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用相似变换与三角网格“铺设”绘制表面

用相似变换与三角网格“铺设”绘制表面
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2020-08-06 04:56
测试改名0多边形立体几何变换几何作图
该范例演示如何从基础几何元素出发，通过相似变换（含伸缩/平移与镜像等操作）构造并“铺设”可绘制的表面效果。核心数学概念包括：相似变换与相似比（dilation/缩放）、三角形构造、几何元素的中点与重心、以及向量与坐标关联的几何关系；同时利用射线、直线交点等确定局部几何。作图要点：在 GeoGebra 中使用 Vector、Translate、Circle、Point、Ray、OrthogonalLine、Mirror、Segment、facecachée3points、Polygon、fachequadrlatèrecachée、Midpoint、Line、Intersect、Centroid 等命令；对象方面通过高度/长度参数（如 Hauteur、Length）、基点与中间量（如 O、A1、B、P、Q、R、S、T、U、V、W、Z_1）以及视图/缩放变量（如 Zoom、Vue、k、l、m、n）来组织网格与表面片元，并用面类对象（facecachée3points、fachequadrlatèrecachée）呈现表面。适合中学高年级到大学初级阶段的几何建模、相似变换与立体/曲面可视化的教学演示，也适合作为 GeoGebra 几何作图练习与项目式学习素材。

