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五棱锥

五棱锥
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2020-05-22 20:19
多边形矩阵变换
该范例演示了五棱锥的几何构造。涉及的核心数学概念包括多边形、平面图形、矩阵和变换。通过使用GeoGebra命令如Point、Polygon、Translate、Rotate等，可以构建五棱锥的各个面和棱。适合高中数学教学和学习。

GeoGebra 命令及参数

  1. Point(yAxis) → OO
  2. Rotate(OO, (360°) / 16) → B
  3. Mirror(B, yAxis) → A
  4. Line(A, B) → n
  5. Polygon(A, B, 8) → veelhoek1, f, D, E, F, G, H, g, h, i, j, k, l, m, C
  6. Line(G, H) → r
  7. Intersect(n, r) → L
  8. Translate(L, Vector[(0, 0, -1.13)]) → L'_1
  9. Line(F, E) → q
  10. Intersect(q, r) → J
  11. Translate(J, Vector[(0, 0, -1.13)]) → J'_1
  12. Line(D, C) → p
  13. Intersect(p, q) → K
  14. Translate(K, Vector[(0, 0, -1.13)]) → K'_1
  15. Intersect(n, p) → I
  16. Translate(I, Vector[(0, 0, -1.13)]) → I'_1
  17. Polygon(I'_1, K'_1, J'_1, L'_1) → vierhoek15'_1, i'_1, k'_1, j'_1, l'_1
  18. Midpoint(B, C) → E_2
  19. Segment(E_2, I) → e_3
  20. Segment(J, L) → s
  21. Segment(L, I) → t
  22. Segment(I, K) → a
  23. Segment(K, J) → b
  24. Prism(veelhoek1, 2) → c, M, zijvlakBCPN, zijvlakCDQP, zijvlakDERQ, zijvlakEFSR, zijvlakFGTS, zijvlakGHUT, zijvlakAHUM, zijvlakMNPQRSTU, RibbeAM, RibbeBN, N, RibbeCP, RibbeDQ, RibbeER, RibbeFS, RibbeGT, RibbeHU, RibbeMN, RibbeNP, RibbePQ, RibbeQR, P, RibbeRS, RibbeST, RibbeTU, RibbeMU, Q, R, S, T, U, zijvlakABNM
  25. Midpoint(RibbePQ) → V
  26. Midpoint(RibbeST) → W
  27. Midpoint(T, P) → Z
  28. Segment(Z, P) → straal
  29. CircleArc(Z, P, A_1) → d
  30. Point(d) → C_1
  31. Point(d) → D_1
  32. Point(d) → E_1
  33. Point(d) → F_1
  34. Point(d) → G_1
  35. Rotate(L_1, (360°) / 8, zAxis) → M_1
  36. Rotate(K_1, (360°) / 8, zAxis) → N_1
  37. Rotate(J_1, (360°) / 8, zAxis) → O_1
  38. Rotate(H_1, (360°) / 8, zAxis) → P_1
  39. Rotate(I_1, (360°) / 8, zAxis) → Q_1
  40. Polygon(I_1, P, Q, Q_1) → vierhoek1, i_1, p_1, q_1, e
  41. Polygon(I_1, Q_1, P_1, H_1) → vierhoek2, f_1, g_1, h_1, j_1
  42. Polygon(J_1, H_1, P_1, O_1) → vierhoek3, k_1, l_1, m_1, o_1
  43. Polygon(K_1, J_1, O_1, N_1) → vierhoek4, n_1, r_1, s_1, t_1
  44. Polygon(L_1, K_1, N_1, M_1) → vierhoek5, a_1, b_1, c_1, d_1
  45. Polygon(L_1, M_1, B_1) → driehoek1, e_1, f_2, g_2
  46. Rotate(vierhoek1, (360°) / 8, zAxis) → vierhoek1'
  47. Rotate(vierhoek2, (360°) / 8, zAxis) → vierhoek2'
  48. Rotate(vierhoek3, (360°) / 8, zAxis) → vierhoek3'
  49. Rotate(vierhoek4, (360°) / 8, zAxis) → vierhoek4'
  50. Rotate(vierhoek5, (360°) / 8, zAxis) → vierhoek5'
  51. Rotate(driehoek1, (360°) / 8, zAxis) → driehoek1'
  52. Rotate(vierhoek1, (2 * (360°) / 8), zAxis) → vierhoek1'_1
  53. Rotate(vierhoek2, (2 * (360°) / 8), zAxis) → vierhoek2'_1
  54. Rotate(vierhoek3, (2 * (360°) / 8), zAxis) → vierhoek3'_1
