MathPlotAI Logo MathPlotAI

Transformation and Composition Exploration

Transformation and Composition Exploration
Loading the math board and drawing, please wait…
2015-01-04 01:27
PolygonVectorCoordinate GeometryTransformationGeometric Construction
This example demonstrates the process of altering geometric shapes by selecting and composing different transformations such as rotation, reflection, and translation. Core math concepts include transformations, composition of transformations, geometric construction, and vector operations. It utilizes commands like Rotate, Line, Mirror, Vector, Translate, Polygon, and Point in GeoGebra for construction. Suitable for high school and above students to explore the properties and effects of combined geometric transformations.

GeoGebra Commands & Parameters

  1. Rotate(A, Angle, O_0) → A'_3
  2. Line(P_1, P_2) → r_0
  3. Mirror(A, r_0) → A'_2
  4. Vector(A, A'_1) → v
  5. Translate(A, v) → B_8
  6. Polygon(A, B, C, D, E, F) → poly0, a, b, c, d, e, f
  7. Translate(B, v) → B'
  8. Translate(C, v) → C'
  9. Translate(D, v) → D'
  10. Translate(E, v) → E'
  11. Translate(F, v) → F'
  12. Polygon(B_8, B', C', D', E', F') → poly1, a', b', c', d', e', f'
  13. Mirror(B, r_0) → B'_1
  14. Mirror(C, r_0) → C'_1
  15. Mirror(D, r_0) → D'_1
  16. Mirror(E, r_0) → E'_1
  17. Mirror(F, r_0) → F'_1
  18. Polygon(A'_2, B'_1, C'_1, D'_1, E'_1, F'_1) → poly2, a'_1, b'_1, c'_1, d'_1, e'_1, f'_1
  19. Rotate(B, Angle, O_0) → B'_2
  20. Rotate(C, Angle, O_0) → C'_2
  21. Rotate(D, Angle, O_0) → D'_2
  22. Rotate(E, Angle, O_0) → E'_2
  23. Rotate(F, Angle, O_0) → F'_2
  24. Polygon(A'_3, B'_2, C'_2, D'_2, E'_2, F'_2) → poly3, a'_2, b'_2, c'_2, d'_2, e'_2, f'_2
  25. Vector(A'_1, A''_1) → v_1
  26. Translate(B_8, v_1) → B_7
  27. Translate(B', v_1) → B''
  28. Translate(C', v_1) → C''
  29. Translate(D', v_1) → D''
  30. Translate(E', v_1) → E''
  31. Translate(F', v_1) → F''
  32. Polygon(B_7, B'', C'', D'', E'', F'') → poly1', a'', b'', c'', d'', e'', f''
  33. Vector(A'_3, A''_7) → v_3
  34. Translate(A'_3, v_3) → J
  35. Translate(B'_2, v_3) → K
  36. Translate(C'_2, v_3) → L
  37. Translate(D'_2, v_3) → M
  38. Translate(E'_2, v_3) → N
  39. Translate(F'_2, v_3) → O_4
  40. Polygon(J, K, L, M, N, O_4) → poly3', j, k, l, m, n, o
  41. Vector(A'_2, A''_4) → v_2
  42. Translate(A'_2, v_2) → R
  43. Translate(B'_1, v_2) → S
  44. Translate(C'_1, v_2) → T
  45. Translate(D'_1, v_2) → U
  46. Translate(E'_1, v_2) → V
  47. Translate(F'_1, v_2) → W
  48. Polygon(R, S, T, U, V, W) → poly2', r, s, t, u_1, v_5, w_1
  49. Line(P_3, P_4) → r_1
  50. Line(P_5, P_6) → r_3
  51. Line(P_8, P_7) → r_2
  52. Mirror(B_8, r_1) → A''_2
  53. Mirror(B', r_1) → B''_1
  54. Mirror(C', r_1) → C''_1
  55. Mirror(D', r_1) → D''_1
  56. Mirror(E', r_1) → E''_1
  57. Mirror(F', r_1) → F''_1
  58. Polygon(A''_2, B''_1, C''_1, D''_1, E''_1, F''_1) → poly1'_1, a''_1, b''_1, c''_1, d''_1, e''_1, f''_1
