MathPlotAI Logo MathPlotAI

TR KAYCubic Cube 2: Dynamic 3D Cube with Trigonometric Optimization

TR KAYCubic Cube 2: Dynamic 3D Cube with Trigonometric Optimization
Loading the math board and drawing, please wait…
2016-12-01 05:54
VectorTrigonometrySolid GeometryTransformationOptimization
① This example demonstrates a three-dimensional cube constructed through geometric transformations such as vector translation, mirroring, midpoint, and perpendicular lines, with trigonometric parameters (sinphi, cosphi, sintheta, costheta) controlling rotation and scaling, forming an optimization scenario. ② Core mathematical concepts include solid geometry (cube and general quadrilaterals), vector operations, coordinate transformations, trigonometric functions, and optimization problems. ③ Construction highlights: GeoGebra commands used include Vector, Translate, Mirror, Midpoint, OrthogonalLine, Segment, and Polygon; objects such as 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 are combined to project the 3D shape and allow dynamic adjustments. ④ Suitable for high school or early college students to explore the interplay between solid geometry and trigonometry, and useful for teaching dynamic geometry or mathematical modeling, especially for studying how parameter changes affect shape and optimal solutions.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Create a dynamic 3D cube using geometric transformations like translation, mirroring, and midpoint. Control rotation and scaling with trigonometric parameters such as sinphi, cosphi, sintheta, and costheta. Start by defining key points and vectors, then build the cube's faces using translation, mirroring, and midpoint operations. Finally, use translation and mirroring to create multiple copies of the cube for dynamic effects.
The prompt above was compiled from the original drawing commands and is for reference only.