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TR KAYCubic Cube 2：三维立方体与三角函数的动态优化构造

TR KAYCubic Cube 2：三维立方体与三角函数的动态优化构造
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2016-12-01 05:54
向量三角函数立体几何变换最值优化
① 本范例演示了一个由向量平移、镜像、中点、垂线等几何变换构造的三维立方体模型，并融入三角函数参数（sinphi、cosphi、sintheta、costheta）控制其旋转与缩放，形成动态的优化问题场景。② 核心数学概念包括立体几何中的立方体与四边形、向量运算、坐标变换、三角函数以及最值优化（通过滑块调整参数观察图形变化）。③ 作图要点：使用GeoGebra命令如Vector（向量）、Translate（平移）、Mirror（镜像）、Midpoint（中点）、OrthogonalLine（垂线）、Segment（线段）、Polygon（多边形）构建基础图形；对象包括B_1、Dümen、xa、sinphi、cosphi、ya、sintheta、costheta、Büyüt、Z、w、Y、v、g、O、B、D、o、A1、B1、C1、D1、xy、X、u、k、a、b、c、A、d、C、e、f、E、F、h、i、j、G，这些对象协同实现三维到二维的投影与动态调整。④ 适合高中或大学低年级学生用于理解立体几何与三角函数的结合，也可作为数学建模或动态几何教学的演示案例，尤其适用于探讨参数变化对图形形状与最优解的影响。

GeoGebra 命令及参数

  1. Vector(B_1, Z) → w
  2. Vector(B_1, Y) → v
  3. Translate(O, Vector[g * v]) → B
  4. Mirror(B, O) → D
  5. Translate(D, Vector[-1.0 * 1.0 * 2.0 * o * v]) → A1
  6. Translate(A1, Vector[4.0 * o * w]) → B1
  7. Midpoint(B1, A1) → C1
  8. Midpoint(A1, C1) → D1
  9. Vector(B_1, X) → u
  10. OrthogonalLine(Dümen, yAxis) → a
  11. OrthogonalLine(Dümen, xAxis) → b
  12. Translate(O, Vector[g * u]) → A
  13. Segment(O, A) → d
  14. Translate(O, Vector[k * w]) → C
  15. Segment(O, B) → e
  16. Segment(O, C) → f
  17. Mirror(A, O) → E
  18. Mirror(C, O) → F
  19. Segment(D, O) → h
  20. Segment(E, O) → i
  21. Segment(O, F) → j
  22. Translate(B, u) → G
  23. Translate(G, w) → H
  24. Mirror(G, A) → I
  25. Mirror(I, D) → J
  26. Mirror(J, E) → K
  27. Translate(K, w) → L
  28. Translate(J, w) → M
  29. Translate(I, w) → N
  30. Polygon(M, L, H, N) → poly1, m, l, h_1, n
  31. Mirror(N, I) → P
  32. Mirror(M, J) → Q
  33. Mirror(H, G) → R
  34. Mirror(L, K) → S
  35. Polygon(M, Q, P, N) → poly2, m_1, q, p, n_1
  36. Polygon(N, H, R, P) → poly3, n_2, h_2, r, p_1
  37. Polygon(L, M, Q, S) → poly4, l_1, m_2, q_1, s
  38. Polygon(H, L, S, R) → poly5, h_3, l_2, s_1, r_1
  39. Polygon(R, P, Q, S) → poly6, r_2, p_2, q_2, s_2
  40. Translate(B, Vector[2.0 * o * v]) → T
  41. Vector(O, T) → z
  42. Translate(M, z) → M'
  43. Translate(L, z) → L'
  44. Translate(H, z) → H'
  45. Translate(N, z) → N'
  46. Polygon(M', L', H', N') → poly1', m', l', h', n'
  47. Translate(L, z) → L'_1
  48. Translate(M, z) → M'_1
  49. Translate(Q, z) → Q'
  50. Translate(S, z) → S'
  51. Polygon(L'_1, M'_1, Q', S') → poly4', l'_1, m'_1, q', s'
  52. Translate(H, z) → H'_1
  53. Translate(L, z) → L'_2
  54. Translate(S, z) → S'_1
  55. Translate(R, z) → R'
  56. Polygon(H'_1, L'_2, S'_1, R') → poly5', h'_1, l'_2, s'_1, r'
