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Graphical Models of the Projective Extension of Real 3-Space (Visualizing RP3 in

Graphical Models of the Projective Extension of Real 3-Space (Visualizing RP3 in
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2020-10-24 06:19
VectorSolid GeometryTransformationGeometric ConstructionLocus
① This example demonstrates graphical models of the projective extension of real 3-space, i.e., the real projective space RP3, visualized in four dimensions, showing points, lines, planes, and surfaces that represent projective geometry concepts including points at infinity and duality. ② Core mathematical concepts include projective geometry, projective spaces, points/lines/planes at infinity, four-dimensional geometric visualization, and incidence/orthogonality relations among geometric objects. ③ Construction highlights: from the construction summary, GeoGebra commands such as Line, Segment, Vector, Ray, Surface, Point, Locus, Plane, Intersect, OrthogonalLine, IntersectPath, Polygon, Sphere, and OrthogonalPlane are used. Objects include O_3, Z_3, A_3, B_3, C_3, D_3 (3D) and A_4, B_4, C_4, D_4, Z_4 (4D), along with vectors like a_3v, b_3v, c_3v, d_3v, a_4v, etc., and surfaces to build the model. ④ Suitable for advanced undergraduate or graduate courses in geometry, projective geometry, or mathematical visualization, and for researchers interested in intuitive understanding of 4D geometry.

