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2D Vectors and Reflection Symmetry

2D Vectors and Reflection Symmetry
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2020-02-12 05:34
VectorTransformationGeometric Construction
①This example demonstrates 2D vectors, reflection symmetry, and polygon construction; ②Core math concepts include vector operations, reflection transformations, symmetry, polygon construction, and analytical geometry; ③Construction highlights: utilizes commands/objects like LineBisector (angle bisector), Vector, OrthogonalVector, OrthogonalLine, Line, Intersect, Mirror, Polygon, Segment, CurveCartesian, CircleArc; ④Suitable for high school mathematics education, helping students understand vectors, symmetry transformations, and geometric constructions.

GeoGebra Commands & Parameters

  1. LineBisector(A, B) → m
  2. Vector(A, B) → u
  3. OrthogonalVector(u) → v
  4. OrthogonalLine(J, m) → i
  5. OrthogonalLine(R, i) → j
  6. Line(A, B) → s
  7. OrthogonalLine(A, s) → b
  8. OrthogonalLine(F, b) → f
  9. OrthogonalLine(T, b) → d
  10. Intersect(j, d) → L
  11. Mirror(L, m) → L'
  12. Polygon(L, L', 3) → poly1, k, l, n, M
  13. Line(R, l) → p
  14. Mirror(p, m) → p'
  15. Intersect(p', f) → S_1
  16. Line(W_2, p') → k_4
  17. Line(O_2, d) → t
  18. OrthogonalLine(G_2, b) → a_3
  19. OrthogonalLine(F_1, i) → j_1
  20. Intersect(j_1, t) → V
  21. Intersect(p, d) → I
  22. Line(I, j) → a
  23. Line(L, M) → i_3
  24. Line(V, i_3) → f_1
  25. Intersect(a_3, f_1) → K
  26. Segment(Mirror[M, f_1], Mirror[L, f_1]) → q'
  27. OrthogonalLine(K, q') → h_1
  28. Intersect(j_1, h_1) → D_1
  29. OrthogonalLine(V_2, a) → j_4
  30. Mirror(f, i) → f'
  31. OrthogonalLine(G, f') → h
  32. Intersect(a, t) → N_2
  33. Mirror(i_3, f_1) → j_3
  34. Intersect(q', p) → U
  35. Mirror(d, i) → d'
  36. Intersect(h_1, j_3) → Z
  37. Intersect(h_1, i_3) → B_1
  38. Intersect(h_1, h) → W
  39. Mirror(W, f_1) → W'
  40. Mirror(D_1, f_1) → E_1
  41. Line(D_1, d') → g_1
  42. Intersect(g_1, f_1) → G_1
  43. Mirror(j, m) → j'
  44. Line(H, p) → q
  45. Line(Q, q) → t_1
  46. OrthogonalLine(H, t_1) → f_2
  47. Intersect(f_2, t_1) → L_1
  48. LineBisector(L_1, H) → h_2
  49. Intersect(h_2, f_2) → N_1
  50. Intersect(h_2, g_1) → Q_1
  51. Vector(Q_1, N_1) → w
  52. OrthogonalLine(C_1, b) → m_1
  53. Intersect(i_3, j_4) → P
  54. OrthogonalLine(P, q') → q_1
  55. OrthogonalLine(B, u, xOyPlane) → g
  56. OrthogonalLine(A_1, b) → r
  57. Intersect(g, r) → C
  58. Polygon(A_1, C, B, A) → q1, a_1, c, b_1, a_2
  59. Intersect(g, i) → F_3
  60. Intersect(m, i) → D
  61. Mirror(h, m) → h'
  62. Intersect(h, f') → E
  63. Intersect(t, i_3) → S
  64. Polygon(G_1, W, D_1, Z, U, N_2, S, B_1, E_1, W') → poly2, e, t_3, k_1, l_1, s_1, c_1, d_1, e_1, r_3, s_3
  65. Polygon(O, N, R, I) → q2, o_1, n_1, r_1, i_1
  66. Mirror(a, m) → a'
  67. OrthogonalLine(E_1, i) → p_1
  68. Intersect(t_1, t) → H_1
  69. Intersect(q, m_1) → I_1
  70. Intersect(t_1, f') → J_1
  71. Intersect(q, f') → K_1
  72. OrthogonalLine(M_1, m) → g_2
  73. Intersect(f_2, g_2) → O_1
  74. Mirror(O_1, h_2) → P_1
  75. Polygon(J_1, K_1, H, O_1, Q_1, P_1, L_1) → poly3, i_2, j_2, h_3, k_2, l_2, m_2, n_2
  76. LineBisector(O, N) → p_2
  77. OrthogonalLine(T_1, f) → q_2
  78. Mirror(U_1, p_2) → V_1
  79. Mirror(T_1, p_2) → W_1
  80. LineBisector(N, T_1) → r_2
  81. Intersect(k_4, d) → R_1
  82. Polygon(R_1, W_1, V_1, Z_1, U_1, T_1, W_2) → poly4, s_2, t_2, b_2, c_2, d_2, e_2, l_4
  83. Line(M, t_1) → f_3
  84. Line(B_2, t_1) → g_3
  85. Intersect(i_3, g_3) → C_2
  86. Intersect(f_1, g_3) → D_2
  87. OrthogonalLine(D_2, p_1) → k_3
  88. Intersect(k_3, j_3) → E_2
  89. Intersect(j_3, q_1) → A_2
  90. Polygon(A_2, P, C_2, D_2, E_2) → poly5, l_3, p_3, m_3, n_3, q_3
  91. Line(H_2, f_2) → b_3
  92. Intersect(b_3, q) → I_2
  93. Intersect(b_3, t_1) → J_2
  94. Intersect(t_1, t) → K_2
  95. Line(K_2, b_3) → c_3
  96. Line(H_2, b_3) → d_3
  97. Intersect(c_3, q) → L_2
  98. Intersect(k_4, t) → M_2
  99. Polygon(I_2, J_2, K_2, M_2, L_2) → poly6, e_3, f_4, g_4, h_4, i_4
  100. CurveCartesian(((1 - t)^(4) * U) + (((4 * (1 - t)^(3)) * t) * P_2) + (((6 * (1 - t)^(2)) * t^(2)) * Q_2) + (((4 * (1 - t)) * t^(3)) * R_2) + (t^(4) * N_2), t, 0, 1) → u_1
  101. Intersect(j_3, f') → S_2
  102. Intersect(g_3, f') → U_2
  103. CurveCartesian(((1 - t)^(4) * E_2) + (((4 * (1 - t)^(3)) * t) * S_2) + (((6 * (1 - t)^(2)) * t^(2)) * T_2) + (((4 * (1 - t)) * t^(3)) * U_2) + (t^(4) * D_2), t, 0, 1) → v_1
  104. LineBisector(L_2, M_2) → m_4
  105. Intersect(m_4, a') → Z_2
  106. CircleArc(Z_2, M_2, L_2) → c_4

AI Drawing Prompt Reference

Draw a 2D geometric figure illustrating vectors and reflection symmetry. Start with a base segment and construct its angle bisector as the axis of symmetry. Create a vector from one endpoint to the other, then generate its orthogonal vector. Through several points, draw perpendicular and parallel lines to the symmetry axis. Use intersections to form a triangle, then reflect it across the axis. Continue adding auxiliary lines, including perpendiculars through intersection points, reflected lines, and segments connecting intersections, to build a symmetric network. Finally, use angle bisectors to create new intersections and complete the symmetric figure.
The prompt above was compiled from the original drawing commands and is for reference only.