  55. Rotate(vierhoek4, (2 * (360°) / 8), zAxis) → vierhoek4'_1
  56. Rotate(vierhoek5, (2 * (360°) / 8), zAxis) → vierhoek5'_1
  57. Rotate(driehoek1, (2 * (360°) / 8), zAxis) → driehoek1'_1
  58. Rotate(vierhoek1, (3 * (360°) / 8), zAxis) → vierhoek1'_2
  59. Rotate(vierhoek2, (3 * (360°) / 8), zAxis) → vierhoek2'_2
  60. Rotate(vierhoek3, (3 * (360°) / 8), zAxis) → vierhoek3'_2
  61. Rotate(vierhoek4, (3 * (360°) / 8), zAxis) → vierhoek4'_2
  62. Rotate(vierhoek5, (3 * (360°) / 8), zAxis) → vierhoek5'_2
  63. Rotate(driehoek1, (3 * (360°) / 8), zAxis) → driehoek1'_2
  64. Plane(V, W, W + (0, 0, 1)) → p_2
  65. Intersect(l_1, p_2) → S_1
  66. Intersect(f_1, p_2) → R_1
  67. Intersect(r_1, p_2) → T_1
  68. Intersect(b_1, p_2) → U_1
  69. Intersect(c_1, p_2) → V_1
  70. Intersect(Segment[Rotate[N_1, (360°) / 8, zAxis], Rotate[M_1, (360°) / 8, zAxis]], p_2) → W_1
  71. Intersect(Segment[Rotate[N_1, (2 * (360°) / 8), zAxis], Rotate[M_1, (2 * (360°) / 8), zAxis]], p_2) → Z_1
  72. Intersect(Segment[Rotate[K_1, (3 * (360°) / 8), zAxis], Rotate[N_1, (3 * (360°) / 8), zAxis]], p_2) → A_2
  73. Intersect(Segment[Rotate[J_1, (3 * (360°) / 8), zAxis], Rotate[O_1, (3 * (360°) / 8), zAxis]], p_2) → B_2
  74. Intersect(Segment[Rotate[H_1, (3 * (360°) / 8), zAxis], Rotate[P_1, (3 * (360°) / 8), zAxis]], p_2) → C_2
  75. Intersect(Segment[Rotate[I_1, (3 * (360°) / 8), zAxis], Rotate[Q_1, (3 * (360°) / 8), zAxis]], p_2) → D_2
  76. Segment(V, R_1) → h_2
  77. Segment(R_1, S_1) → i_2
  78. Segment(S_1, T_1) → j_2
  79. Segment(T_1, U_1) → k_2
  80. Segment(U_1, V_1) → l_2
  81. Segment(V_1, W_1) → m_2
  82. Segment(W_1, Z_1) → n_2
  83. Segment(Z_1, A_2) → q_2
  84. Segment(A_2, B_2) → r_2
  85. Segment(B_2, C_2) → s_2
  86. Segment(C_2, D_2) → t_2
  87. Segment(D_2, W) → a_2
  88. Sequence(Rotate[l1, (nn * (360°) / 8), zAxis], nn, 1, 11) → l2
  89. Segment(O, I) → b_2
  90. Sequence(Rotate[vierhoek1, (nn * (360°) / 8), zAxis], nn, 4, 7) → l3
  91. Sequence(Rotate[vierhoek2, (nn * (360°) / 8), zAxis], nn, 4, 7) → l4
  92. Sequence(Rotate[vierhoek3, (nn * (360°) / 8), zAxis], nn, 4, 7) → l5
  93. Sequence(Rotate[vierhoek4, (nn * (360°) / 8), zAxis], nn, 4, 7) → l6
  94. Sequence(Rotate[vierhoek5, (nn * (360°) / 8), zAxis], nn, 4, 7) → l7
  95. Sequence(Rotate[driehoek1, (nn * (360°) / 8), zAxis], nn, 4, 7) → l8
  96. Segment(N, P) → e_2
  97. Sequence(Segment[Rotate[N, (nn * (360°) / 8), zAxis], Rotate[P, (nn * (360°) / 8), zAxis]], nn, 1, 7) → l9
  98. Segment(B, C) → f_3
  99. Segment(O, K) → t_3
  100. Segment(J_2, E_2) → a_3
  101. OrthogonalLine(I, a_3, space) → d_3
  102. Midpoint(a_3) → U_2
  103. Intersect(zAxis, d_3) → V_2
