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Projections of a Cube

Projections of a Cube
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2021-12-10 00:36
Solid GeometryTransformationGeometric Construction
This applet demonstrates the projections of a cube onto three planes. Key mathematical concepts include solid geometry and projection transformations. Using GeoGebra commands such as Point, Cube, Rotate, Vector, Translate, etc., the construction of the cube and its projections is achieved. Suitable for high school or university-level solid geometry studies.

GeoGebra Commands & Parameters

  1. Point(Circle[C, Distance[B, C], Segment[B, C]]) → A
  2. Cube(B, C, A) → a, D, faceADHG, faceEFGH, edgeBD, edgeBC, edgeAC, edgeAD, edgeBE, edgeDH, edgeEH, edgeEF, E, edgeCF, edgeFG, edgeAG, edgeGH, F, G, H, faceACBD, faceBDHE, faceBCFE, faceACFG
  3. Rotate(a, (-α), xAxis) → a'
  4. Rotate(H, (-α), xAxis) → H'
  5. Rotate(E, (-α), xAxis) → E'
  6. Rotate(B, (-α), xAxis) → B'
  7. Rotate(D, (-α), xAxis) → D'
  8. Rotate(A, (-α), xAxis) → A'
  9. Rotate(G, (-α), xAxis) → G'
  10. Rotate(F, (-α), xAxis) → F'
  11. Rotate(C, (-α), xAxis) → C'
  12. Rotate(a', (-β), yAxis) → a''
  13. Rotate(E', (-β), yAxis) → E''
  14. Rotate(B', (-β), yAxis) → B''
  15. Rotate(C', (-β), yAxis) → C''
  16. Rotate(F', (-β), yAxis) → F''
  17. Rotate(H', (-β), yAxis) → H''
  18. Rotate(G', (-β), yAxis) → G''
  19. Rotate(D', (-β), yAxis) → D''
  20. Rotate(A', (-β), yAxis) → A''
  21. Rotate(a'', (-γ), zAxis) → a'''
  22. Rotate(G'', (-γ), zAxis) → G'''
  23. Rotate(F'', (-γ), zAxis) → F'''
  24. Rotate(E'', (-γ), zAxis) → E'''
  25. Rotate(H'', (-γ), zAxis) → H'''
  26. Rotate(B'', (-γ), zAxis) → B'''
  27. Rotate(D'', (-γ), zAxis) → D'''
  28. Rotate(A'', (-γ), zAxis) → A'''
  29. Rotate(C'', (-γ), zAxis) → C'''
  30. Vector(K, L) → u
  31. Translate(I, u) → I'
  32. Translate(J, u) → J'
  33. Vector(K, M) → v
  34. Translate(I, v) → I'_{1}
  35. Translate(J, v) → J'_{1}
  36. Translate(N, u) → N'
  37. Translate(N, v) → N'_{1}
  38. Polygon(I'_{1}, J'_{1}, N'_{1}, M) → q1, i'_{1}, j'_{1}, n'_{1}, m
  39. Polygon(J', I', I'_{1}, J'_{1}) → q2, j', i', f, g
  40. Polygon(I', L, M, I'_{1}) → q3, h, l, m_1, i
  41. Midpoint(J, J'_{1}) → O
  42. Midpoint(I, I'_{1}) → P
  43. Midpoint(K, M) → Q
  44. Segment(F''', C''') → j
  45. Segment(C''', A''') → k
  46. Segment(A''', D''') → n
  47. Segment(D''', B''') → q
  48. Segment(B''', C''') → r
  49. Segment(F''', G''') → s
  50. Segment(G''', A''') → t
  51. Segment(G''', H''') → b
  52. Segment(H''', E''') → c
  53. Segment(E''', F''') → d
  54. Segment(H''', D''') → e
  55. Segment(E''', B''') → f_1
  56. OrthogonalLine(A''', q1) → p
  57. OrthogonalLine(G''', q1) → g_1
  58. OrthogonalLine(D''', q1) → h_1
  59. OrthogonalLine(H''', q1) → i_1
  60. Intersect(h_1, q1) → R
  61. Intersect(p, q1) → S
  62. Intersect(g_1, q1) → T
  63. Intersect(i_1, q1) → U
  64. Segment(R, U) → j_1
  65. Segment(U, T) → k_1
  66. Segment(T, S) → l_1
  67. Segment(S, R) → n_1
  68. OrthogonalLine(B''', q1) → p_1
  69. OrthogonalLine(C''', q1) → q_1
