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多边形与向量演示

多边形与向量演示
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2020-05-22 20:26
多边形向量解析几何几何作图
①本范例演示了如何通过构造多边形和向量展示二维空间中的几何关系；②涉及的核心数学概念包括多边形构造、向量运算、中点计算、旋转与镜像变换；③作图要点提炼自构造摘要，使用了Segment、Intersect、Point、CircleArc、Line、Mirror、Rotate、Polygon、Sequence、Midpoint、Distance、Plane等命令及对象；④适合高中数学教学，用于理解向量与几何变换的关联，以及解析几何作图方法。

GeoGebra 命令及参数

  1. Segment(A, B) → f
  2. Segment(B, C) → g
  3. Segment(D, A) → i
  4. Segment(C, D) → h
  5. Intersect(i, yAxis) → E
  6. Intersect(f, xAxis) → F
  7. Intersect(g, yAxis) → G
  8. Point(xAxis) → H
  9. CircleArc(G, C, I) → c
  10. CircleArc(G, I, B) → d
  11. Point(c) → J
  12. Point(c) → K
  13. Point(c) → L
  14. Point(c) → M
  15. Point(c) → N
  16. Line(J, g) → j
  17. Line(K, g) → k
  18. Line(L, g) → l
  19. Line(M, g) → m
  20. Line(N, g) → n
  21. Mirror(I_1, vgl1) → N_1
  22. Mirror(J_1, vgl1) → O_1
  23. Mirror(K_1, vgl1) → P_1
  24. Mirror(L_1, vgl1) → Q_1
  25. Mirror(M_1, vgl1) → R_1
  26. Rotate(I_1, 90°, zAxis) → S_1
  27. Rotate(J_1, 90°, zAxis) → P
  28. Rotate(L_1, 90°, zAxis) → Q
  29. Rotate(M_1, 90°, zAxis) → R
  30. Rotate(I_1, (-(90°)), zAxis) → S
  31. Line(M_1, h) → p
  32. Line(L_1, h) → q
  33. Line(K_1, h) → r
  34. Line(J_1, h) → s
  35. Line(I_1, h) → t
  36. Line(H_1, h) → a
  37. Line(R, i) → b
  38. Line(Q, i) → e
  39. Line(P, i) → f_1
  40. Line(S_1, i) → g_1
  41. Intersect(g_1, t) → T
  42. Intersect(f_1, s) → U
  43. Intersect(r, e) → V
  44. Intersect(p, b) → W
  45. Rotate(K_1, 90°, zAxis) → Z
  46. Line(Z, i) → h_1
  47. Intersect(r, h_1) → A_1
  48. Intersect(q, e) → T_1
  49. Mirror(T, vgl1) → T'
  50. Mirror(U, vgl1) → U'
  51. Mirror(A_1, vgl1) → U_1
  52. Mirror(T_1, vgl1) → V_1
  53. Mirror(W, vgl1) → W'
  54. Polygon(D, A, W', W) → vierhoek1, d_1, a_1, w', w
  55. Polygon(W, W', V_1, T_1) → vierhoek2, w_1, i_1, v_1, t_1
  56. Polygon(T_1, V_1, U_1, A_1) → vierhoek3, j_1, k_1, u_1, l_1
  57. Polygon(A_1, U_1, U', U) → vierhoek4, m_1, n_1, u', u
  58. Polygon(U, U', T', T) → vierhoek5, u_2, p_1, t', t_2
  59. Polygon(T, T', H_1) → driehoek1, q_1, t_3, r_1
  60. Segment(H_1, T) → s_1
  61. Segment(T, U) → b_1
  62. Segment(U, A_1) → c_1
  63. Segment(A_1, T_1) → e_1
  64. Segment(T_1, W) → f_2
  65. Segment(W, D) → g_2
  66. Segment(H_1, T') → h_2
  67. Segment(T', U') → i_2
  68. Segment(U', U_1) → j_2
  69. Segment(U_1, V_1) → k_2
  70. Segment(V_1, W') → l_2
  71. Segment(W', A) → m_2
  72. Sequence(Rotate[vierhoek1, (nn * 90°), zAxis], nn, 1, 4) → l1
  73. Sequence(Rotate[vierhoek2, (nn * 90°), zAxis], nn, 1, 4) → l2
  74. Sequence(Rotate[vierhoek3, (nn * 90°), zAxis], nn, 1, 4) → l3
  75. Sequence(Rotate[vierhoek4, (nn * 90°), zAxis], nn, 1, 4) → l4
  76. Sequence(Rotate[vierhoek5, (nn * 90°), zAxis], nn, 1, 4) → l5
  77. Sequence(Rotate[driehoek1, (nn * 90°), zAxis], nn, 1, 4) → l6
