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Geometric Construction and Transformation of the TEKA Logo

Geometric Construction and Transformation of the TEKA Logo
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2020-05-07 17:27
CirclePolygonVectorTransformationGeometric Construction
This example demonstrates how to construct a complex logo named 'TEKA' using GeoGebra's geometric construction and transformation tools, starting from basic elements. The core mathematical concepts include two-dimensional vectors, circles and circular arcs, polygons, line segments, mirror reflection, geometric transformations, as well as the visualization of abstract concepts such as expected value and logarithm within the graphic. The construction highlights the use of commands like Line, OrthogonalLine, Vector, OrthogonalVector, LineBisector, Mirror, Intersect, Midpoint, Segment, CircumcircleArc, Polygon, Semicircle, and CircleArc, creating numerous objects (e_2, e_1, B, A, s, b, u, v, F, f, E_1, s_1, A_1, h, f', G, k, H, l, V, f_1, G_1, t_1, f_2, D, i, L, M, I, n, K, F_2, t, n_3, g, m, E, N_1, C_2, D_2) that are combined through precise geometric relationships. It is suitable for high school and above students or teachers to explore the integrated application of geometric transformations, vector operations, and figure construction, and can also serve as a reference for mathematical art design.

GeoGebra Commands & Parameters

  1. Line(A, B) → s
  2. OrthogonalLine(A, s) → b
  3. Vector(A, B) → u
  4. OrthogonalVector(u) → v
  5. OrthogonalLine(F, b) → f
  6. Line(E_1, f) → s_1
  7. LineBisector(A_1, A) → h
  8. Mirror(f, h) → f'
  9. Line(G, f') → k
  10. Line(H, k) → l
  11. OrthogonalLine(V, b) → f_1
  12. Line(G_1, f_1) → t_1
  13. LineBisector(G_1, V) → f_2
  14. OrthogonalLine(D, s) → i
  15. Intersect(i, l) → L
  16. Intersect(i, f) → I
  17. Line(M, I) → n
  18. Line(K, n) → t
  19. Line(F_2, t) → n_3
  20. OrthogonalLine(B, u, xOyPlane) → g
  21. LineBisector(A, B) → m
  22. Intersect(m, f) → E
  23. Mirror(i, m) → i'
  24. Intersect(i', k) → N
  25. Line(N, n) → p
  26. Line(C_2, p) → m_3
  27. Intersect(m_3, s_1) → H_2
  28. Intersect(p, f) → P
  29. Line(W, p) → g_1
  30. Intersect(f_2, g_1) → I_1
  31. Intersect(s_1, g_1) → J_1
  32. Midpoint(J_1, I_1) → K_1
  33. OrthogonalLine(K_1, g, xOyPlane) → k_2
  34. Intersect(n_3, k_2) → G_2
  35. Intersect(k_2, m_3) → E_2
  36. Segment(I_2, E_2) → p_3
  37. CircumcircleArc(D_2, G_2, E_2) → c_3
  38. Intersect(t_1, c_3) → J_2, K_2
  39. Line(J_2, p_3) → q_3
  40. Intersect(q_3, c_3) → L_2, M_2
  41. Midpoint(M_2, J_2) → N_2
  42. Vector(J_2, N_2) → u_2
  43. OrthogonalLine(A_2, g, xOyPlane) → l_3
  44. Line(E, t) → r_2
  45. Intersect(r_2, l_3) → B_2
  46. Intersect(k_2, r_2) → W_1
  47. Intersect(i, f') → J
  48. OrthogonalLine(A_1, b) → r
  49. Intersect(g, r) → C
  50. Mirror(J, m) → J'
  51. Mirror(J, h) → J'_1
  52. Mirror(J', h) → J''
  53. Polygon(A, B, C, A_1) → q1, a_1, b_1, c, j
  54. Intersect(p, l) → O
  55. Line(L, n) → q
  56. Intersect(q, k) → Q
  57. Polygon(Q, L, O, N) → q2, q_1, l_1, o, n_1
  58. Intersect(t, l) → R
  59. Polygon(K, I, M, R) → q3, k_1, i_1, m_1, r_1
  60. Intersect(t, k) → S
  61. OrthogonalLine(T, k) → d
  62. Intersect(d, f') → U
  63. Semicircle(T, U) → e
  64. Mirror(e, d) → e'
  65. Intersect(f_1, g_1) → Z
  66. Intersect(f_1, p) → B_1
  67. Polygon(Z, B_1, P, W) → q4, z_1, h_1, p_1, w
  68. Line(C_1, g_1) → j_1
  69. Intersect(j_1, f_1) → D_1
  70. Semicircle(C_1, D_1) → c_1
  71. Intersect(j_1, s_1) → F_1
  72. Intersect(t_1, j_1) → H_1
  73. Semicircle(F_1, H_1) → d_1
  74. Polygon(W, C_1, F_1, J_1) → q5, w_1, g_2, h_2, i_2
  75. LineBisector(D_1, I_1) → j_2
  76. Intersect(k_2, j_2) → L_1
  77. CircleArc(L_1, I_1, D_1, xOyPlane) → c_2
  78. LineBisector(K_1, H_1) → l_2
  79. Intersect(l_2, k_2) → M_1
  80. Polygon(D_1, I_1, K_1, H_1) → q6, m_2, n_2, p_2, q_2
  81. Line(N_1, r_2) → s_2
  82. Intersect(s_2, f_1) → O_1
  83. Intersect(r_2, f_1) → P_1
  84. OrthogonalLine(R_1, k) → t_2
  85. OrthogonalLine(Q_1, l) → a_2
  86. Intersect(a_2, f) → S_1
  87. Intersect(t_2, f) → T_1
  88. Intersect(t_1, r_2) → U_1
  89. Mirror(B_2, W_1) → Z_1
  90. Polygon(O_1, N_1, P_1, E, Z_1, S_1, T_1, V_1, R_1, Q_1, U_1) → poly1, o_1, u_1, k_3, b_2, e_3, f_3, g_3, h_3, v_1, i_3, j_3
  91. Mirror(O_2, N_2) → P_2
  92. Semicircle(P_2, O_2, xOyPlane) → d_3
  93. Polygon(J_2, E_2, I_2, H_2, C_2, D_2, M_2) → poly2, r_3, s_3, t_3, a_3, b_3, f_4, g_4
  94. Semicircle(M_2, J_2, xOyPlane) → c_4
  95. LineBisector(D_2, M_2, xOyPlane) → h_4
  96. Intersect(h_4, k_2) → Q_2
  97. CircleArc(Q_2, M_2, D_2, xOyPlane) → d_4
  98. Intersect(l_2, s_1) → R_2
  99. CircleArc(R_2, K_1, H_1) → d_2
  100. Segment(U, T) → i_4

AI Drawing Prompt Reference

Start with two base points and draw a line through them, then construct a perpendicular line through one point. Use the vector between the points to create an orthogonal vectorholine, and draw parallel lines through other points. Apply angle bisectors and mirror reflections to generate multiple auxiliary lines and intersection points, building a complex geometric framework. Add circles, arcs, polygons, and segments, connecting elements via midpoints, intersections, and perpendiculars, resulting in a symmetric, layered logo with nested shapes and precise crossings.
The prompt above was compiled from the original drawing commands and is for reference only.