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Mathematical Construction of the Mercedes Logo

Mathematical Construction of the Mercedes Logo
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2019-04-30 00:26
CirclePolygonVector
This example demonstrates how to construct the Mercedes-Benz logo using 2D vectors and geometric transformations. Key math concepts include vectors, circles, ellipses, polygons, rotation, reflection, midpoints, and angle bisectors. Construction highlights involve commands like Circle, Vector, Line, Rotate, LineBisector, Polygon, Mirror, Segment, Midpoint, AngularBisector, Intersect, CircleArc, and CircleSector in GeoGebra. Suitable for high school mathematics education to showcase practical applications of geometric transformations and figure construction.

GeoGebra Commands & Parameters

  1. Circle(O, A) → c
  2. Vector(O, A) → u
  3. Circle(O, B) → d
  4. Line(A, O) → f
  5. Rotate(B, 120°, O) → B'
  6. Rotate(B, (-(120°)), O) → B'_1
  7. Rotate(B, 1°, O) → B'_2
  8. Rotate(B, -1°, O) → B'_3
  9. Rotate(B', 1°, O) → B''
  10. Rotate(B', -1°, O) → B''_1
  11. Rotate(B'_1, 1°, O) → C
  12. Rotate(B'_1, -1°, O) → D
  13. LineBisector(B, B') → g
  14. LineBisector(B, B'_1) → h
  15. Rotate(E, 120°, O) → E'
  16. Rotate(E, (-(120°)), O) → E'_1
  17. Polygon(E'_1, O, B, B'_2) → c1, e'_1, o, b, b'_2
  18. Mirror(E'_1, g) → F
  19. Mirror(O, g) → O'
  20. Mirror(B, g) → B'_4
  21. Mirror(B'_2, g) → G
  22. Polygon(F, O', B'_4, G) → c1', f_1, o', b'_4, g_1
  23. Mirror(F, f) → F'
  24. Mirror(O', f) → O''
  25. Mirror(B'_4, f) → H
  26. Mirror(G, f) → G'
  27. Polygon(F', O'', H, G') → c1'', f', o'', h_1, g'
  28. Polygon(B'_3, B, O, E') → c2, b'_3, b_1, o_1, e'
  29. Mirror(B'', f) → B'''
  30. Mirror(B', f) → B''_2
  31. Mirror(O, f) → O'_1
  32. Mirror(E, f) → E'_2
  33. Segment(O, E) → i
  34. Segment(O, F') → j
  35. Segment(O, F) → k
  36. Circle(O, I) → p
  37. Midpoint(F, O) → J
  38. Midpoint(F, J) → K
  39. Midpoint(J, O) → L
  40. Polygon(I, K, L) → t1, l, i_1, k_1
  41. Mirror(I, g) → I'
  42. Mirror(K, g) → K'
  43. Mirror(L, g) → L'
  44. Polygon(I', K', L') → t1', l', i', k'
  45. Mirror(I', f) → I''
  46. Mirror(K', f) → K''
  47. Mirror(L', f) → L''
  48. Polygon(I'', K'', L'') → t1'', l'', i'', k''
  49. Midpoint(O, F') → M
  50. Polygon(I, O, M) → t2, m, i_2, o_3
  51. AngularBisector(F', O, B'_1) → n
  52. Intersect(d, n) → N, P
  53. Mirror(O, o_3) → O'_2
  54. Mirror(M, o_3) → M'
  55. Segment(O'_2, M') → j_1
  56. Intersect(p, n) → Q, R
  57. CircleArc(O, B, B') → q
  58. CircleArc(O, I, I') → r
  59. CircleArc(O, I, B') → s
  60. Rotate(B, 4°, O) → B'_5
  61. CircleArc(O, P, B'_5) → t
  62. CircleArc(O, R, I) → e_1
  63. CircleArc(O, R, I) → p_1
  64. CircleArc(O, P, B') → q_1
  65. Intersect(c, n) → G_1, G_2
  66. Intersect(c, h) → U, V
  67. Intersect(c, f) → W, Z
  68. AngularBisector(U, O, Z) → l_1
  69. Intersect(c, l_1) → A_1, B_1
  70. AngularBisector(U, O, B_1) → m_1
  71. Intersect(c, m_1) → C_1, D_1
  72. Intersect(c, g) → E_1, F_1
  73. CircleArc(O, U, B'_1) → c_1
  74. CircleArc(O, B', B'_1) → d_1
  75. CircleArc(O, I', I'') → r_1
  76. CircleArc(O, U, F_1) → s_1
  77. CircleArc(O, I', I'') → t_1
  78. CircleArc(O, G_1, B') → c_2
  79. Polygon(G_1, N, B', U) → c4, n_1, n_2, a_1, u_1
  80. CircleArc(O, F_1, G_2) → d_2
  81. Polygon(G_2, P, B'_1, F_1) → c5, g_2, p_2, b'_1, f_2
  82. CircleArc(O, B'_1, P) → e_2
  83. CircleArc(O, I'', R) → k_2
  84. CircleArc(O, I'', R) → q_2
  85. CircleArc(O, G_2, V) → s_2
  86. Intersect(d, t, 1) → T
  87. Intersect(d, h) → I_1, J_1
  88. Polygon(V, J_1, T, G_2) → c7, v, l_2, t_2, m_2
  89. CircleArc(O, V, C_1) → c_3
  90. Intersect(d, m_1) → K_1, L_1
  91. Polygon(C_1, K_1, J_1, V) → c8, a_2, b_2, f_3, v_1
  92. Intersect(d, l_1) → S, H_1
  93. Polygon(A_1, S, K_1, C_1) → c9, g_3, s_3, h_3, i_3
  94. CircleArc(O, A_1, A) → d_3
  95. Polygon(A, A_1, S, B) → c10, a_3, j_3, s_4, b_3
  96. Rotate(P, 8°, O) → P'
  97. Rotate(R, 8°, O) → R'
  98. Rotate(I'', (-(4°)), O'') → I'''
  99. Line(O'', B'') → e
  100. Intersect(c, e) → M_1, N_1
  101. Rotate(N_1, -1°, O) → O_1
  102. Polygon(O_1, N_1, E, O) → c3, j_2, k_3, e_3, o_2
  103. Rotate(B'', 2°, O) → B'''_1
  104. CircleSector(O, B''_1, B'''_1) → q_3
  105. Mirror(q_3, f) → t_3
  106. Mirror(O_1, f) → P_1
  107. Mirror(N_1, f) → Q_1
  108. Mirror(E, f) → E'_3
  109. Mirror(O, f) → O'_3
  110. Polygon(P_1, Q_1, E'_3, O'_3) → c3', l_3, m_3, e'_3, o'_3
  111. Polygon(R, G_2, B'_1, I'') → c6, r_2, h_2, n_3, p_3
  112. CircleArc(O, I'', R) → c_4
  113. Segment(I, R) → r_3
  114. Vector(O, F') → w
  115. Intersect(c_4, n_3, 1) → S_1
  116. Polygon(R_1, O, S_1) → t3, f_4, g_4, o_4
  117. Polygon(R, P, R') → t4, r', r_4, p_4
  118. CircleArc(O, S_1, R) → d_4
  119. Intersect(r', r_4) → T_1
  120. CircleArc(O, C, T_1) → e_4