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Geometric Construction and Transformation of the Valdés Investment Logo

Geometric Construction and Transformation of the Valdés Investment Logo
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2020-02-04 06:39
PolygonAngleVectorTransformationGeometric Construction
① This GeoGebra example constructs a logo named 'Inversiones Valdés' by combining multiple geometric transformations and construction tools, resulting in a composite emblem featuring triangles, rectangles, a parabola, circles, and 3D-like elements. ② Core mathematical concepts include 2D vectors, angles, similarity transformations, reflection, geometric transformations (translation, rotation, mirroring), angle bisectors, perpendicular bisectors, and parametric Cartesian curves, demonstrating systematic geometric construction and composite transformations. ③ Construction highlights: From the summary, commands such as LineBisector, Vector, OrthogonalVector, Midpoint, OrthogonalLine, Line, Intersect, Polygon, Mirror, CurveCartesian, Circle, Translate, Rotate, and AngularBisector are used, along with numerous auxiliary points (A, B, J, K, F, etc.) and text/image objects, ultimately forming the logo through transformation combinations. ④ Target audience: Suitable for high school and above, useful for teaching geometric transformations, vectors, and coordinate geometry, as well as a creative example linking mathematics with graphic design.

GeoGebra Commands & Parameters

  1. LineBisector(A, B) → m
  2. Vector(A, B) → u
  3. OrthogonalVector(u) → v
  4. Midpoint(A_1, A) → J
  5. OrthogonalLine(B, u, xOyPlane) → g
  6. Line(A, B) → s
  7. OrthogonalLine(A, s) → b
  8. OrthogonalLine(A_1, b) → r
  9. Intersect(g, r) → C
  10. OrthogonalLine(J, m) → i
  11. OrthogonalLine(K, i) → l
  12. OrthogonalLine(F, l) → n
  13. Intersect(l, n) → Q
  14. Polygon(A_1, C, B, A) → q1, a_1, c, b_1, a_2
  15. Intersect(n, m) → U
  16. Mirror(n, i) → n'
  17. Mirror(l, m) → l'
  18. Intersect(l', n) → E
  19. OrthogonalLine(G, n) → f
  20. CurveCartesian(((1 - t)^(6) * D) + (((6 * (1 - t)^(5)) * t) * O) + (((15 * (1 - t)^(4)) * t^(2)) * P) + (((20 * (1 - t)^(3)) * t^(3)) * S) + (((15 * (1 - t)^(2)) * t^(4)) * T) + (((6 * (1 - t)) * t^(5)) * R) + (t^(6) * H), t, 0, 1) → d
  21. CurveCartesian(((1 - t)^(6) * D) + (((6 * (1 - t)^(5)) * t) * I) + (((15 * (1 - t)^(4)) * t^(2)) * U) + (((20 * (1 - t)^(3)) * t^(3)) * L) + (((15 * (1 - t)^(2)) * t^(4)) * M) + (((6 * (1 - t)) * t^(5)) * N) + (t^(6) * H), t, 0, 1) → e
  22. Intersect(l, n') → V
  23. Intersect(m, n') → W
  24. LineBisector(V, W) → h
  25. Intersect(h, n') → Z
  26. Circle(B_1, Z) → k
  27. Intersect(h, d) → C_1, D_1, E_1, F_1, G_1, H_1
  28. Intersect(h, i) → J_1
  29. CurveCartesian(((1 - t)^(4) * I_1) + (((4 * (1 - t)^(3)) * t) * L_1) + (((6 * (1 - t)^(2)) * t^(2)) * J_1) + (((4 * (1 - t)) * t^(3)) * K_1) + (t^(4) * E_1), t, 0, 1) → a
  30. CurveCartesian(((1 - t)^(4) * M_1) + (((4 * (1 - t)^(3)) * t) * N_1) + (((6 * (1 - t)^(2)) * t^(2)) * O_1) + (((4 * (1 - t)) * t^(3)) * P_1) + (t^(4) * E_1), t, 0, 1) → j
  31. CurveCartesian(((1 - t)^(4) * M_1) + (((4 * (1 - t)^(3)) * t) * V_1) + (((6 * (1 - t)^(2)) * t^(2)) * U_1) + (((4 * (1 - t)) * t^(3)) * T_1) + (t^(4) * Q_1), t, 0, 1) → q