  104. CircleArc(V_2, E_2, J_2) → c_4
  105. Rotate(c_4, 90°, zAxis) → d_4
  106. Point(c_4) → W_2
  107. Point(c_4) → Z_2
  108. Point(c_4) → A_3
  109. Polygon(J_2, I, B) → driehoek2, b_4, f_4, i_4
  110. Segment(B, C) → g_4
  111. Line(W_2, g_4) → h_4
  112. Line(Z_2, g_4) → j_4
  113. Line(A_3, g_4) → k_4
  114. Polygon(C, I, J_2) → driehoek3, l_4, c_5, i_5
  115. Midpoint(a) → K_3
  116. Segment(K_3, V) → n_5
  117. Point(n_5) → L_3
  118. PointIn(zijvlakCDQP) → M_3
  119. PointIn(zijvlakCDQP) → N_3
  120. CircumcircleArc(M_3, L_3, N_3) → d_5
  121. Line(N_3, n_5) → p_5
  122. Line(M_3, n_5) → q_5
  123. Intersect(a, zijvlakCDQP) → O_3
  124. Segment(C, D) → r_5
  125. Intersect(p_5, r_5) → P_3
  126. Intersect(r_5, q_5) → Q_3
  127. Segment(N_3, P_3) → s_5
  128. Segment(M_3, Q_3) → t_5
  129. Translate(d_5, Vector[(0, 0, -0.75)]) → c_2
  130. Translate(C, Vector[(1.5 * vert)]) → C'
  131. Translate(D, Vector[(1.5 * vert)]) → D'
  132. Segment(C', D') → d_2
  133. Translate(P_3, Vector[(1.5 * vert)]) → F_2
  134. Translate(Q_3, Vector[(1.5 * vert)]) → G_2
  135. Translate(N_3, vert) → H_2
  136. Translate(M_3, vert) → I_2
  137. Segment(H_2, F_2) → g_3
  138. Segment(I_2, G_2) → h_3
  139. Point(n_5) → O_2
  140. Translate(veelhoek1, Vector[(0, 0, z(O_2))]) → veelhoek1'
  141. Translate(veelhoek1, Vector[(0, 0, z(C'))]) → veelhoek1'_1
  142. Polygon(C', D', Q, P) → vierhoek6, c', d', q_3, p_3
  143. Sequence(Rotate[vierhoek6, (nn * (360°) / 8), zAxis], nn, 1, 7) → l10
  144. Sequence(Rotate[c_2, (nn * (360°) / 8), zAxis], nn, 1, 7) → l11
  145. Sequence(Segment[Rotate[H_2, (nn * (360°) / 8), zAxis], Rotate[F_2, (nn * (360°) / 8), zAxis]], nn, 1, 7) → l12
  146. Sequence(Segment[Rotate[I_2, (nn * (360°) / 8), zAxis], Rotate[G_2, (nn * (360°) / 8), zAxis]], nn, 1, 7) → l13
  147. Line(R_2, O_2) → i_3
  148. Rotate(P_2, 90°, i_3) → S_2
  149. Midpoint(P, Q) → B_3
  150. Translate(Q, Vector[(0, 0, -0.81)]) → Q'
  151. Translate(P, Vector[(0, 0, -0.81)]) → P'
  152. Segment(Q', P') → j_3
  153. Rotate(Q', 90°, i_3) → Q''
  154. CircleArc(O_2, Q', Q'') → c_3
  155. Point(c_3) → C_3
  156. Point(c_3) → D_3
  157. Point(c_3) → E_3
  158. Point(c_3) → F_3
  159. Segment(V, G_3) → k_3
  160. Segment(V, G_3) → l_3
  161. Segment(G_3, H_3) → m_3
  162. Segment(H_3, I_3) → n_3
  163. Segment(I_3, J_3) → r_3
  164. Segment(J_3, Q') → s_3
  165. Mirror(G_3, p_4) → R_3
  166. Mirror(H_3, p_4) → S_3
  167. Mirror(I_3, p_4) → T_3
  168. Mirror(J_3, p_4) → U_3
  169. Line(V_3, Q') → b_3
  170. Line(W_3, P') → m_4
  171. Line(J_3, b_3) → n_4
  172. Line(I_3, b_3) → q_4
  173. Line(H_3, b_3) → r_4
  174. Line(G_3, b_3) → s_4
  175. Line(U_3, m_4) → t_4
  176. Line(T_3, m_4) → a_4
