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Monge Projection of the Cube

Monge Projection of the Cube
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2012-09-15 19:33
Coordinate GeometrySolid GeometryTransformationGeometric Construction
① This example demonstrates how to obtain the top view and front view of a cube using Monge (orthographic) projection when the plane, centre, and a vertex of one face are known. ② Core mathematical concepts include orthographic projection in solid geometry, the principles of Monge projection in descriptive geometry, spatial vectors and plane relationships, and the geometric properties of projection transformations. ③ Construction highlights: the GeoGebra commands Point, Ray, OrthogonalLine, Segment, Mirror, Intersect, Line, Circle, CircleSector, CircleArc are employed to create objects such as 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, using rays, perpendicular lines, segments, reflections, intersections, and circular arcs to build the two projection planes. ④ Suitable for high‑school solid geometry and university descriptive geometry courses, as well as for geometric construction practice and understanding projection transformations.

GeoGebra Commands & Parameters

  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