MathPlotAI Logo MathPlotAI

Geometric Construction of the Vall Companys Logo with Vectors and Transformation

Geometric Construction of the Vall Companys Logo with Vectors and Transformation
Loading the math board and drawing, please wait…
2020-01-13 05:12
CirclePolygonVectorTransformationGeometric Construction
This GeoGebra example reconstructs the 'Vall Companys' logo through a step-by-step geometric construction, demonstrating how to combine vectors, perpendicular lines, angle bisectors, polygons, midpoints, orthogonals, intersections, centroids, reflections, and circular arcs to form a complex planar design. The core mathematical concepts include two-dimensional vector operations, geometric transformations such as translation and reflection, congruence and symmetry, and the integration of polygonal and circular elements. The construction begins by defining points A and B and vectors u and v, using Vector and OrthogonalVector to establish directions. Then LineBisector, Midpoint, OrthogonalLine, and Intersect create auxiliary lines and key points. Polygon and Centroid build the main shape, while Mirror applies symmetry, and CircleArc adds curved details. Text objects label the final figure. This example is suitable for high school or early college students learning geometric construction, vector applications, and transformation geometry, and it also serves as an interdisciplinary project linking mathematics with graphic design.

GeoGebra Commands & Parameters

  1. Vector(A, B) → u
  2. OrthogonalVector(u) → v
  3. LineBisector(A, B) → m
  4. Polygon(A, B, 4) → poly1, f, h, j, k, D, E
  5. Midpoint(E, A) → J
  6. OrthogonalLine(J, m) → i
  7. OrthogonalLine(K, i) → l
  8. OrthogonalLine(F, l) → n
  9. Intersect(l, n) → Q
  10. OrthogonalLine(M, i) → a
  11. Intersect(a, i) → N
  12. Centroid(poly1) → F_3
  13. Line(A, F_3) → c
  14. Line(A, B) → s
  15. OrthogonalLine(A, s) → b
  16. Intersect(m, n) → H
  17. Mirror(n, i) → n'
  18. Intersect(m, n') → W
  19. OrthogonalLine(B, u, xOyPlane) → g
  20. Mirror(J, m) → J'
  21. Mirror(l, m) → l'
  22. OrthogonalLine(G, l) → p
  23. LineBisector(R, P) → q
  24. OrthogonalLine(M_3, b) → g_4
  25. OrthogonalLine(R, s) → j_4
  26. OrthogonalLine(W_3, q) → k_4
  27. Intersect(l, g_4) → Z_3
  28. Intersect(k_4, l) → C_4
  29. Mirror(p, i) → p'
  30. Mirror(j_4, m) → f_1
  31. Mirror(g_4, i) → g_1
  32. OrthogonalLine(C, b) → r
  33. OrthogonalLine(L, i) → t
  34. Mirror(t, m) → t'
  35. Mirror(M_3, i) → U
  36. Line(V, i) → h_1
  37. Mirror(S, c) → S'
  38. Intersect(c, t) → A_1
  39. Vector(A_1, F_3) → i_1
  40. CircleArc(B_1, S', S) → c_1
  41. OrthogonalLine(S, s) → j_1
  42. Intersect(j_1, p) → C_1
  43. Mirror(C_1, c) → D_1
  44. CircleArc(B_1, D_1, C_1) → p_1
  45. Intersect(j_4, i) → E_1
  46. Polygon(E_1, K, S', S, H, F_3) → poly2, l_1, k_2, s', s_1, h_2, f_3
  47. Intersect(m, p) → F_1
  48. Polygon(E_1, D_1, C_1, F_1, F_3) → poly3, m_1, n_1, q_1, r_1, t_1