  78. Sequence(Segment[Rotate[H_1, (nn * 90°), zAxis], Rotate[T', (nn * 90°), zAxis]], nn, 1, 2) → l7
  79. Sequence(Segment[Rotate[T', (nn * 90°), zAxis], Rotate[U', (nn * 90°), zAxis]], nn, 1, 2) → l8
  80. Sequence(Segment[Rotate[U', (nn * 90°), zAxis], Rotate[U_1, (nn * 90°), zAxis]], nn, 1, 2) → l9
  81. Sequence(Segment[Rotate[U_1, (nn * 90°), zAxis], Rotate[V_1, (nn * 90°), zAxis]], nn, 1, 2) → l10
  82. Sequence(Segment[Rotate[V_1, (nn * 90°), zAxis], Rotate[W', (nn * 90°), zAxis]], nn, 1, 2) → l11
  83. Sequence(Segment[Rotate[W', (nn * 90°), zAxis], Rotate[A, (nn * 90°), zAxis]], nn, 1, 2) → l12
  84. Midpoint(H, E) → W_1
  85. Line(W_1, zAxis) → n_2
  86. Intersect(c_1, n_2) → Z_1
  87. Distance(W_1, E) → horstraal
  88. Plane(H, E, Z_1) → p_2
  89. Intersect(p_2, w') → A_2
  90. Intersect(p_2, v_1) → B_2
  91. Intersect(p_2, u_1) → C_2
  92. Plane(B, D, H_1) → diag
  93. Mirror(C_2, diag) → D_2
  94. Mirror(B_2, diag) → E_2
  95. Mirror(A_2, diag) → F_2
  96. Segment(H, F_2) → q_2
  97. Segment(F_2, E_2) → r_2
  98. Segment(E_2, D_2) → s_2
  99. Segment(D_2, Z_1) → a_2
  100. Segment(Z_1, C_2) → b_2
  101. Segment(C_2, B_2) → c_2
  102. Segment(B_2, A_2) → d_2
  103. Segment(A_2, E) → e_2
  104. Sequence(Segment[Rotate[D_2, (nn * 90°), zAxis], Rotate[Z_1, (nn * 90°), zAxis]], nn, 1, 3) → l13
  105. Sequence(Segment[Rotate[Z_1, (nn * 90°), zAxis], Rotate[C_2, (nn * 90°), zAxis]], nn, 1, 3) → l14
  106. Sequence(Segment[Rotate[C_2, (nn * 90°), zAxis], Rotate[B_2, (nn * 90°), zAxis]], nn, 1, 3) → l15
  107. Sequence(Segment[Rotate[B_2, (nn * 90°), zAxis], Rotate[A_2, (nn * 90°), zAxis]], nn, 1, 3) → l16
  108. Sequence(Segment[Rotate[A_2, (nn * 90°), zAxis], Rotate[E, (nn * 90°), zAxis]], nn, 1, 3) → l17
  109. Sequence(Segment[Rotate[H, (nn * 90°), zAxis], Rotate[F_2, (nn * 90°), zAxis]], nn, 1, 3) → l18
  110. Sequence(Segment[Rotate[F_2, (nn * 90°), zAxis], Rotate[E_2, (nn * 90°), zAxis]], nn, 1, 3) → l19
  111. Sequence(Segment[Rotate[E_2, (nn * 90°), zAxis], Rotate[D_2, (nn * 90°), zAxis]], nn, 1, 3) → l20
  112. Point(b_1) → G_2
  113. Segment(H_1, T) → f_3
  114. Segment(T, G_2) → g_3
  115. Segment(G_2, Z_1) → h_3
  116. Sequence(Segment[Rotate[H_1, (nn * 90°), zAxis], Rotate[T, (nn * 90°), zAxis]], nn, 1, 3) → l21
  117. Sequence(Segment[Rotate[T, (nn * 90°), zAxis], Rotate[G_2, (nn * 90°), zAxis]], nn, 1, 3) → l22
  118. Sequence(Segment[Rotate[G_2, (nn * 90°), zAxis], Rotate[Z_1, (nn * 90°), zAxis]], nn, 1, 3) → l23
  119. Segment(E, F) → i_3
  120. Segment(F, G) → j_3
  121. Segment(E, H) → k_3
  122. Segment(H, G) → l_3
  123. Segment(W_1, O) → m_3
  124. Point(i_3) → H_2
  125. Segment(H_2, O) → n_3
  126. Point(j_3) → I_2
  127. Segment(O, I_2) → p_3
  128. Point(l_3) → J_2
  129. Segment(O, J_2) → q_3
  130. Segment(B, I_2) → r_3
  131. Segment(A, H_2) → s_3
  132. Segment(W_1, D) → a_3
  133. Segment(J_2, C) → b_3
  134. Rotate(Z_1, 90°, zAxis) → K_2
