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Cube and Shadows with a Slanted Image Plane

Cube and Shadows with a Slanted Image Plane
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2020-11-23 21:03
Coordinate GeometrySolid GeometryTransformation
This example demonstrates a cube and its projection in a slanted image plane. Key mathematical concepts include solid geometry and projection transformations. Using GeoGebra commands such as OrthogonalLine, Circle, Intersect, etc., the construction of the cube and its projection is achieved. Suitable for high school or university solid geometry courses.

GeoGebra Commands & Parameters

  1. OrthogonalLine(F, eq1) → i
  2. Circle(F, d) → c
  3. Intersect(c, i, 2) → A
  4. Segment(F, A) → j
  5. Intersect(h, eq1) → B
  6. Segment(B, A) → k
  7. OrthogonalLine(A, k) → l
  8. Intersect(eq1, l) → V_3
  9. Segment(V_3, A) → m
  10. Angle(B, A, V_3) → β
  11. Polygon(B, A, V_3) → t1, i_5, b_6, a_6
  12. Circle(B, A) → e
  13. Rotate(F, 30°, A) → F'
  14. Angle(F, A, F') → α
  15. Line(A, F') → f_5
  16. Intersect(eq1, f_5) → G_1
  17. Segment(G_1, A) → g_5
  18. Circle(F, G_1) → c_5
  19. Intersect(e, eq1, 2) → C
  20. CircleArc(B, C, A) → p
  21. Point(Circle[D, a]) → E
  22. Segment(D, E) → n
  23. Polygon(D, E, 4) → poligon1, q, r, s, t, G, H
  24. Line(C, n) → b
  25. Line(C, r) → g_1
  26. Intersect(h, g_1) → V_2
  27. Segment(C, V_2) → h_1
  28. Intersect(h, b) → V_1
  29. Segment(C, V_1) → i_1
  30. Line(G, E) → j_1
  31. Line(H, D) → k_1
  32. Line(E, D) → l_1
  33. Line(H, G) → m_1
  34. Intersect(f_1, k_1) → K
  35. Segment(K, D) → n_1
  36. Intersect(f_1, j_1) → L
  37. Segment(L, E) → p_1
  38. Intersect(f_1, l_1) → M
  39. Segment(M, E) → q_1
  40. Intersect(f_1, m_1) → N
  41. Segment(G, N) → r_1
  42. Line(L, V_2) → s_1
  43. Line(K, V_2) → t_1
  44. Line(N, V_1) → a_1
  45. Line(M, V_1) → b_1
  46. Intersect(t_1, b_1) → O
  47. Intersect(t_1, a_1) → P
  48. Intersect(s_1, a_1) → Q
  49. Intersect(s_1, b_1) → R
  50. Segment(K, O) → c_1
  51. Segment(O, P) → d_1
  52. Segment(P, V_2) → e_1
  53. Segment(L, R) → f_2
  54. Segment(R, Q) → g_2
  55. Segment(Q, V_2) → h_2
  56. Segment(M, R) → i_2
  57. Segment(R, O) → j_2
  58. Segment(O, V_1) → k_2
  59. Segment(N, Q) → l_2
  60. Segment(P, V_2) → i_6
  61. Segment(Q, P) → m_2
  62. Segment(P, V_1) → n_2
  63. Polygon(O, P, Q, R) → q1, o, p_5, q_5, r_5
  64. Segment(D, C) → p_2
  65. Segment(O, D) → r_2
  66. Segment(E, R) → s_2
  67. Segment(E, C) → t_2
  68. Segment(H, C) → a_2
  69. Segment(H, P) → b_2
  70. Segment(G, C) → c_2
  71. Segment(G, Q) → d_2
  72. Line(K, m) → e_2
  73. Line(L, m) → f_3
  74. Circle(K, a) → c_3
  75. Circle(L, a) → d_3
  76. Intersect(c_3, e_2, 2) → S
  77. Segment(S, K) → g_3
  78. Intersect(d_3, f_3, 2) → T
  79. Segment(T, L) → h_3
  80. Segment(A, V_3) → g
  81. Line(V_3, L) → i_3
  82. Line(V_3, K) → j_3
  83. Line(A, T) → k_3
  84. Line(A, S) → l_3
  85. Intersect(i_3, k_3) → U
  86. Segment(L, U) → m_3
  87. Segment(L, V_3) → f_4
  88. Intersect(j_3, l_3) → V