  177. Line(S_3, m_4) → e_4
  178. Line(R_3, m_4) → f_5
  179. Intersect(m_4, b_3) → Z_3
  180. Intersect(t_4, n_4) → A_4
  181. Intersect(a_4, q_4) → B_4
  182. Intersect(e_4, r_4) → C_4
  183. Intersect(s_4, f_5) → D_4
  184. Polygon(Z_3, Q', J_3, A_4) → vierhoek7, z_3, q', g_5, h_5
  185. Polygon(A_4, J_3, I_3, B_4) → vierhoek8, j_5, k_5, l_5, m_5
  186. Polygon(B_4, I_3, H_3, C_4) → vierhoek9, a_5, b_5, e_5, f_6
  187. Polygon(C_4, H_3, G_3, D_4) → vierhoek10, g_6, h_6, i_6, j_6
  188. Polygon(D_4, G_3, V) → driehoek4, v, k_6, l_6
  189. Polygon(P', Z_3, A_4, U_3) → vierhoek11, p', m_6, n_6, u_3
  190. Polygon(U_3, A_4, B_4, T_3) → vierhoek12, p_6, q_6, r_6, s_6
  191. Polygon(T_3, B_4, C_4, S_3) → vierhoek13, t_6, a_6, b_6, c_6
  192. Polygon(S_3, C_4, D_4, R_3) → vierhoek14, d_6, e_6, f_7, g_7
  193. Polygon(R_3, D_4, V) → driehoek5, v_1, h_7, i_7
  194. Segment(V, D_4) → j_7
  195. Segment(D_4, C_4) → k_7
  196. Segment(C_4, B_4) → l_7
  197. Segment(B_4, A_4) → m_7
  198. Segment(A_4, Z_3) → n_7
  199. Mirror(G_3, p_4) → E_4
  200. Mirror(H_3, p_4) → F_4
  201. Segment(E_4, F_4) → p_7
  202. Mirror(H_3, p_4) → G_4
  203. Mirror(I_3, p_4) → H_4
  204. Segment(G_4, H_4) → q_7
  205. Mirror(I_3, p_4) → I_4
  206. Mirror(J_3, p_4) → J_4
  207. Segment(I_4, J_4) → r_7
  208. Mirror(J_3, p_4) → K_4
  209. Mirror(Q', p_4) → Q''_1
  210. Segment(K_4, Q''_1) → s_7
  211. Sequence(Rotate[l14, (nn * 90°), zAxis], nn, 1, 3) → l15
  212. Sequence(Rotate[l16, (nn * 90°), zAxis], nn, 1, 3) → l17
  213. Sequence(Rotate[l18, (nn * 90°), zAxis], nn, 1, 3) → l19
  214. Sequence(Rotate[l20, (nn * 90°), zAxis], nn, 1, 3) → l21
  215. Point(n_7) → L_4
  216. Segment(L_4, Q') → t_7
  217. Segment(P', L_4) → a_7
  218. Sequence(Rotate[l22, (nn * 90°), zAxis], nn, 1, 3) → l23
  219. Polygon(I, K, J, L) → vierhoek15, i_8, k_8, j_8, l_8
  220. Translate(I, Vector[(0, 0, -1)]) → I'
  221. Translate(K, Vector[(0, 0, -1)]) → K'
  222. Translate(J, Vector[(0, 0, -1)]) → J'
  223. Translate(L, Vector[(0, 0, -1)]) → L'
  224. Rotate(vierhoek15'_1, (360°) / 8, zAxis) → vierhoek15''
  225. Segment(L_4, M_4) → b_7
  226. Sequence(Segment[Rotate[L_4, (nn * 90°), zAxis], Rotate[M_4, (nn * 90°), zAxis]], nn, 1, 3) → l24
  227. Polygon(A, B, 8) → veelhoek2, c_7, O_4, P_4, Q_4, R_4, S_4, d_7, e_7, f_8, g_8, h_8, m_8, n_8, N_4
  228. Segment(L, I) → p_8
  229. Segment(I, K) → q_8
  230. Midpoint(c_7) → T_4
  231. Midpoint(a) → U_4
  232. Midpoint(E, D) → V_4
  233. Midpoint(F, G) → W_4
  234. Midpoint(S_4, A) → Z_4
  235. Midpoint(B, C) → A_5
  236. Segment(W_4, T_4) → r_8
  237. Segment(W_4, U_4) → s_8
  238. Intersect(s, xAxis) → B_5
  239. Segment(A_5, B_5) → b_8
  240. Segment(B_5, V_4) → c_8
