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Geometric Construction of Banco de Santander Logo

Geometric Construction of Banco de Santander Logo
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2019-10-04 00:30
PolygonVectorFunctionTransformationGeometric Construction
This example demonstrates how to geometrically construct a simplified version of the Banco de Santander logo. It involves core mathematical concepts such as 2D vectors, angles, area, function graphs, similarity transformations, incircle, tangent function, logarithm, median line, and equilateral triangles. The construction utilizes GeoGebra commands and objects like Polygon, Midpoint, Line, Vector, OrthogonalVector, Mirror, OrthogonalLine, Intersect, Rotate, CurveCartesian, Segment, Angle, LineBisector, and CircleArc. It is suitable for high school mathematics education, showcasing the integration of geometric transformations and function graphing.

GeoGebra Commands & Parameters

  1. Polygon(A, B, 4) → poly1, f, g, h, i, C, D
  2. Midpoint(D, A) → E
  3. Midpoint(D, B) → G
  4. Line(E, G) → j
  5. Vector(A, B) → u
  6. OrthogonalVector(u) → v
  7. Line(L, j) → k
  8. Mirror(k, j) → k'
  9. Line(A, B) → m
  10. OrthogonalLine(G, m) → n
  11. Intersect(n, k') → C_1
  12. Rotate(C_1, 30°, K) → Q
  13. Line(Q, K) → s
  14. OrthogonalLine(F, m) → l
  15. Intersect(l, k') → O_2
  16. Intersect(l, j) → I
  17. CurveCartesian(((1 - t)^(3) * C_1) + (((3 * (1 - t)^(2)) * t) * N) + (((3 * (1 - t)) * t^(2)) * M) + (t^(3) * I), t, 0, 1) → c
  18. Intersect(s, c) → R, S, U
  19. Midpoint(U, C_1) → A_1
  20. Line(A_1, K) → f_1
  21. Intersect(f_1, c) → B_1, D_1, E_1
  22. Mirror(E_1, n) → F_1
  23. Mirror(U, n) → U'
  24. Midpoint(U, U') → P
  25. Mirror(K, P) → K'_1
  26. Segment(K'_1, F_1) → d_1
  27. CurveCartesian(((1 - t)^(3) * K'_1) + (((3 * (1 - t)^(2)) * t) * P_1) + (((3 * (1 - t)) * t^(2)) * O) + (t^(3) * U'), t, 0, 1) → h_1
  28. Intersect(d_1, h_1) → Q_1, T_1, U_1
  29. Midpoint(U, K'_1) → Z
  30. Mirror(L, j) → L'
  31. Mirror(F, n) → F'
  32. Mirror(H, n) → H'
  33. OrthogonalLine(H, m) → p
  34. OrthogonalLine(H, f) → q
  35. OrthogonalLine(F', m) → a
  36. OrthogonalLine(H', m) → b
  37. Mirror(I, n) → I'
  38. Mirror(C_1, j) → J
  39. Mirror(c, n) → c'
  40. Mirror(c, j) → c'_1
  41. Mirror(c'_1, n) → d
  42. Polygon(C_1, I, J, I') → q1, c_1, i_1, j_1, i'
  43. Angle(C_1, K, Q) → α
  44. Segment(U, K) → t
  45. Mirror(U, n) → U'_1
  46. Mirror(K, n) → K'
  47. Segment(U'_1, K') → t'
  48. Segment(K'_1, U) → e
  49. Mirror(K'_1, n) → T
  50. Segment(T, U') → e'
  51. Midpoint(K, U') → W
  52. Vector(K, W) → r
  53. Line(C, B) → g_2
  54. Line(O_1, b) → j_2
  55. CurveCartesian(((1 - t)^(3) * U) + (((3 * (1 - t)^(2)) * t) * H_1) + (((3 * (1 - t)) * t^(2)) * G_1) + (t^(3) * K), t, 0, 1) → w
  56. CurveCartesian(((1 - t)^(3) * E_1) + (((3 * (1 - t)^(2)) * t) * I_1) + (((3 * (1 - t)) * t^(2)) * J_1) + (t^(3) * K), t, 0, 1) → a_1
  57. CurveCartesian(((1 - t)^(3) * Z) + (((3 * (1 - t)^(2)) * t) * K_1) + (((3 * (1 - t)) * t^(2)) * V) + (t^(3) * P), t, 0, 1) → b_1
  58. CurveCartesian(((1 - t)^(3) * K'_1) + (((3 * (1 - t)^(2)) * t) * L_1) + (((3 * (1 - t)) * t^(2)) * M_1) + (t^(3) * F_1), t, 0, 1) → e_1
  59. CurveCartesian(((1 - t)^(2) * F_1) + (((2 * (1 - t)) * t) * N_1) + (t^(2) * W), t, 0, 1) → g_1
  60. Polygon(U, E_1, W_1) → t1, m_1, i_2, k_2
  61. Polygon(Z, E_1, P) → t2, p_1, z_1, n_1
  62. Polygon(U_1, U', F_1) → t4, t_1, f_2, h_2
  63. Polygon(C_1, F_1, W) → t3, q_1, r_1, s_1
  64. Segment(P, W) → k_1
  65. CurveCartesian(((1 - t)^(3) * U_1) + (((3 * (1 - t)^(2)) * t) * V_1) + (((3 * (1 - t)) * t^(2)) * O) + (t^(3) * U'), t, 0, 1) → o_1
  66. Intersect(j, w) → Z_1, A_2, B_2
  67. LineBisector(B_2, U) → l_2
  68. Intersect(l_2, g_2) → D_2
  69. CircleArc(D_2, U, B_2) → c_2
  70. CurveCartesian(((1 - t)^(2) * Z) + (((2 * (1 - t)) * t) * S_1) + (t^(2) * C_2), t, 0, 1) → l_1
  71. Segment(P, W) → n_2
  72. Intersect(l_2, k_1) → E_2
  73. LineBisector(E_2, W) → m_2
  74. Segment(E_2, W) → p_2
  75. Intersect(m_2, j) → F_2
  76. CircleArc(F_2, W, E_2) → d_2
  77. Segment(E_2, W) → q_2

AI Drawing Prompt Reference

Draw a simplified Banco de Santander logo consisting of a quadrilateral and various helper lines, vectors. Start by creating a quadrilateral, then find the midpoints of opposite sides and draw lines connecting them. Based on one side of the quadrilateral, create a vector and its orthogonal vector, using these to draw auxiliary lines. Apply transformations like rotation and mirroring, along with midpoint and intersection calculations, to generate the core curves and shapes of the logo. The final image should exhibit symmetry and geometric elegance.
The prompt above was compiled from the original drawing commands and is for reference only.