  57. Translate(R, z) → R'_1
  58. Translate(P, z) → P'
  59. Translate(Q, z) → Q'_1
  60. Translate(S, z) → S'_2
  61. Polygon(R'_1, P', Q'_1, S'_2) → poly6', r'_1, p', q'_1, s'_2
  62. Translate(M, z) → M'_2
  63. Translate(Q, z) → Q'_2
  64. Translate(P, z) → P'_1
  65. Translate(N, z) → N'_1
  66. Polygon(M'_2, Q'_2, P'_1, N'_1) → poly2', m'_2, q'_2, p'_1, n'_1
  67. Translate(N, z) → N'_2
  68. Translate(H, z) → H'_2
  69. Translate(R, z) → R'_2
  70. Translate(P, z) → P'_2
  71. Polygon(N'_2, H'_2, R'_2, P'_2) → poly3', n'_2, h'_2, r'_2, p'_2
  72. Translate(A, Vector[2.0 * o * u]) → U
  73. Vector(O, U) → t
  74. Translate(M, t) → M'_3
  75. Translate(L, t) → L'_3
  76. Translate(H, t) → H'_3
  77. Translate(N, t) → N'_3
  78. Polygon(M'_3, L'_3, H'_3, N'_3) → poly1'_1, m'_3, l'_3, h'_3, n'_3
  79. Translate(L, t) → L'_4
  80. Translate(M, t) → M'_4
  81. Translate(Q, t) → Q'_3
  82. Translate(S, t) → S'_3
  83. Polygon(L'_4, M'_4, Q'_3, S'_3) → poly4'_1, l'_4, m'_4, q'_3, s'_3
  84. Translate(M, t) → M'_5
  85. Translate(Q, t) → Q'_4
  86. Translate(P, t) → P'_3
  87. Translate(N, t) → N'_4
  88. Polygon(M'_5, Q'_4, P'_3, N'_4) → poly2'_1, m'_5, q'_4, p'_3, n'_4
  89. Translate(N, t) → N'_5
  90. Translate(H, t) → H'_4
  91. Translate(R, t) → R'_3
  92. Translate(P, t) → P'_4
  93. Polygon(N'_5, H'_4, R'_3, P'_4) → poly3'_1, n'_5, h'_4, r'_3, p'_4
  94. Translate(R, t) → R'_4
  95. Translate(P, t) → P'_5
  96. Translate(Q, t) → Q'_5
  97. Translate(S, t) → S'_4
  98. Polygon(R'_4, P'_5, Q'_5, S'_4) → poly6'_1, r'_4, p'_5, q'_5, s'_4
  99. Translate(H, t) → H'_5
  100. Translate(L, t) → L'_5
  101. Translate(S, t) → S'_5
  102. Translate(R, t) → R'_5
  103. Polygon(H'_5, L'_5, S'_5, R'_5) → poly5'_1, h'_5, l'_5, s'_5, r'_5
  104. Translate(T, Vector[4.0 * o * u]) → V
  105. Vector(T, V) → u1
  106. Vector(O, V) → v1
  107. Translate(M, v1) → M'_6
  108. Translate(L, v1) → L'_6
  109. Translate(H, v1) → H'_6
  110. Translate(N, v1) → N'_6
  111. Polygon(M'_6, L'_6, H'_6, N'_6) → poly1'_2, m'_6, l'_6, h'_6, n'_6
  112. Translate(L, v1) → L'_7
  113. Translate(M, v1) → M'_7
  114. Translate(Q, v1) → Q'_6
  115. Translate(S, v1) → S'_6
  116. Polygon(L'_7, M'_7, Q'_6, S'_6) → poly4'_2, l'_7, m'_7, q'_6, s'_6
  117. Translate(M, v1) → M'_8
  118. Translate(Q, v1) → Q'_7
  119. Translate(P, v1) → P'_6
  120. Translate(N, v1) → N'_7
  121. Polygon(M'_8, Q'_7, P'_6, N'_7) → poly2'_2, m'_8, q'_7, p'_6, n'_7
  122. Translate(R, v1) → R'_6
  123. Translate(P, v1) → P'_7
  124. Translate(Q, v1) → Q'_8
  125. Translate(S, v1) → S'_7
