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Imaginary Planes: Spatial Geometric Construction and Transformation

Imaginary Planes: Spatial Geometric Construction and Transformation
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2019-11-12 06:57
PolygonVectorSolid GeometryTransformationGeometric Construction
This example demonstrates the construction of multiple interconnected planes in 3D space using points, segments, and polygons, followed by transformations such as mirroring, translation, and rotation to create symmetric solid structures, forming a visual model called 'Imaginary Planes'. The core mathematical concepts involve spatial geometry of planes and prisms, vector transformations, differential equations, and statistical characteristics (geometric mean, mode) represented geometrically, along with symmetry and transformation groups. The construction uses GeoGebra commands including Point, Segment, Polygon, Mirror, Intersect, Vector, Translate, Rotate, Line, and Angle to build the base shapes and apply transformations. This example is suitable for high school to university-level teaching in solid geometry, linear algebra, or mathematical modeling, and can be used to explore intuitive connections between geometric transformations and statistical concepts.

GeoGebra Commands & Parameters

  1. Point(xAxis) → A
  2. Segment(A, B) → f
  3. Point(xAxis) → C
  4. Segment(C, D) → g
  5. Segment(D, B) → h
  6. Segment(A, C) → i
  7. Point(i) → E
  8. Point(h) → F
  9. Segment(E, F) → j
  10. Segment(G, F) → k
  11. Point(yAxis) → H
  12. Segment(G, H) → l
  13. Segment(H, E) → m
  14. Point(h) → I
  15. Polygon(H, E, F, G) → q1, h_1, e, f_1, g_1
  16. Polygon(B, D, C, A) → q2, b, d, c, a
  17. Mirror(F, j) → F'
  18. Intersect(p, yAxis, 1) → J
  19. Vector(K, L) → u
  20. Translate(B, u) → B'
  21. Translate(D, u) → D'
  22. Translate(C, u) → C'
  23. Translate(A, u) → A'
  24. Polygon(B', D', C', A') → q2', b', d', c', a'
  25. Rotate(B, 90°, E) → B'_1
  26. Rotate(D, 90°, E) → D'_1
  27. Rotate(C, 90°, E) → C'_1
  28. Rotate(A, 90°, E) → A'_1
  29. Polygon(B'_1, D'_1, C'_1, A'_1) → q2'_1, b'_1, d'_1, c'_1, a'_1
  30. Rotate(B, (-(90°)), E) → B'_2
  31. Rotate(D, (-(90°)), E) → D'_2
  32. Rotate(C, (-(90°)), E) → C'_2
  33. Rotate(A, (-(90°)), E) → A'_2
  34. Polygon(B'_2, D'_2, C'_2, A'_2) → q2'_2, b'_2, d'_2, c'_2, a'_2
  35. Mirror(B'_2, yAxis) → M
  36. Mirror(D'_2, yAxis) → N
  37. Mirror(C'_2, yAxis) → O
  38. Mirror(A'_2, yAxis) → P
  39. Polygon(M, N, O, P) → q3, m_1, n, o, p_1
  40. Mirror(B'_1, yAxis) → Q
  41. Mirror(D'_1, yAxis) → R
  42. Mirror(C'_1, yAxis) → S
  43. Mirror(A'_1, yAxis) → T
  44. Polygon(Q, R, S, T) → q4, q, r, s, t
  45. Point(t) → V
  46. Line(V, t) → i_1
  47. Point(r) → Z
  48. Line(Z, r) → j_1
  49. Point(i_1) → U
  50. Point(j_1) → W
  51. Segment(U, W) → k_1
  52. Polygon(V, U, W, Z) → q5, v, u_1, w, z_1
  53. Segment(A_1, B_1) → l_1
  54. Segment(B_1, C_1) → n_1
  55. Polygon(C_1, A_1, B_1) → t1, b_1, c_1, a_1
  56. Point(yAxis) → D_1
  57. Polygon(D_1, J, E_1, F_1) → q6, d_1, j_2, e_1, q_1
  58. Rotate(D_1, (-(180°)), J) → G_1
  59. Rotate(J, (-(180°)), J) → J'
  60. Rotate(E_1, (-(180°)), J) → H_1
  61. Rotate(F_1, (-(180°)), J) → I_1
  62. Polygon(G_1, J', H_1, I_1) → q6', r_1, j', s_1, t_1
  63. Vector(J_1, K_1) → v_1
  64. Translate(G_1, v_1) → L_1
  65. Translate(J', v_1) → J''
  66. Translate(H_1, v_1) → M_1
  67. Translate(I_1, v_1) → N_1
  68. Polygon(L_1, J'', M_1, N_1) → q6'', f_2, j'', g_2, h_2
  69. Polygon(O_1, P_1, Q_1, R_1) → q7, o_1, i_2, k_2, l_2
  70. Point(Circle[E, 1]) → S_1
  71. Segment(E, S_1) → m_2
  72. Point(Circle[E, -1]) → T_1
  73. Segment(E, T_1) → n_2
  74. Rotate(E, 180°, E) → E'
  75. Rotate(S_1, 180°, E) → U_1
  76. Segment(E', U_1) → p_2
  77. Rotate(O_1, 180°, J) → V_1
  78. Rotate(P_1, 180°, J) → W_1
  79. Rotate(Q_1, 180°, J) → Z_1
  80. Rotate(R_1, 180°, J) → A_2
  81. Polygon(V_1, W_1, Z_1, A_2) → q7', q_2, w_1, r_2, a_2
  82. Vector(B_2, C_2) → u_2
  83. Translate(V_1, u_2) → D_2
  84. Translate(W_1, u_2) → E_2
  85. Translate(Z_1, u_2) → F_2
  86. Translate(A_2, u_2) → G_2
  87. Polygon(D_2, E_2, F_2, G_2) → q7'', d_2, e_2, s_2, t_2
  88. Vector(H_2, I_2) → v_2
  89. Translate(B, v_2) → B'_3
  90. Translate(D, v_2) → D'_3
  91. Translate(C, v_2) → C'_3
  92. Translate(A, v_2) → A'_3
  93. Polygon(B'_3, D'_3, C'_3, A'_3) → q2'_3, b'_3, d'_3, c'_3, a'_3
  94. Intersect(d', c') → J_2
  95. Segment(J_2, K_2) → b_2
  96. Segment(J_2, L_2) → c_2
  97. Segment(J_2, M_2) → f_3
  98. Point(p_2) → N_2
  99. Point(m_2) → O_2
  100. Angle(c', i_2) → α
  101. Segment(J, O_2) → g_3
  102. Angle(g_3, m_2) → β

AI Drawing Prompt Reference

Construct multiple interconnected planes in 3D space using points and line segments, then apply transformations like mirroring, translation, and rotation to create symmetric solid structures. Begin with a quadrilateral and perform transformations to generate interconnected geometric shapes, demonstrating spatial geometry and symmetry of transformation groups.
The prompt above was compiled from the original drawing commands and is for reference only.