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Arsalan盒子：折纸盒的几何构造与动态演示

Arsalan盒子：折纸盒的几何构造与动态演示
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2020-05-19 13:37
多边形立体几何变换几何作图滑块动画
该范例演示了如何通过GeoGebra的几何作图工具，从平面图形逐步折叠出一个三维盒子的过程，直观展示折纸盒的立体几何形态。核心数学概念包括点、线段、多边形、垂线、圆、角平分线、中点、距离测量以及旋转和变换等，涉及平面几何与立体几何的衔接。作图要点包括：使用Point、Segment、OrthogonalLine、Circle、Intersect、Polygon、LineBisector、Distance、If、Rotate、Line、Midpoint等命令，构造了多个辅助对象（如a、b、A、B、f、g、c、C、h、i、D、q1等），并通过变量xvar控制折叠角度，利用距离变量（distanzaAE、distanzaDJ、distanzaJH）动态调整盒子的尺寸与形状。适合中学高年级或大学低年级学生学习几何作图、变换与空间想象，也可用于数学实验或折纸几何的探究活动。

GeoGebra 命令及参数

  1. Point(Circle[A, b]) → B
  2. Segment(A, B) → f
  3. OrthogonalLine(B, f) → g
  4. Circle(B, a) → c
  5. Intersect(c, g, 2) → C
  6. OrthogonalLine(C, g) → h
  7. OrthogonalLine(A, f) → i
  8. Intersect(h, i) → D
  9. Polygon(A, B, C, D) → q1, a_1, b_1, c_1, d
  10. Circle(D, xvar) → e
  11. Intersect(f, a_1) → F
  12. Circle(A, xvar) → k
  13. Intersect(k, d, 1) → E
  14. OrthogonalLine(E, d) → j
  15. Intersect(j, b_1) → G
  16. Segment(E, G) → l
  17. LineBisector(D, E) → m
  18. Intersect(m, d) → H
  19. Intersect(m, b_1) → I
  20. Segment(H, I) → n
  21. Circle(D, xvar) → p
  22. Intersect(p, d, 1) → J
  23. OrthogonalLine(J, d) → q
  24. Intersect(q, b_1) → K
  25. Segment(J, K) → r
  26. LineBisector(J, A) → s
  27. Intersect(d, s) → L
  28. Intersect(s, b_1) → M
  29. Segment(L, M) → t
  30. Distance(A, E) → distanzaAE
  31. Distance(D, J) → distanzaDJ
  32. Distance(J, H) → distanzaJH
  33. Distance(H, L) → distanzaHL
  34. Distance(L, E) → distanzaLE
  35. Circle(C, b - a + xvar) → d_1
  36. Intersect(d_1, c_1, 1) → N
  37. OrthogonalLine(N, c_1) → f_1
  38. Intersect(f_1, f) → O
  39. Segment(N, O) → g_1
  40. LineBisector(D, N) → h_1
  41. Intersect(h_1, c_1) → P
  42. Intersect(h_1, f) → Q
  43. Segment(P, Q) → i_1
  44. Circle(D, b - a + xvar) → e_1
  45. Intersect(e_1, c_1, 1) → R
  46. OrthogonalLine(R, c_1) → j_1
  47. Intersect(j_1, f) → S_2
  48. LineBisector(R, C) → k_1
  49. Intersect(k_1, c_1) → T
  50. Intersect(k_1, f) → U
  51. Segment(T, U) → l_1
  52. Segment(R, S_2) → m_1
  53. Distance(D, P) → distanzaDP
  54. Distance(H, E) → distanzaHE
  55. Distance(J, L) → distanzaJL
  56. Distance(T, C) → distanzaTC
  57. Distance(D, R) → distanzaDR
  58. Intersect(n, i_1) → V_1
  59. Intersect(i_1, t) → W
  60. Intersect(t, l_1) → Z
  61. Intersect(l_1, n) → A_1
  62. LineBisector(T, C) → n_1
  63. LineBisector(D, P) → p_1
  64. LineBisector(D, H) → q_1
  65. LineBisector(L, A) → r_1
  66. If(xvar < a, Polygon[W, Z, A_1, V_1], q1) → q2
  67. Intersect(q_1, d) → B_1
  68. Intersect(q_1, b_1) → C_1
  69. Intersect(r_1, d) → D_1
  70. Intersect(b_1, r_1) → E_1
  71. Intersect(p_1, c_1) → F_1
  72. Intersect(p_1, f) → G_1
  73. Intersect(n_1, c_1) → H_1
  74. Intersect(n_1, f) → I_1
  75. Segment(B_1, C_1) → t_1
  76. Segment(D_1, E_1) → f_2
  77. Segment(F_1, G_1) → g_2
  78. Segment(H_1, I_1) → h_2
  79. Intersect(n, g_2) → J_1
  80. Intersect(t, g_2) → K_1
  81. Intersect(i_1, f_2) → L_1
  82. Intersect(l_1, f_2) → M_1
  83. Intersect(t, q1) → N_1, N_2
  84. Intersect(h_2, t) → O_1
  85. Intersect(n, h_2) → P_1
  86. Intersect(l_1, t_1) → Q_1
  87. Intersect(i_1, t_1) → R_1
  88. Polygon(L_1, M_1, Z, W) → q3, i_2, j_2, z_2, w_1
  89. Polygon(Z, O_1, P_1, A_1) → q4, z_3, o_1, k_2, l_2
  90. Polygon(A_1, Q_1, R_1, V_1) → q5, m_2, n_2, p_2, v_3
  91. Polygon(V_1, J_1, K_1, W) → q6, v_2, q_2, r_2, w_2
  92. Segment(A_1, C) → s_2
  93. Segment(Z, B) → t_2
  94. Segment(W, A) → a_2
  95. Segment(V_1, D) → b_2
  96. Distance(A_1, P_1) → distanzaA1P1
  97. Rotate(A_1, (-(90°)), m) → S_1
  98. Rotate(Q_1, (-(90°)), m) → T_1
  99. Rotate(R_1, (-(90°)), m) → U_1
  100. Rotate(V_1, (-(90°)), m) → W_1
  101. Polygon(S_1, T_1, U_1, W_1) → q5', c_2, d_2, u_1, e_2
  102. Rotate(L_1, 90°, s) → Z_1
  103. Rotate(M_1, 90°, s) → A_2
  104. Rotate(Z, 90°, s) → Z'
  105. Rotate(W, 90°, s) → W'
  106. Polygon(Z_1, A_2, Z', W') → q3', g_3, h_3, z', w'
  107. Line(P, Q) → i_3
  108. Line(T, U) → j_3
  109. Rotate(V_1, (-(90°)), i_3) → B_2
  110. Rotate(J_1, (-(90°)), i_3) → C_2
  111. Rotate(K_1, (-(90°)), i_3) → D_2
  112. Rotate(W, (-(90°)), i_3) → W'_1
  113. Polygon(B_2, C_2, D_2, W'_1) → q6', k_3, l_3, m_3, w'_1
  114. Rotate(Z, 90°, j_3) → Z'_1
  115. Rotate(O_1, 90°, j_3) → E_2
  116. Rotate(P_1, 90°, j_3) → F_2
  117. Rotate(A_1, 90°, j_3) → G_2
  118. Polygon(Z'_1, E_2, F_2, G_2) → q4', z'_1, n_3, p_3, q_3
  119. Midpoint(V_1, A_1) → H_2
  120. Midpoint(V_1, W) → I_2
  121. Midpoint(A_1, P_1) → J_2