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鞍形曲面的阴影与投影（一）

鞍形曲面的阴影与投影（一）
几何画板及绘图内容加载中，请稍等……
2020-04-22 03:53
函数立体几何变换几何作图轨迹
① 本范例演示了三维鞍形曲面在光照下的阴影与投影效果，通过二维平面上的几何构造模拟曲面的立体视觉。② 核心数学概念涉及曲面（鞍面）、投影几何、阴影生成以及空间点线面的位置关系，同时结合了函数图像与变换思想。③ 作图要点：从构造摘要可见，使用了线段（Segment）、向量（Vector）、平移（Translate）、直线（Line）、点（Point）、交点（Intersect）、多边形（Polygon）和轨迹（Locus）等GeoGebra命令，通过多个辅助点（如A、B、C、D等）和辅助线（如g、h、i等）逐步构建出阴影边界与投影轮廓。④ 适合高中或大学低年级学生用于理解三维图形在二维表示中的投影与阴影原理，也可作为立体几何或解析几何的拓展教学素材。

GeoGebra 命令及参数

  1. Segment(C, A) → f
  2. Segment(A, Bp) → g
  3. Vector(A, C) → u
  4. Translate(A, u) → A'
  5. Translate(Bp, u) → B'
  6. Segment(A', B') → g'
  7. Segment(Bp, B') → h
  8. Vector(A, D) → v
  9. Translate(Bp, v) → B'_1
  10. Translate(B', v) → B''
  11. Translate(C, v) → C'
  12. Translate(B'_1, v) → E
  13. Translate(D, v) → D'
  14. Translate(B'', v) → B'''
  15. Translate(C', v) → C''
  16. Segment(C', D) → i
  17. Segment(D, B'_1) → j
  18. Segment(B'_1, B'') → k
  19. Segment(B'', C') → l
  20. Segment(D', C'') → m
  21. Segment(C'', B''') → n
  22. Segment(B''', E) → p
  23. Segment(E, D') → q
  24. Segment(A, F) → r
  25. Segment(D, G) → s
  26. Segment(D', H) → t
  27. Line(B', r) → a
  28. Point(a) → I
  29. Segment(B', I) → b
  30. Segment(C, F) → c
  31. Segment(F, Bp) → d
  32. Segment(Bp, I) → e
  33. Segment(I, C) → f_1
  34. Line(B'', s) → k_1
  35. Point(k_1) → L
  36. Segment(L, B'') → l_1
  37. Segment(G, C') → m_1
  38. Segment(C', L) → n_1
  39. Segment(L, B'_1) → p_1
  40. Segment(B'_1, G) → q_1
  41. Line(B''', t) → r_1
  42. Point(r_1) → M
  43. Segment(B''', M) → s_1
  44. Segment(H, C'') → t_1
  45. Segment(C'', M) → a_1
  46. Segment(M, E) → b_1
  47. Segment(E, H) → c_1
  48. Line(F, Bp) → g_1
  49. Line(F, C) → h_1
  50. Intersect(f_1, g_1) → J
  51. Intersect(e, h_1) → K
  52. Segment(K, F) → i_1
  53. Segment(F, J) → j_1
  54. Intersect(m_1, p_1) → N
  55. Line(E, M) → d_1
  56. Line(M, C'') → e_1
  57. Intersect(c_1, e_1) → O
  58. Intersect(t_1, d_1) → P
  59. Segment(O, M) → f_2
  60. Segment(M, P) → g_2
  61. Line(K, r) → h_2
  62. Line(J, h_2) → i_2
  63. Line(N, s) → j_2
  64. Line(O, t) → k_2
  65. Line(P, t) → l_2
  66. Intersect(h_2, h) → R
  67. Intersect(i_2, g') → S
  68. Intersect(j_2, k) → Q
  69. Intersect(j_2, i) → T
  70. Intersect(k_2, q) → U
  71. Intersect(l_2, m) → V
  72. Segment(R, K) → m_2
  73. Segment(J, S) → n_2
  74. Segment(T, N) → p_2
  75. Segment(V, P) → q_2
  76. Segment(U, O) → r_2
  77. Line(R, g) → s_2
  78. Line(T, g) → t_2
  79. Line(Q, j) → a_2
  80. Line(V, q) → b_2
  81. Line(U, p) → c_2
  82. Line(S, f) → d_2
  83. Intersect(g, d_2) → W
  84. Intersect(s_2, f) → Z
  85. Intersect(t_2, k) → A_1
  86. Intersect(a_2, i) → B_1
  87. Intersect(b_2, p) → C_1
  88. Intersect(c_2, n) → D_1
  89. Segment(Z, R) → e_2
  90. Segment(W, S) → f_3
  91. Segment(B_1, Q) → g_3
  92. Segment(T, A_1) → h_3
  93. Segment(D_1, U) → i_3
  94. Segment(V, C_1) → j_3
  95. Segment(Z, W) → k_3
  96. Segment(B_1, A_1) → l_3
