  90. Angle(T, C, T') → ε
  91. Ray(C, T') → q_1
  92. Dilate(C, If[21 < e ≤ 22, 22 - e, 0], D) → C'_8
  93. Vector(C, C'_8) → u_2
  94. Dilate(D, If[23 < e ≤ 24, 24 - e, 0], W) → D'_3
  95. Dilate(C, If[23 < e ≤ 24, 24 - e, 0], W) → C'_9
  96. Dilate(B_2, If[25 < e ≤ 26, 26 - e, 0], W) → G_1
  97. Dilate(A, If[25 < e ≤ 26, 26 - e, 0], W) → A'_2
  98. Rotate(B_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → H_1
  99. Rotate(C'_2, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → I_1
  100. Rotate(C, If[26.75 < e ≤ 27.75, (π / 2 * (e - 26.75)), π / 2], B_2) → C'_{10}
  101. Midpoint(D, R) → D_1
  102. Rotate(A, ((-π)) / 2, B_2) → A'_3
  103. Rotate(B_2, π / 2, A) → E_1
  104. Midpoint(A'_3, E_1) → F_1
  105. Segment(F_1, A) → n_1
  106. Segment(B_2, F_1) → r_1
  107. Point(a) → M_3
  108. Angle(A'_1, B_2, A) → ζ
  109. Angle(D, A'_1, B_2) → η
  110. Angle(B_2, A, D) → θ
  111. Angle(M_3, D, A'_1) → κ
  112. Rotate(A'_1, (-(κ / 2)), D) → N_3
  113. Angle(N_3, D, A'_1) → ι
  114. Ray(D, N_3) → s
  115. Line(A'_1, B_2) → t
  116. Intersect(s, t) → O_3
  117. Line(D, M_3) → n_4
  118. OrthogonalLine(O_3, n_4) → p_4
  119. Intersect(n_4, p_4) → P_3
  120. Angle(O_3, P_3, D) → λ
  121. Rotate(B_2, 45°, A) → Q_3
  122. Angle(B_2, A, Q_3) → μ
  123. Ray(A, Q_3) → q_4
  124. Circle(A, a / 6) → c_4
  125. Intersect(c_4, f) → R_3,
  126. Ray(D, O_3) → r_4
  127. Ray(A, P_3) → s_4
  128. Intersect(r_4, s_4) → S_3
  129. Circle(D, a / 6) → e_4
  130. Intersect(e_4, n_4, 2) → T_3
  131. Dilate(D, If[6 < e ≤ 7, 7 - e, 0], M_3) → D'_4
  132. Segment(D, D'_4) → t_4
  133. Polygon(A, B_2, 4) → poly6, p_1, s_1, t_1, c_1, C_1, J_1
  134. Dilate(D, If[7 < e ≤ 8, 8 - e, 0], O_3) → D'_5
  135. Dilate(D, If[7 < e ≤ 8, 8 - e, 0], O_3) → D'_6
  136. Segment(D, D'_6) → e_1
  137. Rotate(T_3, If[8 < e ≤ 9, ((e - 8) * κ / 2), κ / 2], D) → K_1
  138. CircleSector(D, T_3, K_1) → e_2
  139. Rotate(e_2, If[10 < e ≤ 11, ((e - 10) * κ / 2), κ / 2], D) → f_2
  140. Dilate(O_3, If[11 < e ≤ 12, 12 - e, 0], P_3) → L_1
  141. Segment(O_3, L_1) → i_2
  142. Dilate(A, If[13 < e ≤ 14, 14 - e, 0], P_3) → A'_4
  143. Segment(A, A'_4) → j_2
  144. Dilate(P_3, If[14 < e ≤ 15, 15 - e, 0], S_3) → M_1
  145. Dilate(O_3, If[14 < e ≤ 15, 15 - e, 0], S_3) → N_1
  146. Segment(P_3, M_1) → k_2
  147. Segment(O_3, N_1) → l_2
  148. Line(A, A'_1) → m_2
  149. Circle(A'_1, a / 6) → p_2
  150. Intersect(p_2, f'', 1) → O_1
  151. Circle(A'_1, a / 3) → q_2
  152. Intersect(q_2, m_2, 1) → P_1
  153. Rotate(O_1, If[16 < e ≤ 17, ((e - 16) * π / 4), π / 4], A'_1) → Q_1
  154. CircleSector(A'_1, O_1, Q_1) → r_2
  155. Dilate(A'_1, If[16 < e ≤ 17, 17 - e, 0], P_1) → R_1
  156. Segment(A'_1, R_1) → n_2
  157. Rotate(r_2, If[18 < e ≤ 19, ((e - 18) * π / 4), π / 4], A'_1) → s_2
  158. Dilate(A'_1, If[20 < e ≤ 21, 21 - e, 0], S_3) → S_1
  159. Vector(A'_1, S_1) → v_2
  160. Translate(s_2, v_2) → t_2
  161. Line(A'_1, e_1) → f_3
  162. Point(f_3) → T_1
  163. Angle(T_1, A'_1, R_1) → τ
  164. Rotate(t_2, If[20 < e ≤ 21, ((e - 20) * (-τ)), (-τ)], S_1) → e_3