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蒙日投影下的立方体

蒙日投影下的立方体
几何画板及绘图内容加载中，请稍等……
2012-09-15 19:33
解析几何立体几何变换几何作图
① 本范例演示如何利用蒙日投影（正交投影）得到立方体的俯视图与主视图，并已知某一面的平面、中心与顶点。② 涉及的核心数学概念包括立体几何中的正交投影、描述几何的蒙日投影原理、空间向量与平面的相对位置、以及投影变换的几何性质。③ 作图要点：使用 Point、Ray、OrthogonalLine、Segment、Mirror、Intersect、Line、Circle、CircleSector、CircleArc 等命令，构造点 x12、A、B、β2、S2、A2、b、c、S1、A1、d、e、C1、a、C2、f、a4、g、h、i、So、Ao、k、p、p3、A0、A02、r3、j、q3、S0、s3、F、G、c4、t3、l、C0、n、q 等对象，通过射线、垂线、线段、镜像、相交、圆弧等步骤完成两个投影面的构建。④ 适合高中立体几何、大学描述几何课程教学演示，也可作为几何作图训练与投影变换理解的教学素材。

GeoGebra 命令及参数

  1. Point(x_{12}) → B
  2. Ray(B, A) → β_2
  3. Point(β_2) → S_2
  4. Point(β_2) → A_2
  5. OrthogonalLine(S_2, x_{12}) → b
  6. OrthogonalLine(A_2, x_{12}) → c
  7. Point(b) → S_1
  8. Point(c) → A_1
  9. Segment(S_2, S_1) → d
  10. Segment(A_2, A_1) → e
  11. Mirror(A_1, S_1) → C_1
  12. OrthogonalLine(C_1, x_{12}) → a
  13. Intersect(a, β_2) → C_2
  14. Segment(C_1, C_2) → f
  15. Segment(A_1, C_1) → a_4
  16. Line(S_1, x_{12}) → g
  17. Line(A_1, x_{12}) → h
  18. OrthogonalLine(B, x_{12}) → i
  19. Intersect(g, i) → S_o
  20. Intersect(h, i) → A_o
  21. Circle(S_o, Segment[B, S_2]) → k
  22. Circle(A_o, Segment[B, A_2]) → p
  23. Circle(B, A_2) → p_3
  24. Intersect(p, h, 1) → A_0
  25. Intersect(p_3, x_{12}, 1) → A_{02}
  26. CircleSector(B, A_2, A_{02}) → r_3
  27. Segment(A_0, A_1) → j
  28. Segment(A_{02}, A_0) → q_3
  29. Intersect(k, g, 1) → S_0
  30. Circle(B, S_2) → s_3
  31. Intersect(s_3, x_{12}) → F, G
  32. CircleArc(B, S_2, F) → c_4
  33. Segment(F, S_0) → t_3
  34. Segment(S_0, S_1) → l
  35. Mirror(A_0, S_0) → C_0
  36. Segment(A_0, C_0) → n
  37. OrthogonalLine(S_0, n) → q
  38. Circle(S_0, C_0) → r
  39. Intersect(r, q, 2) → D_0
  40. Intersect(r, q, 1) → B_0
  41. Segment(B_0, D_0) → s
  42. Segment(A_0, B_0) → t
  43. Segment(B_0, C_0) → a_1
  44. Segment(C_0, D_0) → b_1
  45. Segment(D_0, A_0) → c_1
  46. OrthogonalLine(B_0, i) → d_1
  47. Segment(C_0, C_1) → m
  48. OrthogonalLine(D_0, i) → e_1
  49. Intersect(i, e_1) → D_o
  50. Intersect(i, d_1) → B_o
  51. Circle(B, Segment[D_0, D_o]) → f_1
  52. Intersect(f_1, β_2, 1) → D_2
  53. OrthogonalLine(D_2, x_{12}) → g_1
  54. Intersect(e_1, g_1) → D_1
  55. Segment(D_0, D_1) → h_1
  56. Segment(D_1, D_2) → i_1
  57. Circle(B, Segment[B_0, B_o]) → k_1
  58. Intersect(k_1, β_2, 1) → B_2
  59. OrthogonalLine(B_2, x_{12}) → j_1
  60. Intersect(d_1, j_1) → B_1
  61. Segment(B_0, B_1) → l_1
  62. Segment(B_1, B_2) → m_1
  63. Segment(B_1, A_1) → n_1
  64. Segment(A_1, D_1) → p_1
  65. Segment(D_1, C_1) → q_1
  66. Segment(C_1, B_1) → r_1
  67. Segment(B_1, D_1) → s_1
  68. Segment(A_1, C_1) → t_1
  69. OrthogonalLine(C_2, β_2) → a_2
  70. OrthogonalLine(B_2, β_2) → b_2
  71. OrthogonalLine(S_2, β_2) → c_2
  72. OrthogonalLine(D_2, β_2) → d_2
  73. OrthogonalLine(A_2, β_2) → e_2
  74. Circle(S_2, Segment[A_0, B_0]) → f_2
  75. Intersect(f_2, c_2, 2) → S'_2
  76. OrthogonalLine(S'_2, c_2) → g_2
  77. Intersect(a_2, g_2) → C'_2
  78. Intersect(b_2, g_2) → B'_2
  79. Intersect(d_2, g_2) → D'_2
  80. Intersect(e_2, g_2) → A'_2
  81. Segment(C_2, C'_2) → h_2
  82. Segment(B_2, B'_2) → i_2
  83. Segment(D_2, D'_2) → j_2
  84. Segment(A_2, A'_2) → k_2
  85. Segment(A'_2, C'_2) → l_2
  86. OrthogonalLine(A'_2, x_{12}) → m_2
  87. OrthogonalLine(D'_2, x_{12}) → n_2
  88. OrthogonalLine(S'_2, x_{12}) → p_2
  89. OrthogonalLine(B'_2, x_{12}) → q_2
  90. OrthogonalLine(C'_2, x_{12}) → r_2
  91. Intersect(r_2, m) → C'_1
  92. Intersect(q_2, d_1) → B'_1
  93. Intersect(p_2, g) → S'_1
  94. Intersect(n_2, e_1) → D'_1
  95. Intersect(m_2, h) → A'_1
  96. Segment(C'_2, C'_1) → s_2
  97. Segment(B'_2, B'_1) → t_2
  98. Segment(S'_2, S'_1) → a_3
  99. Segment(D'_2, D'_1) → b_3
  100. Segment(A'_2, A'_1) → c_3
  101. Segment(D_1, D'_1) → d_3
  102. Segment(S_1, S'_1) → e_3
  103. Segment(C_1, C'_1) → f_3
  104. Segment(B_1, B'_1) → g_3
  105. Segment(A_1, A'_1) → h_3
  106. Segment(D'_1, C'_1) → i_3
  107. Segment(C'_1, B'_1) → j_3
  108. Segment(B'_1, A'_1) → k_3
  109. Segment(B'_1, D'_1) → l_3
  110. Segment(C'_1, A'_1) → m_3
  111. Segment(D'_1, A'_1) → n_3
  112. Intersect(r_3, x_{12}, 1) → C
  113. Intersect(r_3, x_{12}, 1) → D
  114. Intersect(r_3, s_3, 1) → E