  97. Segment(D_1, C_1) → m_3
  98. Polygon(A, W, Z) → t1, z_1, a_3, w
  99. Polygon(B_1, A_1, B'_1, D) → q1, n_3, p_3, b'_1, d_3
  100. Polygon(D_1, C_1, E, D', C'') → poligon1, q_3, r_3, e_3, d', c''
  101. Intersect(e_2, f_3) → E_1
  102. Polygon(Z, E_1, W) → t2, w_1, z_2, s_3
  103. Polygon(B_1, A_1, Q) → t3, q_4, t_3, b_3
  104. Polygon(D_1, C_1, B''') → t4, b''', c_3, f_4
  105. Line(W, m_2) → g_4
  106. Line(Z, g_4) → h_4
  107. Line(A_1, s) → i_4
  108. Line(B_1, s) → j_4
  109. Line(D_1, q_2) → k_4
  110. Line(C_1, k_4) → l_4
  111. Intersect(g_4, d) → F_1
  112. Intersect(h_4, c) → G_1
  113. Segment(W, F_1) → m_4
  114. Segment(G_1, Z) → n_4
  115. Intersect(p_1, i_4) → H_1
  116. Intersect(j_4, m_1) → I_1
  117. Segment(B_1, I_1) → p_4
  118. Segment(A_1, H_1) → r_4
  119. Intersect(b_1, l_4) → J_1
  120. Intersect(a_1, k_4) → K_1
  121. Segment(D_1, K_1) → s_4
  122. Segment(C_1, J_1) → t_4
  123. Point(k_3) → L_1
  124. Line(L_1, f) → a_4
  125. Intersect(a_4, g) → M_1
  126. Line(M_1, m_2) → b_4
  127. Intersect(a_4, g') → N_1
  128. Line(N_1, b_4) → c_4
  129. Intersect(c_4, f_1) → O_1
  130. Intersect(b_4, d) → P_1
  131. Segment(O_1, P_1) → d_4
  132. Line(L_1, b_4) → e_4
  133. Intersect(d_4, e_4) → Q_1
  134. Locus(Q_1, L_1) → ponthalmaz1
  135. Point(l_3) → R_1
  136. Line(R_1, k) → f_5
  137. Intersect(f_5, j) → S_1
  138. Line(S_1, s) → g_5
  139. Intersect(f_5, l) → T_1
  140. Line(T_1, s) → h_5
  141. Intersect(q_1, g_5) → U_1
  142. Intersect(n_1, h_5) → V_1
  143. Segment(V_1, U_1) → i_5
  144. Line(R_1, g_5) → j_5
  145. Intersect(i_5, j_5) → W_1
  146. Locus(W_1, R_1) → ponthalmaz2
  147. Point(m_3) → Z_1
  148. Line(Z_1, i_3) → k_5
  149. Intersect(k_5, q) → A_2
  150. Line(A_2, r_2) → l_5
  151. Intersect(k_5, n) → B_2
  152. Line(B_2, l_5) → m_5
  153. Intersect(l_5, c_1) → C_2
  154. Intersect(m_5, a_1) → D_2
  155. Segment(C_2, D_2) → n_5
  156. Line(Z_1, l_5) → p_5
  157. Intersect(p_5, n_5) → E_2
  158. Locus(E_2, Z_1) → ponthalmaz3
  159. Point(h) → F_2
  160. Point(ponthalmaz1) → G_2
  161. Point(ponthalmaz1) → H_2
  162. Point(ponthalmaz1) → I_2
  163. Point(ponthalmaz1) → J_2
  164. Point(ponthalmaz1) → K_2
  165. Point(ponthalmaz1) → L_2
  166. Point(ponthalmaz1) → M_2
  167. Point(ponthalmaz1) → N_2
  168. Point(ponthalmaz1) → O_2
  169. Point(ponthalmaz1) → P_2
  170. Point(ponthalmaz1) → Q_2
  171. Point(ponthalmaz1) → R_2
  172. Point(ponthalmaz1) → S_2
  173. Point(ponthalmaz1) → T_2
  174. Point(ponthalmaz1) → U_2
  175. Point(ponthalmaz1) → V_2
  176. Point(ponthalmaz1) → W_2
  177. Point(ponthalmaz1) → Z_2
  178. Point(ponthalmaz1) → A_3
  179. Point(ponthalmaz1) → B_3
  180. Point(ponthalmaz1) → C_3
  181. Point(ponthalmaz1) → D_3
  182. Point(ponthalmaz1) → E_3
  183. Point(ponthalmaz1) → F_3
  184. Point(ponthalmaz1) → G_3
  185. Point(ponthalmaz1) → H_3
  186. Point(ponthalmaz1) → I_3
  187. Point(ponthalmaz1) → J_3
  188. Point(ponthalmaz1) → K_3
  189. Point(ponthalmaz1) → L_3
  190. Point(ponthalmaz1) → M_3