GeoGebra 命令及参数

  1. Vector(B, Z) → w
  2. Vector(B, X) → u
  3. Translate(O, Vector[(Length * u)]) → P
  4. Vector(B, Y) → v
  5. Translate(P, Vector[(Width * v)]) → Q
  6. Circle(Vue, 0.2) → d
  7. Point(d) → C
  8. Ray(Vue, C) → e_2
  9. OrthogonalLine(Vue, yAxis) → a
  10. OrthogonalLine(Vue, xAxis) → b
  11. Translate(Q, Vector[(Hauteur * w)]) → R
  12. Mirror(R, O) → S
  13. Mirror(R, Q) → T
  14. Mirror(T, P) → U
  15. Mirror(R, P) → V
  16. Mirror(U, O) → W
  17. Mirror(T, O) → Z_1
  18. Mirror(V, O) → A1
  19. Segment(Z_1, U) → k
  20. Segment(U, V) → l
  21. Segment(V, T) → m
  22. Segment(T, W) → n
  23. Segment(W, A1) → p
  24. Segment(A1, Z_1) → q
  25. Segment(A1, R) → r
  26. Segment(R, T) → s
  27. Segment(R, U) → t
  28. Segment(Z_1, S) → a1
  29. Segment(S, W) → b1
  30. Segment(S, V) → c1
  31. facecachée3points(W, S, Z_1) → B1
  32. facecachée3points(Z_1, A1, S) → C1
  33. Polygon(C1, A1, B1, S) → poly2, c1_1, a1_1, b1_1, s_1
  34. fachequadrlatèrecachée(Z_1, U, V, S) → poly4, d1, f1, j1, k1
  35. fachequadrlatèrecachée(A1, R, U, Z_1) → poly5, l1, m1, n1, p1
  36. fachequadrlatèrecachée(W, S, V, T) → poly6, q1, r1, s1, t1
  37. fachequadrlatèrecachée(U, R, T, V) → poly7, a2, b2, c2, d2
  38. fachequadrlatèrecachée(R, A1, W, T) → poly8, e2, f2, g2, h2
  39. Polygon(S, W, T, V) → poly1, s_2, w_1, t_1, v_1
  40. Midpoint(Z_1, R) → A
  41. Translate(A, Vector[(Expand * w)]) → D
  42. Translate(R, Vector[(Expand * w)]) → E
  43. Translate(A1, Vector[(Expand * w)]) → F
  44. Translate(Z_1, Vector[(Expand * w)]) → G
  45. Translate(U, Vector[(Expand * w)]) → H
  46. Polygon(F, E, H, G) → poly3, f_1, e_1, h_1, g_1
  47. Mirror(F, O) → F'
  48. Mirror(E, O) → E'
  49. Mirror(H, O) → H'
  50. Mirror(G, O) → G'
  51. Polygon(F', E', H', G') → poly3', f', e', h', g'
  52. Translate(R, Vector[(Expand * u)]) → I
  53. Translate(T, Vector[(Expand * u)]) → J
  54. Translate(U, Vector[(Expand * u)]) → K
  55. Translate(V, Vector[(Expand * u)]) → L
  56. Polygon(K, I, J, L) → poly9, k_1, i_1, j, l_1
  57. Mirror(K, O) → K'
  58. Mirror(I, O) → I'
  59. Mirror(J, O) → J'
  60. Mirror(L, O) → L'
  61. Polygon(K', I', J', L') → poly9', k', i', j', l'
  62. Translate(A1, Vector[(Expand * v)]) → M
  63. Translate(R, Vector[(Expand * v)]) → N
  64. Translate(W, Vector[(Expand * v)]) → D1
  65. Translate(T, Vector[(Expand * v)]) → E1
  66. Polygon(M, D1, E1, N) → poly10, m_1, d1_1, e1, n_1
  67. Mirror(M, O) → M'
  68. Mirror(D1, O) → D1'
  69. Mirror(E1, O) → E1'
  70. Mirror(N, O) → N'
  71. Polygon(M', D1', E1', N') → poly10', m', d1', e1', n'
  72. Midpoint(n') → A_1
  73. Midpoint(A_1, M') → B_1
  74. Midpoint(B_1, M') → C_1
  75. Midpoint(B_1, A_1) → D_1
  76. Midpoint(B_1, C_1) → E_1
  77. Midpoint(C_1, M') → F_1
  78. Line(F_1, m') → e
  79. Line(E_1, e) → g
  80. Midpoint(e1') → G_1
  81. Midpoint(G_1, E1') → H_1
  82. Midpoint(H_1, E1') → I_1
  83. Line(I_1, d1') → h
  84. Intersect(h, g) → J_1
  85. Intersect(e, h) → K_1
  86. Polygon(E1', D1', M', F_1, K_1, J_1, E_1, N') → poly11, e1'_1, d1'_1, m'_1, f_2, k_2, j_1, e_3, n'_1
  87. Midpoint(m_1) → L_1
  88. Midpoint(D1, L_1) → M_1
  89. Line(M_1, d1_1) → i
  90. Midpoint(L_1, M) → N_1
  91. Line(N_1, i) → a_1
  92. Midpoint(D1, E1) → O_1