  135. Mirror(Z_1, zAxis) → L_2
  136. Midpoint(Z_1, L_2) → M_2
  137. Segment(Z_1, L_2) → c_3
  138. Rotate(K_2, (-(90°)), c_3) → N_2
  139. Segment(U_1, K_2) → e_3
  140. Polygon(Z_1, A_1, C_2) → driehoek2, f_4, z_1, g_4
  141. Polygon(A_1, T_1, B_2, C_2) → vierhoek6, h_4, i_4, j_4, k_4
  142. Polygon(T_1, W, A_2, B_2) → vierhoek7, l_4, w_2, m_4, n_4
  143. Polygon(W, D, E, A_2) → vierhoek8, w_3, d_4, e_4, p_4
  144. Mirror(vierhoek6, diag) → vierhoek6'
  145. Mirror(vierhoek7, diag) → vierhoek7'
  146. Mirror(vierhoek8, diag) → vierhoek8'
  147. Mirror(driehoek2, diag) → driehoek2'
  148. Sequence(Rotate[l24, (nn * 90°), zAxis], nn, 1, 3) → l25
  149. Sequence(Rotate[l26, (nn * 90°), zAxis], nn, 1, 3) → l27
  150. Polygon(Z_1, C_2, B_2, A_2, E, D_3) → veelhoek1, q_4, r_4, s_4, t_4, e_5, a_4
  151. Sequence(Rotate[veelhoek1, (nn * 90°), zAxis], nn, 1, 3) → l28
  152. Mirror(veelhoek1, vgl1) → veelhoek1'
  153. Sequence(Rotate[veelhoek1', (nn * 90°), zAxis], nn, 1, 3) → l29
  154. Mirror(D_3, diag) → E_3
  155. Segment(E_3, D_3) → b_4
  156. Mirror(D_3, zAxis) → O_2
  157. Segment(O_2, D_3) → d_3
  158. Rotate(E_3, (-(90°)), d_3) → P_2
  159. CircleArc(M_2, E_3, P_2) → c_4
  160. Point(c_4) → Q_2
  161. Point(c_4) → R_2
  162. Point(c_4) → T_2
  163. Point(c_4) → U_2
  164. Point(c_4) → V_2
  165. Plane(W_2, M_2, K_2) → spieg
  166. Mirror(Z_2, spieg) → G_3
  167. Mirror(A_3, spieg) → H_3
  168. Mirror(B_3, spieg) → I_3
  169. Mirror(C_3, spieg) → J_3
  170. Mirror(F_3, spieg) → K_3
  171. Polygon(G_3, Z_2, W_2) → driehoek3, f_5, g_5, z_2
  172. Polygon(G_3, H_3, A_3, Z_2) → vierhoek9, h_5, i_5, j_5, k_5
  173. Polygon(H_3, I_3, B_3, A_3) → vierhoek10, l_5, m_5, n_5, p_5
  174. Polygon(I_3, J_3, C_3, B_3) → vierhoek11, q_5, r_5, s_5, t_5
  175. Polygon(J_3, K_3, F_3, C_3) → vierhoek12, a_5, b_5, c_5, d_5
  176. Polygon(K_3, O_2, E_3, F_3) → vierhoek13, f_6, o_2, g_6, h_6
  177. Sequence(Rotate[l30, (nn * 90°), zAxis], nn, 1, 3) → l31
  178. Sequence(Segment[Rotate[E_3, (nn * 90°), zAxis], Rotate[D_3, (nn * 90°), zAxis]], nn, 1, 3) → l32
  179. Segment(O_2, K_3) → i_6
  180. Segment(K_3, J_3) → j_6
  181. Segment(J_3, I_3) → k_6
  182. Segment(I_3, H_3) → l_6
  183. Segment(H_3, G_3) → m_6
  184. Segment(G_3, W_2) → n_6
  185. Sequence(Rotate[l33, (nn * 90°), zAxis], nn, 1, 3) → l34
  186. Segment(Z_1, A_1) → p_6
  187. Sequence(Segment[Rotate[Z_1, (nn * 90°), zAxis], Rotate[A_1, (nn * 90°), zAxis]], nn, 1, 3) → l35
  188. Midpoint(o_2) → L_3
  189. Midpoint(K_3, F_3) → M_3
  190. Midpoint(J_3, C_3) → N_3
  191. Midpoint(I_3, B_3) → O_3
  192. Midpoint(H_3, A_3) → P_3
  193. Midpoint(G_3, Z_2) → Q_3
  194. Segment(L_3, M_3) → q_6
  195. Segment(M_3, N_3) → r_6
  196. Segment(N_3, O_3) → s_6
  197. Segment(O_3, P_3) → t_6
  198. Segment(P_3, Q_3) → a_6
  199. Segment(Q_3, W_2) → b_6
  200. Sequence(Rotate[l36, (nn * 90°), zAxis], nn, 1, 3) → l37