  70. Intersect(p_1, q1) → V
  71. Intersect(q_1, q1) → W
  72. Segment(V, W) → r_1
  73. Segment(W, S) → s_1
  74. Segment(V, R) → t_1
  75. OrthogonalLine(E''', q1) → a_1
  76. OrthogonalLine(F''', q1) → b_1
  77. Intersect(a_1, q1) → Z
  78. Intersect(b_1, q1) → A_1
  79. Segment(Z, A_1) → c_1
  80. Segment(W, A_1) → d_1
  81. Segment(V, Z) → e_1
  82. Segment(Z, U) → f_2
  83. Segment(T, A_1) → g_2
  84. OrthogonalLine(G''', q3) → h_2
  85. OrthogonalLine(A''', q3) → i_2
  86. OrthogonalLine(H''', q3) → j_2
  87. OrthogonalLine(D''', q3) → k_2
  88. OrthogonalLine(C''', q3) → l_2
  89. OrthogonalLine(B''', q3) → m_2
  90. OrthogonalLine(E''', q3) → n_2
  91. OrthogonalLine(F''', q3) → p_2
  92. Intersect(l_2, q3) → B_1
  93. Intersect(i_2, q3) → C_1
  94. Segment(B_1, C_1) → q_2
  95. Intersect(p_2, q3) → D_1
  96. Intersect(h_2, q3) → E_1
  97. Segment(D_1, E_1) → r_2
  98. Segment(E_1, C_1) → s_2
  99. Segment(D_1, B_1) → t_2
  100. Intersect(k_2, q3) → F_1
  101. Intersect(m_2, q3) → G_1
  102. Intersect(n_2, q3) → H_1
  103. Intersect(j_2, q3) → I_1
  104. Segment(F_1, I_1) → a_2
  105. Segment(F_1, G_1) → b_2
  106. Segment(I_1, H_1) → c_2
  107. Segment(H_1, G_1) → d_2
  108. Segment(F_1, C_1) → e_2
  109. Segment(I_1, E_1) → f_3
  110. Segment(H_1, D_1) → g_3
  111. Segment(G_1, B_1) → h_3
  112. OrthogonalLine(A''', q2) → i_3
  113. OrthogonalLine(G''', q2) → j_3
  114. OrthogonalLine(D''', q2) → k_3
  115. OrthogonalLine(H''', q2) → l_3
  116. Intersect(i_3, q2) → J_1
  117. Intersect(k_3, q2) → K_1
  118. Intersect(l_3, q2) → L_1
  119. Intersect(j_3, q2) → M_1
  120. Segment(M_1, L_1) → m_3
  121. Segment(K_1, L_1) → n_3
  122. Segment(J_1, M_1) → p_3
  123. Segment(J_1, K_1) → q_3
  124. OrthogonalLine(F''', q2) → r_3
  125. OrthogonalLine(E''', q2) → s_3
  126. OrthogonalLine(C''', q2) → t_3
  127. OrthogonalLine(B''', q2) → a_3
  128. Intersect(a_3, q2) → N_1
  129. Intersect(t_3, q2) → O_1
  130. Intersect(r_3, q2) → P_1
  131. Intersect(s_3, q2) → Q_1
  132. Segment(Q_1, N_1) → b_3
  133. Segment(Q_1, P_1) → e_3
  134. Segment(Q_1, L_1) → f_4
  135. Segment(P_1, M_1) → g_4
  136. Segment(N_1, K_1) → h_4
  137. Segment(N_1, O_1) → c_3
  138. Segment(O_1, J_1) → d_3
  139. Segment(P_1, O_1) → i_4

AI Drawing Prompt Reference

Create a 3D cube and depict its projections onto three mutually perpendicular planes. Start by constructing the initial reference points from a circle configuration, then define the cube using three adjacent vertices and obtain all other cube vertices. Rotate the cube around the three coordinate axes in sequence to generate the corresponding projected vertex sets for each plane. Use vectors and translations to shift those projected points so the three projection views appear together clearly. Connect the projected vertices with line segments to form the cube’s edges, and draw polygons to represent the visible faces in each projection. Include the key shared reference points by using segment midpoints to indicate correspondence between projections.
The prompt above was compiled from the original drawing commands and is for reference only.