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鞍形曲面的阴影与投影（四）

鞍形曲面的阴影与投影（四）
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2020-04-22 04:14
立体几何变换几何作图轨迹函数图像
本范例演示了三维鞍形曲面在光照下的阴影与投影效果，通过二维几何作图模拟三维曲面的视觉表现。核心数学概念涉及曲面、投影、阴影、切线、法线以及轨迹。作图要点包括：使用线段、直线、交点、正交线、点、向量、平移、轨迹和多边形等GeoGebra命令，构造了多个辅助点（如A'、B、C'等）和辅助线（如f、g、h等），通过平移和轨迹生成阴影边界。适合高中或大学阶段学习立体几何、解析几何和计算机图形学的学生，用于理解三维曲面在二维平面上的投影与阴影形成原理，也可作为数学可视化教学的示例。

GeoGebra 命令及参数

  1. Segment(D, A') → f
  2. Segment(A', B) → g
  3. Line(B, f) → h
  4. Line(D, g) → i
  5. Intersect(h, i) → C'
  6. Segment(D, C') → j
  7. Segment(C', B) → k
  8. OrthogonalLine(A', g) → l
  9. OrthogonalLine(C', j) → m
  10. Point(l) → A
  11. Point(m) → C
  12. Segment(A, A') → n
  13. Segment(C, C') → p
  14. Segment(D, A) → q
  15. Segment(A, B) → r
  16. Segment(B, C) → s
  17. Segment(C, D) → t
  18. Line(D, C) → s_2
  19. Intersect(s_2, r) → E
  20. Line(E, n) → b
  21. Intersect(g, b) → G
  22. Line(B, j) → e
  23. Line(G, k) → i_1
  24. Intersect(j, i_1) → I
  25. Segment(G, I) → j_1
  26. Line(I, n) → l_1
  27. Intersect(t, l_1) → K
  28. Segment(K, I) → q_1
  29. Line(C, B) → c
  30. Intersect(c, q) → F
  31. OrthogonalLine(F, g) → g_1
  32. Intersect(g_1, f) → H
  33. Vector(A', H) → u
  34. Translate(B, u) → B'
  35. OrthogonalLine(B', g) → k_1
  36. Intersect(s, k_1) → J
  37. Segment(J, B') → m_1
  38. Segment(I, B') → n_k
  39. Point(n_k) → L
  40. Line(L, j_1) → p_1
  41. Intersect(p_1, j) → M
  42. Intersect(g, p_1) → N
  43. Intersect(p, p_1) → O
  44. Intersect(p, n) → P
  45. Line(M, n) → t_1
  46. Line(N, t_1) → a_1
  47. Intersect(t, t_1) → Q
  48. Intersect(a_1, r) → R
  49. Segment(Q, R) → c_1
  50. Line(L, t_1) → d_1
  51. Intersect(c_1, d_1) → S
  52. Locus(S, L) → ponthalmaz1
  53. Segment(D, C) → r_1
  54. Segment(C, B) → s_1
  55. Segment(A, D) → b_1
  56. Intersect(r_1, b_1) → T
  57. Intersect(r_1, s_1) → U
  58. Point(r_1) → V
  59. Segment(A, B) → e_1
  60. Polygon(A, D, U, B) → q1, a_2, d_2, u_1, b_2
  61. Point(ponthalmaz1) → W
  62. Point(ponthalmaz1) → Z
  63. Point(ponthalmaz1) → A_1
  64. Point(ponthalmaz1) → B_1
  65. Point(ponthalmaz1) → C_1
  66. Point(ponthalmaz1) → D_1
  67. Point(ponthalmaz1) → E_1
  68. Point(ponthalmaz1) → F_1
  69. Point(ponthalmaz1) → G_1
  70. Point(ponthalmaz1) → H_1
  71. Point(ponthalmaz1) → I_1
  72. Point(ponthalmaz1) → J_1
  73. Polygon(U, K, G_1, H_1, I_1, J_1, J, W, Z, A_1, B_1, C_1, D_1, E_1, F_1) → poligon1, u_2, l_2, m_2, n_2, p_2, q_2, j_3, k_2, w, z_1, f_2, g_2, h_2, i_2, j_2
  74. Segment(A', A_s) → a
  75. Intersect(b_1, e_1) → L_1
  76. Segment(L_1, A_s) → d
  77. Line(C', a) → f_1
  78. Line(U, d) → h_1
  79. Intersect(f_1, h_1) → C_s
  80. Segment(C', C_s) → n_1
  81. Segment(U, C_s) → r_2
  82. Segment(D, A_s) → t_2
  83. Segment(C_s, B) → e_2
  84. Intersect(r_1, b_1) → N_1
  85. Segment(C_s, N_1) → c_2
  86. Segment(B, A_s) → f_3
  87. Intersect(s_1, e_1) → K_1
  88. Line(K_1, C_s) → e_3
  89. Intersect(t_2, e_3) → M_1
  90. Segment(M_1, K_1) → h_4
  91. Intersect(s_1, e_1) → R_1
  92. Line(R_1, A_s) → k_7
  93. Line(A_s, N_1) → l_7
  94. Intersect(c_2, k_7) → S_8
  95. Intersect(e_2, l_7) → T_1
  96. Segment(S_8, A_s) → m_7
  97. Segment(A_s, T_1) → n_7
  98. Line(M_1, a) → i_4
  99. Intersect(f, i_4) → O_1
  100. Segment(M_1, O_1) → j_4
  101. Line(S_8, a) → p_7
  102. Line(T_1, a) → q_7
  103. Intersect(q_7, k) → U_1
