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Geometric Construction of the Air France Logo with Fractal Transformations

Geometric Construction of the Air France Logo with Fractal Transformations
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2019-12-23 06:00
PolygonVectorTransformationParametric EquationGeometric Construction
This GeoGebra example reconstructs the Air France logo through precise geometric construction, demonstrating how lines, polygons, vectors, and geometric transformations can generate a logo with fractal-like features. Core mathematical concepts include 2D vectors, straight lines and perpendiculars, angle bisectors, midpoints, mirror symmetry, polygons, and centroid/barycenter, along with parametric curves and fractal geometry. The construction uses commands such as Line, OrthogonalLine, Vector, OrthogonalVector, LineBisector, Midpoint, Intersect, Mirror, and Polygon, building auxiliary lines, finding intersection points, and applying reflections to progressively assemble the logo's components, with text and image objects for labeling and decoration. This example is suitable for high school or college geometry lessons and for interdisciplinary projects linking mathematics and design, helping students understand how geometric transformations apply to real-world graphic design.

GeoGebra Commands & Parameters

  1. Line(A, B) → s
  2. OrthogonalLine(A, s) → b
  3. Vector(A, B) → u
  4. OrthogonalVector(u) → v
  5. LineBisector(A, C) → d
  6. Midpoint(C, A) → J
  7. OrthogonalLine(K, d) → i
  8. Intersect(i, s) → G
  9. OrthogonalLine(R, b) → l
  10. Mirror(l, d) → l'
  11. Intersect(i, l') → F
  12. LineBisector(A, B) → m
  13. Intersect(m, d) → E
  14. OrthogonalLine(C, v, xOyPlane) → f
  15. Intersect(f, m) → U
  16. OrthogonalLine(B, u, xOyPlane) → g
  17. Intersect(g, f) → D
  18. Intersect(m, d) → I
  19. Polygon(A, B, D, C) → poly2, j, k, q, r
  20. Mirror(i, m) → i'
  21. Mirror(R, I) → R'
  22. Line(V, R) → h
  23. Line(L, h) → n
  24. Line(M, n) → p
  25. Line(O, p) → t
  26. Line(P, t) → a
  27. Line(Q, a) → c
  28. Line(S, c) → e
  29. Line(T, e) → f_1
  30. Line(R', f_1) → g_1
  31. Line(H, h) → h_1
  32. Intersect(l, h_1) → W
  33. Intersect(l, n) → Z
  34. Intersect(l, p) → A_1
  35. Intersect(l, t) → B_1
  36. Intersect(l, a) → C_1
  37. Intersect(l, c) → D_1
  38. Intersect(l, e) → E_1
  39. Intersect(l, f_1) → F_1
  40. Intersect(l, g_1) → G_1
  41. Polygon(V, H, W, R) → q1, v_1, h_2, w, r_1
  42. Polygon(L, M, A_1, Z) → q2, l_1, m_1, a_1, z_1
  43. Polygon(O, P, C_1, B_1) → q3, o_1, p_1, c_1, b_1
  44. Polygon(Q, S, E_1, D_1) → q4, q_1, s_1, e_1, d_1
  45. Polygon(T, R', G_1, F_1) → q5, t_1, r', i_1, j_1