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Geometric Construction of the French Corporate Responsibility Logo

Geometric Construction of the French Corporate Responsibility Logo
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2019-12-22 05:51
PolygonVectorFunctionTransformationGeometric Construction
This example demonstrates how to construct a corporate responsibility logo using geometric transformations and function curves, turning an abstract brand symbol into a precise mathematical drawing. The core mathematical concepts include 2D vectors, rotation, translation, reflection, properties of triangles and rhombi, median lines, hyperbolas, and parametric curves. The construction uses GeoGebra commands such as Line, OrthogonalLine, LineBisector, Vector, OrthogonalVector, Mirror, Intersect, Rotate, Midpoint, Polygon, Circle, CurveCartesian, and CircleArc, involving objects like A, B, s, b, m, u, v, I, h, C, d, h', N, B_1, O, E, C_1, E_1, O', O'', O''', G, G', q_2, N'_1, p_2, V, W, a, R, Q, A_2, K, K', K'', K''', P, P', T, S. It integrates geometric relations and algebraic curves to achieve symmetry and layering in the logo. Suitable for high school and above, it serves as a comprehensive application of geometric transformations, vector operations, and function graphing, and can be used as an interdisciplinary teaching case combining mathematics and design.

GeoGebra Commands & Parameters

  1. Line(A, B) → s
  2. OrthogonalLine(A, s) → b
  3. LineBisector(A, B) → m
  4. Vector(A, B) → u
  5. OrthogonalVector(u) → v
  6. OrthogonalLine(I, b) → h
  7. LineBisector(A, C) → d
  8. Mirror(h, d) → h'
  9. Intersect(m, h') → N
  10. Intersect(m, d) → E
  11. Rotate(B_1, (-(20°)), E) → C_1
  12. Rotate(C_1, (-(20°)), E) → E_1
  13. Rotate(O, (-(15°)), E) → O'
  14. Rotate(O', (-(60°)), E) → O''
  15. Rotate(O'', (-(60°)), E) → O'''
  16. Rotate(O''', (-(60°)), E) → G
  17. Rotate(G, (-(60°)), E) → G'
  18. OrthogonalLine(G', b) → q_2
  19. Rotate(N, (-(60°)), E) → N'_1
  20. LineBisector(E_1, N'_1) → p_2
  21. OrthogonalLine(V, s) → a
  22. Rotate(A_2, (-(30°)), E) → K
  23. Rotate(K, (-(60°)), E) → K'
  24. Rotate(K', (-(60°)), E) → K''
  25. Rotate(K'', (-(60°)), E) → K'''
  26. Rotate(K''', (-(60°)), E) → P
  27. Rotate(P, (-(60°)), E) → P'
  28. Intersect(h, m) → S
  29. OrthogonalLine(C, v, xOyPlane) → f
  30. Intersect(f, m) → U
  31. Midpoint(C, A) → J
  32. OrthogonalLine(M, d) → i
  33. Mirror(i, m) → i'
  34. OrthogonalLine(B, u, xOyPlane) → g
  35. Intersect(g, f) → D
  36. Mirror(M, m) → M'
  37. Midpoint(E, M') → L
  38. Intersect(g, h') → A_1
  39. Polygon(A, B, D, C) → q11, a_3, b_2, d_2, c_3
  40. Circle(E, N) → c
  41. Circle(E, O) → k
  42. OrthogonalLine(F, b) → j
  43. OrthogonalLine(V, a) → e
  44. Rotate(N, 60°, E) → N'
  45. Rotate(N', 60°, E) → N''
  46. Rotate(N'', 60°, E) → N'''
  47. Rotate(N'_1, (-(60°)), E) → J_1
  48. Circle(E, A_2) → p
  49. Rotate(G', (-(60°)), E) → G''
  50. CurveCartesian(((1 - t)^(4) * N) + (((4 * (1 - t)^(3)) * t) * Q) + (((6 * (1 - t)^(2)) * t^(2)) * T) + (((4 * (1 - t)) * t^(3)) * R) + (t^(4) * W), t, 0, 1) → n
  51. Mirror(n, m) → n'
  52. Rotate(n, (-(60°)), E) → n'_1
  53. Rotate(n', (-(60°)), E) → n''
  54. Rotate(n'_1, (-(60°)), E) → l
  55. Rotate(n'', (-(60°)), E) → n'''
  56. Rotate(l, (-(60°)), E) → l'
  57. Rotate(n''', (-(60°)), E) → q
  58. Rotate(q, (-(60°)), E) → q'
  59. Rotate(l', (-(60°)), E) → l''
  60. Mirror(q', d) → q''
  61. Mirror(l'', d) → l'''
