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CIMAT Logo: Geometric Construction with Vectors, Transformations, and Functions

CIMAT Logo: Geometric Construction with Vectors, Transformations, and Functions
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2019-12-08 00:15
VectorFunctionIntegralTransformationGeometric Construction
This GeoGebra example constructs a complex geometric logo named 'CIMAT', demonstrating how to combine vectors, perpendicular lines, midpoints, angle bisectors, reflections, translations, and various function curves (logarithmic and power functions) along with integral elements to create a mathematically aesthetic emblem. Core mathematical concepts include 2D vector operations, geometric transformations (reflection, translation, similarity), function graphs (logarithmic, power), and integral calculus ideas. Construction highlights: from the summary, commands such as Vector, OrthogonalLine, OrthogonalVector, Midpoint, Intersect, LineBisector, Mirror, Line, CurveCartesian, Segment, Polygon, Semicircle, and Translate are used, involving numerous objects (H, B, A, u, g, v, C, J, M, f, D, m, M', U, o, s, b, I, d, E, h, i, i', h', F, G, K, L, K', N, j, k, O, P, Q, P'). This example is suitable for high school and above, serving as an advanced exercise in geometric construction, transformations, and function integration, and as a reference for mathematical art design.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. OrthogonalLine(B, u, xOyPlane) → g
  3. OrthogonalVector(u) → v
  4. Midpoint(C, A) → J
  5. OrthogonalLine(C, v, xOyPlane) → f
  6. Intersect(g, f) → D
  7. LineBisector(A, B) → m
  8. Mirror(M, m) → M'
  9. Intersect(f, m) → U
  10. Line(A, B) → s
  11. OrthogonalLine(A, s) → b
  12. LineBisector(A, C) → d
  13. Intersect(m, d) → E
  14. OrthogonalLine(I, b) → h
  15. OrthogonalLine(M, d) → i
  16. Mirror(i, m) → i'
  17. Mirror(h, d) → h'
  18. Midpoint(E, M') → L
  19. Mirror(K, L) → K'
  20. Intersect(m, h') → N
  21. OrthogonalLine(G, d) → j
  22. OrthogonalLine(F, d) → k
  23. Intersect(j, h') → O
  24. Mirror(P, d) → P'
  25. Mirror(Q, d) → Q'
  26. CurveCartesian(((1 - t)^(3) * P) + (((3 * (1 - t)^(2)) * t) * S) + (((3 * (1 - t)) * t^(2)) * R) + (t^(3) * Q), t, 0, 1) → a
  27. Mirror(a, d) → a'
  28. OrthogonalLine(P, g, xOyPlane) → l
  29. Mirror(W, d) → W'
  30. Mirror(Z, d) → Z'
  31. Intersect(l, m) → V
  32. Line(Q, Z) → n
  33. Line(W, n) → p
  34. Intersect(g, h') → A_1
  35. Line(V, A_1) → q
  36. Intersect(q, p) → B_1
  37. OrthogonalLine(D_1, m) → e
  38. Mirror(B_1, d) → C_1
  39. Mirror(V, d) → V'
  40. Segment(P', W) → t
  41. Mirror(P', d) → P''
  42. Mirror(W, d) → W'_1
  43. Segment(P'', W'_1) → t'
  44. Mirror(Q', d) → Q''
  45. Mirror(Z', d) → Z''
  46. Segment(Q'', Z'') → h_1
  47. Segment(T, Z) → i_1
  48. Intersect(m, i_1) → F_1
  49. Intersect(p, i_1) → G_1
  50. Polygon(F_1, T, P, V) → q1, j_1, t_1, p_1, v_1
  51. Polygon(Q, B_1, G_1, Z) → q2, q_1, b_1, k_1, z_1
  52. Semicircle(V, B_1, xOyPlane) → c
  53. Mirror(c, d) → c'
  54. Polygon(C_1, Q', Z', W') → q3, c_1, q', z', w'
  55. Intersect(t, m) → E_1
  56. Polygon(P', V', E_1) → t1, e_1, p', v'
  57. Line(Z', t') → r
  58. Intersect(r, j) → H_1
  59. Intersect(r, w') → I_1
  60. Intersect(t', m) → J_1
  61. Intersect(r, m) → K_1
  62. Polygon(Z, W, P', T) → q4, z_2, w, f_1, t_2
  63. Polygon(I_1, W', J_1, K_1) → q5, g_1, l_1, m_1, n_1
  64. Vector(Q', P') → u_1
  65. Translate(a', u_1) → a''
  66. Mirror(a'', d) → a'''
  67. Translate(c', u_1) → c''
  68. Mirror(c'', d) → c'''
  69. Translate(P', u_1) → P''_1
  70. Translate(V', u_1) → V''
  71. Translate(C_1, u_1) → L_1
  72. Mirror(P''_1, d) → M_1
  73. Mirror(V'', d) → V'''
  74. Mirror(L_1, d) → N_1
  75. Polygon(M_1, V''', V'', P''_1) → q6, r_1, v''', v'', p''_1
  76. Polygon(T, L_1, P') → t2, s_1, t_3, a_1
  77. Mirror(t2, d) → t2'
  78. Mirror(T, d) → T'
  79. Mirror(L_1, d) → O_1
  80. Segment(T', O_1) → d_1
  81. Polygon(P, Q, B_1, V) → q7, p_2, q_2, f_2, v_2
  82. Polygon(M_1, P, N_1, V''') → q8, g_2, p_3, h_2, i_2
  83. Polygon(V'', L_1, P', P''_1) → q9, j_2, k_2, l_2, m_2
  84. Polygon(P', Q', C_1, V') → q10, n_2, r_2, s_2, a_2
  85. Semicircle(V, B_1, xOyPlane) → c_2
  86. Mirror(c, d) → c'_1
  87. Translate(c', u_1) → c''_1
  88. Mirror(c'', d) → c'''_1
  89. Translate(a', u_1) → a''_1
  90. Mirror(a''_1, d) → o_1
  91. Polygon(A, B, D, C) → q11, a_3, b_2, d_2, c_3

AI Drawing Prompt Reference

Draw a mathematical logo named CIMAT. Start with two points defining a vector, then construct a perpendicular line through one endpoint and another perpendicular vector. Find midpoints and angle bisectors, and use intersections to locate key points. Reflect several points across lines and midpoints to create symmetry. Use cubic Bezier curves to connect specific points, then mirror those curves. Add more perpendicular and parallel lines, find intersections, and connect them with segments. Finally, connect symmetric points with segments to complete the emblem. The design incorporates vectors, perpendiculars, midpoints, angle bisectors, reflections, translations, Bezier curves, and line segments, forming an artistic geometric pattern.
The prompt above was compiled from the original drawing commands and is for reference only.