  49. Mirror(c_1, i) → d_1
  50. Mirror(c_1, m) → e_1
  51. Mirror(d_1, m) → k_1
  52. Mirror(p_1, m) → c_2
  53. Mirror(c_2, i) → d_2
  54. Mirror(c_2, m) → e_2
  55. Mirror(E_1, m) → G_1
  56. Mirror(K, m) → K'
  57. Mirror(S', m) → S''
  58. Mirror(S, m) → S'_1
  59. Mirror(H, m) → H'
  60. Mirror(F_3, m) → H_1
  61. Polygon(G_1, K', S'', S'_1, H', H_1) → poly2', a_1, k', s'', s'_1, h', b_1
  62. Mirror(E_1, m) → I_1
  63. Mirror(D_1, m) → J_1
  64. Mirror(C_1, m) → K_1
  65. Mirror(F_1, m) → L_1
  66. Mirror(F_3, m) → M_1
  67. Polygon(I_1, J_1, K_1, L_1, M_1) → poly3', f_2, g_2, i_2, j_2, l_2
  68. Mirror(E_1, i) → N_1
  69. Mirror(K, i) → K'_1
  70. Mirror(S', i) → S''_1
  71. Mirror(S, i) → S'_2
  72. Mirror(H, i) → H'_1
  73. Mirror(F_3, i) → O_1
  74. Polygon(N_1, K'_1, S''_1, S'_2, H'_1, O_1) → poly2'_1, m_2, k'_1, s''_1, s'_2, h'_1, o_1
  75. Mirror(E_1, i) → P_1
  76. Mirror(D_1, i) → Q_1
  77. Mirror(C_1, i) → R_1
  78. Mirror(F_1, i) → S_1
  79. Mirror(F_3, i) → T_1
  80. Polygon(P_1, Q_1, R_1, S_1, T_1) → poly3'_1, n_2, p_2, q_2, r_2, s_2
  81. Mirror(N_1, m) → U_1
  82. Mirror(K'_1, m) → V_1
  83. Mirror(S''_1, m) → W_1
  84. Mirror(S'_2, m) → Z_1
  85. Mirror(H'_1, m) → A_2
  86. Mirror(O_1, m) → B_2
  87. Polygon(U_1, V_1, W_1, Z_1, A_2, B_2) → poly4, u_1, v_1, w_1, z_1, a_2, b_2
  88. Mirror(P_1, m) → C_2
  89. Mirror(Q_1, m) → D_2
  90. Mirror(R_1, m) → E_2
  91. Mirror(S_1, m) → F_2
  92. Mirror(T_1, m) → G_2
  93. Polygon(C_2, D_2, E_2, F_2, G_2) → poly5, t_2, g_3, h_3, i_3, j_3
  94. OrthogonalLine(Z, p') → k_3
  95. Intersect(t, g_1) → I_2
  96. Intersect(a, h_1) → J_2
  97. Intersect(t, g_4) → K_2
  98. Line(K_2, c) → d
  99. Mirror(d, c) → d'
  100. Intersect(d, g_1) → H_2
  101. Line(T, d) → e
  102. Intersect(e, g_1) → L_2
  103. Polygon(I_2, J_2, T, L_2, H_2, K_2) → poly6, l_3, m_3, t_3, n_3, p_3, q_3
  104. Mirror(e, c) → e'
  105. Intersect(d', k_4) → M_2
  106. OrthogonalLine(N_2, i) → r_3
  107. Intersect(d', r_3) → O_2
  108. Intersect(e', r_3) → P_2
  109. Intersect(k_4, e') → Q_2
  110. Intersect(k_3, g_4) → R_2
  111. Intersect(g_4, r_3) → S_2
  112. OrthogonalLine(L_2, g_1) → s_3
  113. Intersect(s_3, r) → T_2
  114. Intersect(s_3, k_4) → U_2
  115. Polygon(O_2, P_2, Q_2, U_2, S_2, R_2, M_2) → poly7, o_2, a_3, b_3, u_2, c_3, d_3, e_3
  116. Intersect(t', g_4) → V_2
  117. Intersect(t', r) → W_2
  118. Polygon(W_2, T_2, V_2) → t1, v_2, w_2, f_4
  119. LineBisector(I_2, J_2) → h_4
  120. Intersect(h_4, t) → Z_2
  121. CircleArc(Z_2, J_2, I_2) → c_4
  122. LineBisector(M_2, R_2) → i_4
  123. Intersect(m, i_4) → A_3
  124. Intersect(k_3, i_4) → B_3
  125. Vector(A_3, B_3) → l_4
  126. CircleArc(C_3, M_2, R_2) → d_4
  127. LineBisector(T_2, V_2) → m_4
  128. Intersect(m_4, g_4) → D_3
  129. Midpoint(F_1, D_3) → E_3
  130. CircleArc(E_3, V_2, T_2) → e_4
  131. LineBisector(S_2, U_2) → n_4
  132. CircleArc(D_3, S_2, T_2) → p_4
  133. Polygon(G_3, H_3, W_2) → t2, q_4, r_4, s_4