  126. Polygon(R'_6, P'_7, Q'_8, S'_7) → poly6'_2, r'_6, p'_7, q'_8, s'_7
  127. Translate(H, v1) → H'_7
  128. Translate(L, v1) → L'_8
  129. Translate(S, v1) → S'_8
  130. Translate(R, v1) → R'_7
  131. Polygon(H'_7, L'_8, S'_8, R'_7) → poly5'_2, h'_7, l'_8, s'_8, r'_7
  132. Translate(N, v1) → N'_8
  133. Translate(H, v1) → H'_8
  134. Translate(R, v1) → R'_8
  135. Translate(P, v1) → P'_8
  136. Polygon(N'_8, H'_8, R'_8, P'_8) → poly3'_2, n'_8, h'_8, r'_8, p'_8
  137. Translate(C, Vector[2.0 * o * w]) → W
  138. Vector(O, W) → w1
  139. Translate(M, w1) → M'_9
  140. Translate(L, w1) → L'_9
  141. Translate(H, w1) → H'_9
  142. Translate(N, w1) → N'_9
  143. Polygon(M'_9, L'_9, H'_9, N'_9) → poly1'_3, m'_9, l'_9, h'_9, n'_9
  144. Translate(L, w1) → L'_{10}
  145. Translate(M, w1) → M'_{10}
  146. Translate(Q, w1) → Q'_9
  147. Translate(S, w1) → S'_9
  148. Polygon(L'_{10}, M'_{10}, Q'_9, S'_9) → poly4'_3, l'_{10}, m'_{10}, q'_9, s'_9
  149. Translate(H, w1) → H'_{10}
  150. Translate(L, w1) → L'_{11}
  151. Translate(S, w1) → S'_{10}
  152. Translate(R, w1) → R'_9
  153. Polygon(H'_{10}, L'_{11}, S'_{10}, R'_9) → poly5'_3, h'_{10}, l'_{11}, s'_{10}, r'_9
  154. Translate(N, w1) → N'_{10}
  155. Translate(H, w1) → H'_{11}
  156. Translate(R, w1) → R'_{10}
  157. Translate(P, w1) → P'_9
  158. Polygon(N'_{10}, H'_{11}, R'_{10}, P'_9) → poly3'_3, n'_{10}, h'_{11}, r'_{10}, p'_9
  159. Translate(M, w1) → M'_{11}
  160. Translate(Q, w1) → Q'_{10}
  161. Translate(P, w1) → P'_{10}
  162. Translate(N, w1) → N'_{11}
  163. Polygon(M'_{11}, Q'_{10}, P'_{10}, N'_{11}) → poly2'_3, m'_{11}, q'_{10}, p'_{10}, n'_{11}
  164. Translate(R, w1) → R'_{11}
  165. Translate(P, w1) → P'_{11}
  166. Translate(Q, w1) → Q'_{11}
  167. Translate(S, w1) → S'_{11}
  168. Polygon(R'_{11}, P'_{11}, Q'_{11}, S'_{11}) → poly6'_3, r'_{11}, p'_{11}, q'_{11}, s'_{11}
  169. Vector(O, A1) → z1
  170. Translate(M, z1) → M'_{12}
  171. Translate(L, z1) → L'_{12}
  172. Translate(H, z1) → H'_{12}
  173. Translate(N, z1) → N'_{12}
  174. Polygon(M'_{12}, L'_{12}, H'_{12}, N'_{12}) → poly1'_4, m'_{12}, l'_{12}, h'_{12}, n'_{12}
  175. Translate(M, z1) → M'_{13}
  176. Translate(Q, z1) → Q'_{12}
  177. Translate(P, z1) → P'_{12}
  178. Translate(N, z1) → N'_{13}
  179. Polygon(M'_{13}, Q'_{12}, P'_{12}, N'_{13}) → poly2'_4, m'_{13}, q'_{12}, p'_{12}, n'_{13}
  180. Translate(N, z1) → N'_{14}
  181. Translate(H, z1) → H'_{13}
  182. Translate(R, z1) → R'_{12}
  183. Translate(P, z1) → P'_{13}
  184. Polygon(N'_{14}, H'_{13}, R'_{12}, P'_{13}) → poly3'_4, n'_{14}, h'_{13}, r'_{12}, p'_{13}