  241. Segment(T_4, V_4) → d_8
  242. Midpoint(b) → C_5
  243. Segment(C_5, Z_4) → e_8
  244. Segment(C_5, A_5) → f_9
  245. Segment(Z_4, U_4) → t_8
  246. Segment(O, S_4) → a_8
  247. Segment(O, A) → g_9
  248. Segment(O, B) → h_9
  249. Segment(O, C) → i_9
  250. Segment(O, D) → j_9
  251. Segment(O, E) → k_9
  252. Segment(O, F) → l_9
  253. Segment(O, G) → m_9
  254. Segment(V_4, K) → n_9
  255. Segment(W_4, J) → p_9
  256. Segment(Z_4, L) → q_9
  257. Intersect(a, p_8) → D_5
  258. Segment(A_5, D_5) → r_9
  259. Translate(l23, Vector[(0, 0, -2.32)]) → l23'
  260. Translate(T, Vector[(0, 0, -3.13)]) → T'
  261. Translate(S, Vector[(0, 0, -3.13)]) → S'
  262. Segment(T', S') → RibbeST'
  263. Sequence(Segment[Rotate[T', (nn * 90°), zAxis], Rotate[S', (nn * 90°), zAxis]], nn, 1, 3) → l25
  264. Rotate(T', (360°) / 8, zAxis) → T''
  265. Mirror(F_2, zAxis) → E_5
  266. Mirror(G_2, zAxis) → F_5
  267. Segment(T', E_5) → s_9
  268. Segment(F_5, T'') → t_9
  269. Sequence(Segment[Rotate[T', (nn * 90°), zAxis], Rotate[E_5, (nn * 90°), zAxis]], nn, 1, 3) → l26
  270. Sequence(Segment[Rotate[F_5, (nn * 90°), zAxis], Rotate[T'', (nn * 90°), zAxis]], nn, 1, 3) → l27
  271. Mirror(W_1, zAxis) → G_5
  272. Segment(W_1, G_5) → a_9
  273. Midpoint(W_1, G_5) → H_5
  274. Rotate(W_1, 90°, zAxis) → I_5
  275. Rotate(W_1, (-(90°)), zAxis) → J_5
  276. Segment(J_5, I_5) → b_9
  277. Rotate(G_5, 90°, b_9) → K_5
  278. CircleArc(H_5, G_5, K_5) → c_9
  279. Point(c_9) → L_5
  280. Point(c_9) → M_5
  281. Point(c_9) → N_5
  282. Point(c_9) → O_5
  283. Rotate(Q_5, ((-(360°))) / 8, zAxis) → U_5
  284. Rotate(R_5, ((-(360°))) / 8, zAxis) → V_5
  285. Rotate(S_5, ((-(360°))) / 8, zAxis) → W_5
  286. Rotate(T_5, ((-(360°))) / 8, zAxis) → Z_5
  287. Rotate(G_5, ((-(360°))) / 8, zAxis) → A_6
  288. Polygon(P_5, U_5, Q_5) → driehoek6, d_9, e_9, u_5
  289. Polygon(U_5, V_5, R_5, Q_5) → vierhoek16, f_{10}, v_5, g_{10}, h_{10}
  290. Polygon(V_5, W_5, S_5, R_5) → vierhoek17, i_{10}, w_5, j_{10}, k_{10}
  291. Polygon(W_5, Z_5, T_5, S_5) → vierhoek18, l_{10}, z_5, m_{10}, n_{10}
  292. Polygon(A_6, G_5, T_5, Z_5) → vierhoek19, p_{10}, q_{10}, r_{10}, s_{10}
  293. Point(l2) → B_6
  294. Polygon(A_6, B_6, G_5) → driehoek7, t_{10}, a_{10}, b_{10}
  295. Sequence(Rotate[l28, (nn * (360°) / 8), zAxis], nn, 1, 7) → l29
  296. Segment(A_6, Z_5) → c_{10}
  297. Segment(Z_5, W_5) → d_{10}
  298. Segment(W_5, V_5) → e_{10}
  299. Segment(V_5, U_5) → f_{11}
  300. Segment(U_5, P_5) → g_{11}
  301. Sequence(Rotate[l30, (nn * (360°) / 8), zAxis], nn, 1, 7) → l31

AI 绘图提示语参考

绘制一个五角柱，首先创建一个五边形底面，然后通过旋转和平移生成五角柱的各个面和棱。具体步骤包括创建五边形顶点，旋转生成新的顶点，通过这些顶点创建多边形和线段，最后利用这些线段和多边形构建五角柱的各个面。
以上提示语为根据原始绘图命令整理而来，仅供参考。