  62. CurveCartesian(((1 - t)^(4) * O') + (((4 * (1 - t)^(3)) * t) * Q') + (((6 * (1 - t)^(2)) * t^(2)) * T') + (((4 * (1 - t)) * t^(3)) * R') + (t^(4) * W), t, 0, 1) → r
  63. Line(O', W) → t
  64. Mirror(r, t) → r'
  65. Rotate(r, (-(60°)), E) → r'_1
  66. Rotate(r', (-(60°)), E) → r''
  67. Rotate(r'_1, (-(60°)), E) → w
  68. Rotate(r'', (-(60°)), E) → r'''
  69. Rotate(r''', (-(60°)), E) → a_1
  70. Rotate(w, (-(60°)), E) → w'
  71. Rotate(a_1, (-(60°)), E) → b_1
  72. Rotate(w', (-(60°)), E) → w''
  73. Rotate(w'', (-(60°)), E) → w'''
  74. Rotate(b_1, (-(60°)), E) → c_1
  75. CurveCartesian(((1 - t)^(4) * K) + (((4 * (1 - t)^(3)) * t) * Q'_1) + (((6 * (1 - t)^(2)) * t^(2)) * T'_1) + (((4 * (1 - t)) * t^(3)) * R'_1) + (t^(4) * W), t, 0, 1) → d_1
  76. Line(K, W) → f_1
  77. Mirror(d_1, f_1) → e_1
  78. Rotate(d_1, (-(60°)), E) → g_1
  79. Rotate(g_1, (-(60°)), E) → h_1
  80. Rotate(h_1, (-(60°)), E) → i_1
  81. Rotate(i_1, (-(60°)), E) → j_1
  82. Rotate(j_1, (-(60°)), E) → k_1
  83. Rotate(e_1, (-(60°)), E) → l_1
  84. Rotate(l_1, (-(60°)), E) → m_1
  85. Rotate(m_1, (-(60°)), E) → n_1
  86. Rotate(n_1, (-(60°)), E) → o_1
  87. Rotate(o_1, (-(60°)), E) → p_1
  88. Intersect(h', b) → Z
  89. Mirror(I, m) → I'
  90. Polygon(Z, I, I', A_1) → q1, z_1, i_2, q_1, r_1
  91. Circle(E, B_1) → s_1
  92. Rotate(E_1, (-(40°)), E) → F_1
  93. Rotate(F_1, (-(20°)), E) → G_1
  94. Rotate(G_1, (-(40°)), E) → H_1
  95. Rotate(H_1, (-(60°)), E) → I_1
  96. Rotate(I_1, (-(60°)), E) → K_1
  97. Rotate(K_1, (-(60°)), E) → L_1
  98. Rotate(G_1, (-(60°)), E) → M_1
  99. Rotate(M_1, (-(60°)), E) → N_1
  100. Rotate(N_1, (-(60°)), E) → O_1
  101. Rotate(O_1, (-(60°)), E) → P_1
  102. Polygon(N', O_1, L_1) → t1, t_1, f_2, g_2
  103. Mirror(N', E) → N''_1
  104. Mirror(O_1, E) → Q_1
  105. Mirror(L_1, E) → R_1
  106. Polygon(N''_1, Q_1, R_1) → t1', h_2, n''_1, j_2
  107. Rotate(W, (-(60°)), E) → W'
  108. Rotate(W, (-(120°)), E) → W'_1
  109. Rotate(W', (-(120°)), E) → W''
  110. Rotate(W', (-(180°)), E) → W''_1
  111. Rotate(W''_1, (-(60°)), E) → S_1
  112. Polygon(E_1, F_1, N'_1) → t2, k_2, l_2, m_2
  113. Mirror(E_1, E) → T_1
  114. Mirror(F_1, E) → U_1
  115. Mirror(N'_1, E) → V_1
  116. Intersect(p_2, q_2) → W_1
  117. Polygon(T_1, U_1, V_1) → t2', v_1, n_2, u_1
  118. CircleArc(W_1, N'_1, E_1) → c_2
  119. LineBisector(E_1, F_1) → r_2
  120. Mirror(c_2, r_2) → e_2
  121. Mirror(c_2, E) → s_2
  122. Mirror(e_2, E) → t_2
  123. Mirror(c_2, m) → d_3
  124. Mirror(e_2, m) → e_3
  125. Mirror(d_3, E) → f_3
  126. Mirror(e_3, E) → g_3
  127. Polygon(I_1, M_1, S) → t3, s_3, a_2, h_3
  128. Rotate(g_3, (-(60°)), E) → k_3
  129. Mirror(k_3, m) → p_3

AI Drawing Prompt Reference

Draw a symmetric geometric logo. Start with a horizontal baseline and a perpendicular line through its left endpoint, plus a perpendicular bisector of the two endpoints. Use the intersection of the bisector and the perpendicular as a rotation center, and rotate several points counterclockwise by 20, 60, and other angles to create layered symmetry. Connect key points to form a rhombus, and draw two concentric circles centered at its center. From the rhombus vertices, drop perpendiculars to the baseline, intersecting a slanted line. Then rotate these intersection points around the center to generate multiple symmetric sets. Finally, connect some points with smooth curves to suggest hyperbolic arcs, achieving a balanced, layered geometric design.
The prompt above was compiled from the original drawing commands and is for reference only.