GeoGebra Commands & Parameters

  1. Line(O_3, Z_3) → h
  2. Segment(A_3, B_3) → i
  3. Segment(B_3, C_3) → j
  4. Segment(C_3, A_3) → k
  5. Segment(A_3, D_3) → l
  6. Segment(B_3, D_3) → m
  7. Segment(C_3, D_3) → n
  8. Segment(A_4, D_4) → q
  9. Segment(D_4, B_4) → r
  10. Segment(C_4, B_4) → s
  11. Segment(C_4, A_4) → t
  12. Segment(C_4, D_4) → a
  13. Segment(B_4, A_4) → b
  14. Vector(O_3, Z_3) → z_3v
  15. Vector(O_3, Z_4) → z_4v
  16. Vector(O_3, A_3) → a_3v
  17. Vector(O_3, B_3) → b_3v
  18. Vector(O_3, C_3) → c_3v
  19. Vector(O_3, D_3) → d_3v
  20. Vector(O_3, A_4) → a_4v
  21. Vector(O_3, B_4) → b_4v
  22. Vector(O_3, C_4) → c_4v
  23. Vector(O_3, D_4) → d_4v
  24. Ray(O_3, A_3) → c
  25. Ray(O_3, B_3) → d
  26. Ray(O_3, C_3) → e
  27. Ray(O_3, D_3) → f_1
  28. Ray(O_3, A_4) → h_1
  29. Ray(O_3, B_4) → i_1
  30. Ray(O_3, C_4) → j_1
  31. Ray(O_3, D_4) → k_1
  32. Surface((cos(u) * cos(v)), (cos(u) * sin(v)), sin(u), u, (-π), π, v, 0, π) → o
  33. Surface((cos(u) * cos(v)), (cos(u) * sin(v)), sin(u), u, (-π), 0, v, 0, π) → u
  34. Point(i) → PAB_3
  35. Locus(PAB'_3, PAB_3) → AB'_3
  36. Point(j) → PBC_3
  37. Locus(PAB'_4, PAB_3) → AB'_4
  38. Locus(PBC'_3, PBC_3) → BC'_3
  39. Locus(PBC'_4, PBC_3) → BC'_4
  40. Point(k) → PAC_3
  41. Locus(PAC'_3, PAC_3) → AC'_3
  42. Locus(PAC'_4, PAC_3) → AC'_4
  43. Point(n) → PCD_3
  44. Locus(PCD'_3, PCD_3) → CD'_3
  45. Locus(PCD'_4, PCD_3) → CD'_4
  46. Point(l) → PAD_3
  47. Locus(PAD'_3, PAD_3) → AD'_3
  48. Locus(PAD'_4, PAD_3) → AD'_4
  49. Point(m) → PBD_3
  50. Locus(PBD'_4, PBD_3) → BD'_4
  51. Locus(PBD'_3, PBD_3) → BD'_3
  52. Plane(A_3, B_3, C_3) → p_1
  53. Plane(O_3, p_1) → ξ
  54. Line(A_3, C_3) → r_1
  55. Intersect(r_1, xOyPlane) → W_3
  56. OrthogonalLine(W_3, xOyPlane) → s_1
  57. Line(A_4, C_4) → t_1
  58. Intersect(t_1, s_1) → W_4
  59. IntersectPath(ξ, xOyPlane) → a_1
  60. Plane(W_4, a_1) → ω
  61. Segment(W_3, W_4) → q_1
  62. Segment(B_3, B_4) → f_2
  63. Segment(A_3, A_4) → g_2
  64. Segment(C_3, C_4) → h_2
  65. Segment(D_4, D_3) → i_2
  66. Polygon(O_3, A_3, C_3) → lineAB_3, r_2, s_2, t_2
  67. Polygon(O_3, A_4, C_4) → lineAB_4, j_2, a_2, b_2
  68. Polygon(A_3, B_3, C_3) → t1, c_3, k_2, l_2
  69. OrthogonalLine(O_3, ξ) → m_2
  70. Sphere(O_3, 1) → o_1
  71. Intersect(m_2, o_1) → A, B
  72. Vector(O_3, B) → normalXi
  73. OrthogonalLine(O_3, ω) → n_2
  74. Intersect(n_2, o_1) → C, D
  75. Vector(O_3, C) → normalOmega
  76. OrthogonalPlane(O_3, linexyz_1) → linexyz
  77. OrthogonalPlane(O_3, linexyw_1) → linexyw
  78. Intersect(g_2, xOyPlane) → E
  79. Intersect(i_2, xOyPlane) → F
  80. Intersect(h_2, xOyPlane) → G
  81. Intersect(f_2, xOyPlane) → H
  82. Vector(O_3, I) → f
  83. OrthogonalLine(I, xOyPlane) → g
  84. Intersect(g, q_2) → J
  85. Intersect(g, xOyPlane) → K
  86. Segment(I, K) → g_1
  87. Vector(O_3, K) → w_1
  88. IntersectPath(ω, q_2) → l_1
  89. Point(a_1) → L
  90. Point(a_1) → M
  91. OrthogonalPlane(L, a_1) → p_2
  92. Intersect(l_1, p_2) → N
  93. Plane(M, p_2) → p_3
  94. Intersect(l_1, p_3) → O
  95. IntersectPath(p_2, ξ) → m_1
  96. IntersectPath(p_3, ξ) → n_1
  97. Polygon(L, M, O, N) → ct1, l_3, m_3, o_2, n_3
  98. Intersect(n_1, f_3) → P
  99. Intersect(m_1, f_3) → Q
  100. Polygon(L, M, P, Q) → ct2, l_4, m_4, p_4, q_3
  101. Polygon(A_4, B_4, C_4) → t2, c_4, a_4, b_4
  102. OrthogonalLine(O_3, t1) → c_1
  103. Intersect(c_1, o_1) → R, S
  104. Vector(O_3, S) → v_1
  105. OrthogonalLine(S, p) → d_1
  106. Intersect(d_1, p) → T
  107. Vector(O_3, T) → e_1
  108. Segment(S, T) → d_2
  109. Ray(O_3, S) → e_2
  110. Line(O_3, T) → g_3
  111. Ray(O_3, I) → h_3
  112. Line(O_3, K) → i_3
  113. Line(O_3, linexyz_1) → j_3
  114. Line(O_3, linexyw_1) → k_3
  115. Intersect(k_3, o_1) → U, V
  116. Intersect(j_3, o_1) → W, Z
  117. Vector(O_3, V) → u_2
  118. Vector(O_3, W) → v_2
  119. Line(A_3, C_3) → r_3
  120. Intersect(r_3, p) → A_1
  121. Ray(A_1, C_3) → s_3
  122. Line(A_4, C_4) → t_3

AI Drawing Prompt Reference

Create a visualization of the projective extension of real 3-space (real projective space RP3) represented in a four-dimensional view. Start by placing a set of reference points in 3D together with corresponding points in 4D, connect selected point pairs using line-segment edges to form a basic incidence/triangle-like framework in both dimensions. From the same reference point, draw vectors and rays pointing toward the other designated points. Then construct two parametric surfaces resembling spherical parameterizations: use trigonometric expressions with u and v angle parameters, with different u ranges to form two complementary patches. Finally, mark several points derived from the constructed structure and build locus/trajectory relations by linking each marked point to its corresponding primed version along the intended geometric relation, producing the line-generating links associated with the various reference point pairs.
The prompt above was compiled from the original drawing commands and is for reference only.