  89. Segment(K, V) → n_3
  90. Segment(K, V_3) → g_4
  91. Segment(A, U) → p_3
  92. Segment(A, V) → q_3
  93. Intersect(n_3, q_3) → W
  94. Intersect(m_3, p_3) → Z
  95. Line(Z, V_2) → r_3
  96. Line(V_2, W) → s_3
  97. Line(O, V_3) → t_3
  98. Line(R, V_3) → a_3
  99. Line(Q, V_3) → b_3
  100. Line(P, V_3) → e_3
  101. Segment(W, V_2) → h_4
  102. Intersect(s_3, t_3) → A_1
  103. Segment(A_1, V_3) → t_4
  104. Segment(A_1, V_2) → e_5
  105. Intersect(s_3, e_3) → B_1
  106. Segment(A_1, B_1) → i_4
  107. Segment(B_1, V_3) → a_4
  108. Segment(A_1, W) → j_4
  109. Segment(Z, V_2) → n_4
  110. Intersect(s_3, e_3) → C_1
  111. Segment(V_2, C_1) → k_4
  112. Intersect(s_3, t_3) → D_1
  113. Segment(C_1, D_1) → l_4
  114. Segment(D_1, W) → m_4
  115. Segment(E, C) → q_2
  116. Intersect(r_3, a_3) → E_1
  117. Segment(E_1, V_3) → r_4
  118. Segment(Z, E_1) → p_4
  119. Intersect(r_3, b_3) → F_1
  120. Segment(F_1, V_3) → s_4
  121. Segment(F_1, V_2) → g_6
  122. Segment(E_1, F_1) → q_4
  123. Segment(E_1, A_1) → b_4
  124. Segment(A_1, V_1) → c_4
  125. Segment(F_1, B_1) → d_4
  126. Segment(B_1, V_1) → e_4
  127. Polygon(P, B_1, F_1, Q) → q2, p_6, h_5, f_{6}, q_6
  128. Polygon(P, O, A_1, B_1) → q3, p_7, o_1, j_5, k_5
  129. Polygon(A_1, E_1, R, O) → q4, l_5, m_5, r_6, o_2
  130. Polygon(E_1, F_1, Q, R) → q5, n_5, s_5, q_7, r_7
  131. Polygon(A_1, B_1, F_1, E_1) → q6, t_5, a_5, b_5, d_5
  132. Line(V_3, I) → j_6
  133. Intersect(h, j_6) → J
  134. Segment(V_3, J) → k_6
  135. Segment(R, J) → l_6
  136. Segment(E_1, I) → m_6
  137. Intersect(l_6, m_6) → H_1
  138. Segment(Q, J) → n_6
  139. Segment(F_1, I) → s_6
  140. Intersect(n_6, s_6) → I_1
  141. Segment(H_1, V_2) → t_6
  142. Segment(P, J) → c_6
  143. Segment(B_1, I) → d_6
  144. Intersect(c_6, d_6) → J_1
  145. Segment(I_1, V_1) → e_6
  146. Intersect(s_4, e_6) → K_1
  147. Segment(R, H_1) → m_7
  148. Segment(J_1, P) → n_7
  149. Polygon(K_1, I_1, H_1, R, Q) → poligon2, f_7, g_7, h_7, r_8, q_8
  150. Intersect(r_8, q_8) → L_1
  151. Intersect(h_7, r_8) → M_1
  152. Polygon(F_1, L_1, M_1, E_1) → q7, i_7, j_7, k_7, l_7

AI Drawing Prompt Reference

Draw a cube and its shadow on a slanted image plane. Start by defining a slanted plane and a point on it as the projection center. Through this point, draw a line perpendicular to the plane, intersecting a circle to get a projection point. Connect the center to this pointhol, then intersect with the plane to form a triangle. Next, rotate the center point by 30 degrees to create an angle, and draw a line through the rotated point intersecting the plane to get another projection point. Then construct a square as the base of the cube, and connect its vertices to form the cube. Finally, from each vertex of the cube, draw lines to the projection plane, find their intersections with the plane, and connect these intersection points to outline the cube's projected shape on the slanted plane, including the quadrilateral formed by these intersections.
The prompt above was compiled from the original drawing commands and is for reference only.