  93. Midpoint(O_1, E1) → P_1
  94. Line(P_1, e1) → b_1
  95. Line(O_1, m_1) → c_1
  96. Intersect(a_1, c_1) → Q_1
  97. Intersect(c_1, i) → R_1
  98. Intersect(b_1, i) → S_1
  99. Intersect(a_1, b_1) → T_1
  100. Polygon(N, M, D1, O_1, Q_1, T_1, S_1, P_1, E1) → poly12, n_2, m_2, d1_2, o_1, q_1, t_2, s_3, p_1, e1_1
  101. Polygon(S_1, R_1, O_1, P_1) → poly13, s_4, r_1, o_2, p_2
  102. Midpoint(Q_1, T_1) → U_1
  103. Midpoint(R_1, S_1) → V_1
  104. Midpoint(Q_1, R_1) → W_1
  105. Midpoint(W_1, R_1) → A_2
  106. Midpoint(W_1, Q_1) → B_2
  107. Segment(U_1, V_1) → d_1
  108. Midpoint(T_1, S_1) → C_2
  109. Midpoint(C_2, T_1) → D_2
  110. Midpoint(C_2, S_1) → E_2
  111. Segment(B_2, D_2) → a_2
  112. Segment(W_1, C_2) → b_2
  113. Segment(A_2, E_2) → c_2
  114. Midpoint(Q_1, U_1) → F_2
  115. Midpoint(U_1, T_1) → G_2
  116. Midpoint(R_1, V_1) → H_2
  117. Midpoint(V_1, S_1) → I_2
  118. Segment(F_2, H_2) → d_2
  119. Segment(G_2, I_2) → g_2
  120. Polygon(Q_1, T_1, S_1, R_1) → poly14, q_2, t_3, s_5, r_2
  121. Polygon(E_1, J_1, K_1, F_1) → poly15, e_4, j_2, k_3, f_3
  122. Midpoint(I, L) → J_2
  123. Midpoint(E1', M') → K_2
  124. Midpoint(L', I') → L_2
  125. Midpoint(M, E1) → M_2
  126. Midpoint(M, E1) → N_2
  127. Midpoint(E1, M) → O_2
  128. Midpoint(M, E1) → P_2
  129. Midpoint(E_1, M) → Q_2
  130. Midpoint(N, D1) → R_2
  131. Midpoint(n_1) → S_2
  132. Midpoint(d1_1) → T_2
  133. Polygon(A_3, D_3, V_2, U_2) → poly16, a_3, d_3, v_2, u_2
  134. Polygon(B_3, C_3, Z_2, W_2) → poly17, b_3, c_3, z_2, w_2
  135. Polygon(E_3, G_3, J_3, F_3) → poly18, e_5, g_3, j_3, f_4
  136. Polygon(K_3, H_3, I_3, L_3) → poly19, k_4, h_3, i_3, l_3
  137. Midpoint(D_3, U_2) → M_3
  138. Midpoint(C_3, W_2) → N_3
  139. Midpoint(J_3, E_3) → O_3
  140. Midpoint(L_3, H_3) → P_3
  141. Midpoint(U_2, V_2) → Q_3
  142. Midpoint(Q_3, V_2) → R_3
  143. Midpoint(R_3, V_2) → S_3
  144. Midpoint(R_3, Q_3) → T_3
  145. Midpoint(D_3, V_2) → U_3
  146. Midpoint(U_3, D_3) → V_3
  147. Line(V_3, a_3) → l_2
  148. Midpoint(S_3, V_2) → W_3
  149. Midpoint(R_3, W_3) → Z_3
  150. Midpoint(R_3, Z_3) → A_4
  151. Line(A_4, d_3) → h_2
  152. Line(W_3, d_3) → i_2
  153. Intersect(h_2, l_2) → B_4
  154. Intersect(l_2, i_2) → C_4
  155. Polygon(A_4, B_4, C_4, W_3) → poly20, a_4, b_4, c_4, w_3
  156. Intersect(w_2, z_2) → D_4
  157. Midpoint(C_3, Z_2) → E_4
  158. Midpoint(E_4, N_3) → F_4
  159. Midpoint(Z_2, W_2) → G_4
  160. Midpoint(Z_2, G_4) → H_4
  161. Midpoint(E_4, Z_2) → I_4
  162. Midpoint(C_3, E_4) → J_4
  163. Midpoint(b_3) → K_4
  164. Midpoint(C_3, K_4) → L_4
  165. Line(J_4, b_3) → m_3
  166. Midpoint(F_4, E_4) → M_4
  167. Line(M_4, c_3) → n_3
  168. Line(K_4, w_2) → p_3
  169. Line(I_4, z_2) → q_3
  170. Intersect(m_3, p_3) → N_4
  171. Intersect(n_3, m_3) → O_4
  172. Intersect(q_3, n_3) → P_4
  173. Intersect(p_3, q_3) → Q_4
  174. Polygon(N_4, O_4, P_4, Q_4) → poly21, n_4, o_4, p_4, q_4
  175. Polygon(R_4, S_4, U_4, T_4) → q2, r_4, r_3, u_4, t_4
  176. Centroid(poly3) → V_4
  177. Centroid(q2) → W_4
  178. Polygon(Vue, AA, BB, CC) → q3, vue, aa, bb, cc

AI 绘图提示语参考

构建一个由相似变换和三角形组成的可绘制表面。首先，通过点O和向量w、u、v、w生成一系列点和线段，然后利用镜像、旋转和平移操作创建多个四边形和三角形。最后，通过辅助点A、D、E、F等控制形状和比例，形成一个复杂的几何表面。
以上提示语为根据原始绘图命令整理而来，仅供参考。