  59. Mirror(A'_3, r_3) → A''_8
  60. Mirror(B'_2, r_3) → B_9
  61. Mirror(C'_2, r_3) → B_1
  62. Mirror(D'_2, r_3) → C_1
  63. Mirror(E'_2, r_3) → D_1
  64. Mirror(F'_2, r_3) → E_1
  65. Polygon(A''_8, B_9, B_1, C_1, D_1, E_1) → poly3'_1, z, a_2, b_2, c_1, d_1, e_1
  66. Mirror(A'_2, r_2) → A''_5
  67. Mirror(B'_1, r_2) → G_1
  68. Mirror(C'_1, r_2) → H_1
  69. Mirror(D'_1, r_2) → I_1
  70. Mirror(E'_1, r_2) → J_1
  71. Mirror(F'_1, r_2) → K_1
  72. Polygon(A''_5, G_1, H_1, I_1, J_1, K_1) → poly2'_1, f_1, g_4, h_4, i_4, j_1, k_1
  73. Rotate(B', Angle_1, O_1) → B''_2
  74. Rotate(C', Angle_1, O_1) → C''_2
  75. Rotate(D', Angle_1, O_1) → D''_2
  76. Rotate(E', Angle_1, O_1) → E''_2
  77. Rotate(F', Angle_1, O_1) → F''_2
  78. Rotate(B_8, Angle_1, O_1) → A''_3
  79. Polygon(A''_3, B''_2, C''_2, D''_2, E''_2, F''_2) → poly1'_2, a''_2, b''_2, c''_2, d''_2, e''_2, f''_2
  80. Rotate(B'_1, Angle_2, O_2) → M_1
  81. Rotate(C'_1, Angle_2, O_2) → N_1
  82. Rotate(D'_1, Angle_2, O_2) → O_5
  83. Rotate(E'_1, Angle_2, O_2) → Q_1
  84. Rotate(F'_1, Angle_2, O_2) → R_1
  85. Rotate(A'_2, Angle_2, O_2) → A''_6
  86. Polygon(A''_6, M_1, N_1, O_5, Q_1, R_1) → poly2'_2, l_2, m_2, n_2, o_1, q_1, r_4
  87. Rotate(B'_2, Angle_3, O_3) → T_1
  88. Rotate(C'_2, Angle_3, O_3) → U_1
  89. Rotate(D'_2, Angle_3, O_3) → V_1
  90. Rotate(E'_2, Angle_3, O_3) → W_1
  91. Rotate(F'_2, Angle_3, O_3) → Z_1
  92. Rotate(A'_3, Angle_3, O_3) → A''_9
  93. Polygon(A''_9, T_1, U_1, V_1, W_1, Z_1) → poly3'_2, s_1, t_1, u_2, v_4, w_2, z_1
  94. Polygon(G, H, I, O, Q, Z) → poly4, g, h, i, o_2, q, z_2
  95. Polygon(A_1, F_1, L_1, S_1, A_2, B_2) → poly5, a_1, f_2, l_1, s_2, a_3, b_1
  96. Polygon(C_2, D_2, E_2, F_2, G_2, H_2) → poly6, c_2, d_2, e_2, f_3, g_5, h_5
  97. Polygon(I_2, J_2, K_2, L_2, M_2, N_2) → poly7, i_5, j_2, k_2, l_3, m_1, n_1
  98. Polygon(Q_2, R_2, S_2, T_2, U_2, V_2) → poly8, q_2, r_5, s_3, t_2, u_3, v_6
  99. Polygon(W_2, Z_2, A_3, B_3, C_3, D_3) → poly9, w_3, z_3, a_4, b_3, c_3, d_3
  100. Polygon(E_3, F_3, G_3, H_3, I_3, J_3) → poly10, e_3, f_4, g_6, h_6, i_6, j_3
  101. Polygon(K_3, L_3, M_3, N_3, Q_3, R_3) → poly11, k_3, l_4, m_3, n_3, q_3, r_6
  102. Polygon(S_3, T_3, U_3, V_3, W_3, Z_3) → poly12, s_4, t_3, u_4, v_7, w_4, z_4
  103. Polygon(A_4, B_4, C_4, D_4, E_4, F_4) → poly13, a_5, b_4, c_4, d_4, e_4, f_5
  104. Point(xAxis) → I_4
  105. Point(xAxis) → J_4
  106. Polygon(G_4, H_4, I_4, J_4, K_4, L_4) → poly14, g_7, h_7, i_7, j_4, k_4, l_5
  107. Polygon(M_4, N_4, Q_4, R_4, S_4, T_4) → poly15, m_4, n_4, q_4, r_7, s_5, t_4
  108. Polygon(U_4, V_4, W_4, Z_4, A_5, B_5) → poly16, u_5, v_8, w_5, z_5, a_6, b_5
  109. Polygon(C_5, D_5, E_5, F_5, G_5, H_5) → poly17, c_5, d_5, e_5, f_6, g_8, h_8
  110. Polygon(I_5, J_5, K_5, L_5, M_5, N_5) → poly18, i_8, j_5, k_5, l_6, m_5, n_5
  111. Polygon(Q_5, R_5, S_5, T_5, U_5, V_5) → poly19, q_5, r_8, s_6, t_5, u_6, v_9