  191. Point(ponthalmaz1) → N_3
  192. Point(ponthalmaz1) → O_3
  193. Polygon(B', F_2, O_2, P_2, Q_2, R_2, S_2, T_2, U_2, V_2, W_2, Z_2, G_2, A_3, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, H_2, K_3, L_3, M_3, N_3, O_3, F_1, Bp, I, C, I_2, J_2, K_2, L_2, M_2, N_2) → poligon2, b', e_5, o_2, f_6, g_6, h_6, i_6, j_6, u_2, v_2, w_2, q_5, k_6, l_6, m_6, n_6, p_6, q_6, r_6, s_6, t_6, a_6, r_5, b_6, c_6, d_6, e_6, f_7, o_3, g_7, b_7, i_7, c_7, s_5, t_5, a_5, b_5, c_5, d_5
  194. Point(ponthalmaz2) → P_3
  195. Point(ponthalmaz2) → Q_3
  196. Point(ponthalmaz2) → R_3
  197. Point(ponthalmaz2) → S_3
  198. Point(ponthalmaz2) → T_3
  199. Point(ponthalmaz2) → U_3
  200. Point(ponthalmaz2) → V_3
  201. Point(ponthalmaz2) → W_3
  202. Point(ponthalmaz2) → Z_3
  203. Point(ponthalmaz2) → A_4
  204. Point(ponthalmaz2) → B_4
  205. Point(ponthalmaz2) → C_4
  206. Point(ponthalmaz2) → D_4
  207. Point(ponthalmaz2) → E_4
  208. Point(ponthalmaz2) → F_4
  209. Point(ponthalmaz2) → G_4
  210. Point(ponthalmaz2) → H_4
  211. Point(ponthalmaz2) → I_4
  212. Point(ponthalmaz2) → J_4
  213. Point(ponthalmaz2) → K_4
  214. Point(ponthalmaz2) → L_4
  215. Point(ponthalmaz2) → M_4
  216. Point(ponthalmaz2) → N_4
  217. Point(ponthalmaz2) → O_4
  218. Point(ponthalmaz2) → P_4
  219. Point(ponthalmaz2) → Q_4
  220. Point(ponthalmaz2) → R_4
  221. Point(ponthalmaz2) → S_4
  222. Point(ponthalmaz2) → T_4
  223. Point(ponthalmaz2) → U_4
  224. Polygon(B'_1, H_1, Z_3, A_4, B_4, C_4, D_4, E_4, F_4, G_4, H_4, I_4, P_3, J_4, K_4, L_4, M_4, N_4, O_4, P_4, Q_4, R_4, S_4, Q_3, T_4, U_4, B'', C', L, N, G, R_3, S_3, T_3, U_3, V_3, W_3) → poligon3, h_7, w_3, z_3, q_7, r_7, s_7, t_7, a_7, d_7, e_7, f_8, j_7, g_8, h_8, i_8, j_8, k_8, l_8, o_4, m_8, n_8, p_8, k_7, q_8, r_8, u_4, b'', c', l_9, n_9, g_9, l_7, m_7, n_7, p_7, u_3, v_3
  225. Point(ponthalmaz3) → V_4
  226. Point(ponthalmaz3) → W_4
  227. Point(ponthalmaz3) → Z_4
  228. Point(ponthalmaz3) → A_5
  229. Point(ponthalmaz3) → B_5
  230. Point(ponthalmaz3) → C_5
  231. Point(ponthalmaz3) → D_5
  232. Point(ponthalmaz3) → E_5
  233. Point(ponthalmaz3) → F_5
  234. Point(ponthalmaz3) → G_5
  235. Point(ponthalmaz3) → H_5
  236. Point(ponthalmaz3) → I_5
  237. Point(ponthalmaz3) → J_5
  238. Point(ponthalmaz3) → K_5
  239. Point(ponthalmaz3) → L_5
  240. Point(ponthalmaz3) → M_5
  241. Point(ponthalmaz3) → N_5
  242. Point(ponthalmaz3) → O_5
  243. Point(ponthalmaz3) → P_5
  244. Point(ponthalmaz3) → Q_5
  245. Point(ponthalmaz3) → R_5
  246. Point(ponthalmaz3) → S_5
  247. Point(ponthalmaz3) → T_5
  248. Polygon(E, J_1, F_5, G_5, H_5, I_5, J_5, K_5, L_5, M_5, N_5, O_5, V_4, P_5, Q_5, R_5, S_5, T_5, K_1, C'', H, W_4, Z_4, A_5, B_5, C_5, D_5, E_5) → poligon4, e_8, d_8, f_9, h_9, i_9, j_9, k_9, m_9, p_9, q_9, r_9, s_8, o_5, s_9, t_9, a_9, b_9, c_9, d_9, e_9, h_{10}, v_4, w_4, z_4, t_8, a_8, b_8, c_8

AI 绘图提示语参考

画一个三维鞍形曲面，用二维几何构造模拟光照和投影的视觉效果。展示曲面上的阴影生成和投影轮廓，通过多个辅助点和线逐步构建阴影边界和投影线条，体现点、线、面的空间关系及函数图形与变换思想。
以上提示语为根据原始绘图命令整理而来，仅供参考。