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变换与组合探索

变换与组合探索
几何画板及绘图内容加载中，请稍等……
2015-01-04 01:27
多边形向量解析几何变换几何作图
①本范例演示了通过选择和组合不同的几何变换（如旋转、反射、平移）来改变图形形状的过程；②涉及的核心数学概念包括变换、组合变换、几何作图、向量运算；③作图要点：使用了 Rotate（旋转）、Line（线）、Mirror（反射）、Vector（向量）、Translate（平移）、Polygon（多边形）、Point（点）等 GeoGebra 命令和对象；④适合高中及以上学生学习和探索几何变换的性质与组合效果。

GeoGebra 命令及参数

  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