  185. Translate(R, z1) → R'_{13}
  186. Translate(P, z1) → P'_{14}
  187. Translate(Q, z1) → Q'_{13}
  188. Translate(S, z1) → S'_{12}
  189. Polygon(R'_{13}, P'_{14}, Q'_{13}, S'_{12}) → poly6'_4, r'_{13}, p'_{14}, q'_{13}, s'_{12}
  190. Translate(H, z1) → H'_{14}
  191. Translate(L, z1) → L'_{13}
  192. Translate(S, z1) → S'_{13}
  193. Translate(R, z1) → R'_{14}
  194. Polygon(H'_{14}, L'_{13}, S'_{13}, R'_{14}) → poly5'_4, h'_{14}, l'_{13}, s'_{13}, r'_{14}
  195. Translate(L, z1) → L'_{14}
  196. Translate(M, z1) → M'_{14}
  197. Translate(Q, z1) → Q'_{14}
  198. Translate(S, z1) → S'_{14}
  199. Polygon(L'_{14}, M'_{14}, Q'_{14}, S'_{14}) → poly4'_4, l'_{14}, m'_{14}, q'_{14}, s'_{14}
  200. Vector(A1, B1) → a1
  201. Translate(M'_{12}, a1) → M'_{12}'
  202. Translate(L'_{12}, a1) → L'_{12}'
  203. Translate(H'_{12}, a1) → H'_{12}'
  204. Translate(N'_{12}, a1) → N'_{12}'
  205. Polygon(M'_{12}', L'_{12}', H'_{12}', N'_{12}') → poly1'_4', m'_{12}', l'_{12}', h'_{12}', n'_{12}'
  206. Translate(N'_{14}, a1) → N'_{14}'
  207. Translate(H'_{13}, a1) → H'_{13}'
  208. Translate(R'_{12}, a1) → R'_{12}'
  209. Translate(P'_{13}, a1) → P'_{13}'
  210. Polygon(N'_{14}', H'_{13}', R'_{12}', P'_{13}') → poly3'_4', n'_{14}', h'_{13}', r'_{12}', p'_{13}'
  211. Translate(H'_{14}, a1) → H'_{14}'
  212. Translate(L'_{13}, a1) → L'_{13}'
  213. Translate(S'_{13}, a1) → S'_{13}'
  214. Translate(R'_{14}, a1) → R'_{14}'
  215. Polygon(H'_{14}', L'_{13}', S'_{13}', R'_{14}') → poly5'_4', h'_{14}', l'_{13}', s'_{13}', r'_{14}'
  216. Translate(M'_{13}, a1) → M'_{13}'
  217. Translate(Q'_{12}, a1) → Q'_{12}'
  218. Translate(P'_{12}, a1) → P'_{12}'
  219. Translate(N'_{13}, a1) → N'_{13}'
  220. Polygon(M'_{13}', Q'_{12}', P'_{12}', N'_{13}') → poly2'_4', m'_{13}', q'_{12}', p'_{12}', n'_{13}'
  221. Translate(L'_{14}, a1) → L'_{14}'
  222. Translate(M'_{14}, a1) → M'_{14}'
  223. Translate(Q'_{14}, a1) → Q'_{14}'
  224. Translate(S'_{14}, a1) → S'_{14}'
  225. Polygon(L'_{14}', M'_{14}', Q'_{14}', S'_{14}') → poly4'_4', l'_{14}', m'_{14}', q'_{14}', s'_{14}'
  226. Translate(R'_{13}, a1) → R'_{13}'
  227. Translate(P'_{14}, a1) → P'_{14}'
  228. Translate(Q'_{13}, a1) → Q'_{13}'
  229. Translate(S'_{12}, a1) → S'_{12}'
  230. Polygon(R'_{13}', P'_{14}', Q'_{13}', S'_{12}') → poly6'_4', r'_{13}', p'_{14}', q'_{13}', s'_{12}'

AI 绘图提示语参考

绘制一个动态3D立方体，通过平移、镜像、中点和垂直线等几何变换构建。使用sinphi、cosphi、sintheta、costheta等三角参数控制旋转和缩放，形成优化场景。首先定义关键点和向量，然后通过平移、镜像和中点等操作构建立方体的各个面，最后通过平移和镜像操作生成立方体的多个副本，形成动态效果。
以上提示语为根据原始绘图命令整理而来，仅供参考。