  32. CurveCartesian(((1 - t)^(4) * I_1) + (((4 * (1 - t)^(3)) * t) * S_1) + (((6 * (1 - t)^(2)) * t^(2)) * W_1) + (((4 * (1 - t)) * t^(3)) * R_1) + (t^(4) * Q_1), t, 0, 1) → p
  33. Translate(k, Vector[(0.03 * u)]) → k'_1
  34. Polygon(I_1, E_1, Q_1) → t1, q_1, i_1, e_1
  35. Polygon(E_1, Q_1, M_1) → t2, m_1, t, f_1
  36. Mirror(a, m) → a'
  37. Mirror(j, m) → j'
  38. Mirror(p, m) → p'
  39. Mirror(q, m) → q'
  40. Mirror(I_1, m) → Z_1
  41. Mirror(E_1, m) → A_2
  42. Mirror(Q_1, m) → B_2
  43. Polygon(Z_1, A_2, B_2) → t1', b_2, z_1, g_1
  44. Mirror(E_1, m) → C_2
  45. Mirror(Q_1, m) → D_2
  46. Mirror(M_1, m) → E_2
  47. Polygon(C_2, D_2, E_2) → t2', e_2, c_2, d_2
  48. Translate(a', Vector[(-0.04 * v)]) → a''
  49. Translate(j', Vector[(-0.04 * v)]) → j''
  50. Translate(p', Vector[(-0.04 * v)]) → p''
  51. Translate(q', Vector[(-0.04 * v)]) → q''
  52. Translate(Z_1, Vector[(-0.04 * v)]) → F_2
  53. Translate(A_2, Vector[(-0.04 * v)]) → G_2
  54. Translate(B_2, Vector[(-0.04 * v)]) → H_2
  55. Polygon(F_2, G_2, H_2) → t1'', h_2, f_2, g_2
  56. Translate(C_2, Vector[(-0.04 * v)]) → I_2
  57. Translate(D_2, Vector[(-0.04 * v)]) → J_2
  58. Translate(E_2, Vector[(-0.04 * v)]) → K_2
  59. Polygon(I_2, J_2, K_2) → t2'', k_2, i_2, j_2
  60. Rotate(F_2, 45°, H_2) → L_2
  61. Rotate(G_2, 45°, H_2) → M_2
  62. Rotate(H_2, 45°, H_2) → N_2
  63. Rotate(I_2, 45°, H_2) → O_2
  64. Rotate(J_2, 45°, H_2) → P_2
  65. Rotate(K_2, 45°, H_2) → Q_2
  66. Rotate(F_2, 30°, H_2) → R_2
  67. Rotate(G_2, 30°, H_2) → S_2
  68. Rotate(H_2, 30°, H_2) → T_2
  69. Rotate(I_2, 30°, H_2) → U_2
  70. Rotate(J_2, 30°, H_2) → V_2
  71. Rotate(K_2, 30°, H_2) → W_2
  72. Rotate(a'', 15°, H_2) → a'''
  73. Rotate(j'', 15°, H_2) → j'''
  74. Rotate(p'', 15°, H_2) → p'''
  75. Rotate(q'', 15°, H_2) → q'''
  76. Rotate(F_2, 15°, H_2) → Z_2
  77. Rotate(G_2, 15°, H_2) → A_3
  78. Rotate(H_2, 15°, H_2) → B_3
  79. Polygon(Z_2, A_3, B_3) → t1''', b_3, z_2, a_3
  80. Rotate(I_2, 15°, H_2) → C_3
  81. Rotate(J_2, 15°, H_2) → D_3
  82. Rotate(K_2, 15°, H_2) → E_3
  83. Polygon(C_3, D_3, E_3) → t2''', e_3, c_3, d_3
  84. Translate(k, Vector[(0.28 * u) - (0.05 * v)]) → k'
  85. OrthogonalLine(N_2, g, xOyPlane) → j_1
  86. Intersect(g, i) → F_3
  87. Intersect(g, j_1) → G_3
  88. LineBisector(F_3, G_3, xOyPlane) → k_1
  89. Intersect(m, k_1) → H_3
  90. Polygon(I_3, J_3, 4, xOyPlane) → poly1, h_1, l_1, n_1, p_1, K_3, L_3
  91. Mirror(poly1, k_1) → poly1'
  92. Midpoint(I_3, J_3) → M_3
  93. Mirror(N_3, M_3) → O_3
  94. AngularBisector(I_3, J_3, P_3) → r_1
  95. OrthogonalLine(I_3, r_1, xOyPlane) → s_1
  96. Intersect(r_1, s_1) → Q_3
  97. Mirror(N_3, Q_3) → R_3
  98. Mirror(O_3, Q_3) → S_3
  99. Mirror(O_3, s_1) → T_3
  100. Mirror(N_3, s_1) → U_3
  101. Mirror(O_3, r_1) → V_3
  102. Mirror(N_3, r_1) → W_3
  103. Polygon(O_3, N_3, S_3, R_3) → q2, o_3, n_3, s_3, r_3
  104. Polygon(T_3, W_3, V_3, U_3) → q3, t_3, w_3, v_3, u_3

AI Drawing Prompt Reference

Create a composite emblem by first constructing a quadrilateral with perpendicular bisectors and intersections to form a base framework. Then apply mirroring, translation, and rotation to symmetrically duplicate triangles and curved elements on both sides. Include a smooth parabolic curve, a circle, and petal-like regions defined by Bezier curves, resulting in a balanced, layered logo with 3D-like depth.
The prompt above was compiled from the original drawing commands and is for reference only.