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Vector & Mirror-Symmetry Geometric Construction for a Logo-Style Mark

Vector & Mirror-Symmetry Geometric Construction for a Logo-Style Mark
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2020-08-06 23:58
CirclePolygonVectorTransformationGeometric Construction
This GeoGebra example demonstrates how to construct a logo-like figure by combining geometric building blocks and symmetry. It highlights 2D vector representation, constructions related to angle bisectors, orthogonality (perpendicular lines/vectors), mirror (reflection) transformations, and precise placement using intersections. The construction workflow uses LineBisector to obtain bisectors, Vector and OrthogonalVector to define directions, and OrthogonalLine/Line to enforce perpendicular relationships. Mirror is then applied to generate the symmetric counterpart. Key points are determined with Intersect, while Polygon and Segment organize straight boundary parts. Finally, CircumcircleArc and CircleArc create smooth curved edges so the overall outline becomes continuous. Suitable for secondary school geometry extension, transformation-based geometry lessons, and students learning geometric construction in dynamic geometry software.

GeoGebra Commands & Parameters

  1. LineBisector(A, B) → m
  2. Vector(A, B) → u
  3. OrthogonalVector(u) → v
  4. LineBisector(A_1, A) → l
  5. Line(A, B) → s
  6. OrthogonalLine(D, s) → i
  7. Mirror(i, m) → i'
  8. OrthogonalLine(A, s) → b
  9. OrthogonalLine(A_1, b) → r
  10. OrthogonalLine(F, b) → f
  11. Mirror(f, l) → f'
  12. Intersect(i', f') → P
  13. Intersect(i, f') → E
  14. Mirror(I, m) → I'
  15. OrthogonalLine(M, r) → n
  16. Line(L, n) → p
  17. Line(I', p) → q
  18. Line(U, q) → a
  19. OrthogonalLine(N, b) → j
  20. Intersect(a, j) → C_1
  21. LineBisector(C_1, Q) → g_2
  22. Intersect(g_2, a) → S_1
  23. Intersect(q, g_2) → T_1
  24. Vector(S_1, T_1) → u_1
  25. Intersect(m, l) → K
  26. Intersect(i, l) → J
  27. Intersect(m, f') → S
  28. Mirror(f, l) → f'_1
  29. OrthogonalLine(G, f'_1) → h
  30. OrthogonalLine(B, u, xOyPlane) → g
  31. Intersect(g, r) → C
  32. Mirror(F, l) → F'
  33. Polygon(A_1, C, B, A) → c4, a_1, c_3, b_2, a_2
  34. OrthogonalLine(H, f'_1) → k
  35. Intersect(i', f) → J_1
  36. Intersect(f, m) → D_1
  37. Intersect(h, j) → O
  38. Mirror(R, l) → R'
  39. Mirror(T, l) → T'
  40. Intersect(n, j) → V
  41. Intersect(p, j) → W
  42. Intersect(q, j) → Z
  43. Line(I, k) → t
  44. Intersect(t, j) → B_1
  45. Line(T, l) → c
  46. Line(R', c) → e
  47. Line(T', c) → f_1
  48. Line(R, c) → h_1
  49. Intersect(h, l) → E_1
  50. Intersect(k, l) → H_1
  51. Intersect(k, j) → I_1
  52. Polygon(E, I, B_1, I_1, H_1, E_1, O, N) → poly1, e_3, i_1, b_1, d, g_1, j_1, o, n_1
  53. Segment(B_1, D_1) → k_1
  54. Intersect(f, t) → K_1
  55. CircumcircleArc(I, K, K_1) → c_1
  56. Mirror(B_1, l) → L_1
  57. CircumcircleArc(B_1, M_1, L_1) → d_1
  58. Mirror(E, l) → E'
  59. Mirror(I, l) → I'_1
  60. Mirror(B_1, l) → N_1
  61. Mirror(I_1, l) → O_1
  62. Mirror(H_1, l) → P_1
  63. Mirror(E_1, l) → Q_1
  64. Mirror(O, l) → O'
  65. Mirror(N, l) → N'
  66. Polygon(E', I'_1, N_1, O_1, P_1, Q_1, O', N') → poly1', e', i'_1, l_1, o_1, p_1, q_1, o', n'
  67. Line(D_1, Z) → m_1
  68. Line(W, m_1) → r_1
  69. Intersect(r_1, k_1) → R_1
  70. Polygon(D_1, R_1, W, V, M, P, Q, C_1, Z) → poly2, s_1, t_1, w, v_1, m_2, p_2, q_2, f_2, z_1
  71. CircleArc(U_1, Q, C_1) → c_2
  72. Intersect(m_1, e) → W_1
  73. Intersect(m_1, h_1) → Z_1
  74. OrthogonalLine(U_1, j) → h_2
  75. Intersect(h_2, c) → F_1
  76. Intersect(f_1, h_2) → G_1
  77. OrthogonalLine(V_1, f) → i_2
  78. Intersect(i_2, h_1) → A_2
  79. Intersect(i_2, e) → B_2
  80. Polygon(W_1, B_2, G_1, F_1, A_2, Z_1) → poly3, w_1, j_2, k_2, l_2, n_2, r_2
  81. LineBisector(A_2, F_1) → s_2
  82. Intersect(s_2, q) → C_2
  83. CircleArc(C_2, A_2, F_1) → d_2
  84. Segment(B_1, N_1) → f_3
  85. Segment(C_1, Q) → g_3
  86. Segment(F_1, A_2) → h_3
  87. Segment(B_2, G_1) → i_3
  88. Mirror(d_2, l) → t_2
  89. Intersect(i_2, l) → D_2
  90. Intersect(a, j) → E_2

AI Drawing Prompt Reference

On a 2D plane, construct a logo-like geometric figure made from straight and mirrored parts. First determine two main direction lines using an angle-bisector type construction from given endpoints, then create vectors and corresponding perpendicular directions to draw several constraint lines. Use intersections of these perpendicular and direction lines to locate the main corner points. Reflect one set of the constructed lines across the appropriate symmetry lines to obtain the corresponding mirrored lines and points. Connect the resulting intersection points to form a closed polygonal outline, and replace selected straight edges with smooth circular arcs so the overall boundary becomes continuous. The final mark should exhibit mirror symmetry.
The prompt above was compiled from the original drawing commands and is for reference only.