  104. Intersect(p_7, j) → G_2
  105. Segment(G_2, S_8) → r_7
  106. Segment(U_1, T_1) → s_7
  107. Line(O_1, g) → k_4
  108. Intersect(k, k_4) → S_1'
  109. Segment(O_1, S_1') → l_4
  110. Line(U_1, j) → g_3
  111. Line(G_2, k) → h_3
  112. Intersect(g_3, f) → Ö_3
  113. Intersect(g, h_3) → S_2'
  114. Segment(U_1, Ö_3) → i_3
  115. Segment(S_2', G_2) → k_3
  116. Segment(S_2', S_1') → n_s
  117. Segment(Ö_3, S_2') → n_4
  118. Line(S_1', p) → m_4
  119. Intersect(s_1, m_4) → S_1
  120. Segment(S_1, S_1') → p_4
  121. Line(Ö_3, n) → l_3
  122. Line(S_2', n) → m_3
  123. Intersect(b_1, l_3) → Ö_4
  124. Intersect(e_1, m_3) → Ö_2
  125. Segment(Ö_4, Ö_3) → n_3
  126. Segment(S_2', Ö_2) → p_3
  127. Intersect(n_k, n_s) → S_k'
  128. Line(S_k', k_3) → s_4
  129. Intersect(j, s_4) → H_2
  130. Intersect(g, s_4) → I_2
  131. Segment(H_2, I_2) → t_4
  132. Line(H_2, p) → a_4
  133. Line(I_2, a_4) → b_4
  134. Intersect(r_1, a_4) → J_2
  135. Intersect(b_4, e_1) → K_2
  136. Segment(J_2, H_2) → c_4
  137. Segment(K_2, I_2) → d_4
  138. Segment(J_2, K_2) → e_4
  139. Line(S_k', n) → f_5
  140. Intersect(e_4, f_5) → S_k
  141. Segment(S_k, S_k') → g_5
  142. Line(Ö_3, a) → c_3
  143. Point(n_s) → P_1
  144. Line(P_1, k_3) → q_3
  145. Intersect(j, q_3) → Q_1
  146. Intersect(g, q_3) → V_1
  147. Line(Q_1, p) → r_3
  148. Line(V_1, p) → s_3
  149. Intersect(e_1, s_3) → W_1
  150. Intersect(r_3, r_1) → Z_1
  151. Intersect(r_3, r_1) → A_2
  152. Segment(Z_1, W_1) → t_3
  153. Line(P_1, s_3) → a_3
  154. Intersect(t_3, a_3) → B_2
  155. Locus(B_2, P_1) → ponthalmaz2
  156. Line(S_1', n_1) → b_3
  157. Line(S_2', b_3) → d_3
  158. Intersect(f_3, m_7) → C_2
  159. Intersect(d_3, f_3) → D_2
  160. Intersect(b_3, e_2) → E_2
  161. Segment(S_2', D_2) → f_4
  162. Segment(S_1', E_2) → g_4
  163. Line(P_1, g_4) → q_4
  164. Line(B_2, r_2) → r_4
  165. Intersect(q_4, r_4) → F_2
  166. Locus(F_2, P_1) → ponthalmaz3
  167. Point(s_1) → L_2
  168. Point(ponthalmaz2) → M_2
  169. Point(ponthalmaz2) → N_2
  170. Point(ponthalmaz2) → O_2
  171. Point(ponthalmaz2) → P_2
  172. Polygon(Ö_2, K_1, L_2, M_2, N_2, O_2, P_2) → poligon2, ö_2, h_5, i_5, j_5, k_5, o_2, l_5
  173. Point(ponthalmaz2) → Q_2
  174. Point(ponthalmaz2) → R_2
  175. Point(ponthalmaz2) → S_2
  176. Point(ponthalmaz2) → T_2
  177. Point(ponthalmaz2) → U_2
  178. Point(ponthalmaz2) → V_2
  179. Point(ponthalmaz2) → W_2
  180. Point(ponthalmaz2) → Z_2
  181. Point(ponthalmaz2) → A_3
  182. Point(ponthalmaz2) → B_3
  183. Point(ponthalmaz2) → C_3
  184. Point(ponthalmaz2) → D_3
  185. Point(ponthalmaz2) → E_3
  186. Point(ponthalmaz2) → F_3
  187. Point(ponthalmaz2) → G_3
  188. Point(ponthalmaz2) → H_3
  189. Point(ponthalmaz2) → I_3
  190. Point(ponthalmaz2) → J_3
  191. Point(ponthalmaz2) → K_3
  192. Point(ponthalmaz1) → L_3
  193. Point(ponthalmaz1) → M_3
  194. Point(ponthalmaz1) → N_3
  195. Point(ponthalmaz1) → O_3
  196. Point(ponthalmaz1) → P_3
  197. Point(ponthalmaz1) → Q_3
  198. Point(ponthalmaz1) → R_3
  199. Point(ponthalmaz1) → S_3
  200. Point(ponthalmaz1) → T_3
  201. Point(ponthalmaz1) → U_3
  202. Point(s_1) → V_3
  203. Polygon(S_1, Q_2, B_3, C_3, D_3, E_3, F_3, G_3, H_3, I_3, J_3, K_3, R_2, S_k, L_3, M_3, N_3, O_3, P_3, Q_3, R_3, S_3, T_3, S_2, U_3, V_3, T_2, U_2, V_2, W_2, Z_2, A_3) → poligon3, ö_1, s_5, t_5, a_5, b_5, c_5, d_5, e_5, f_6, g_6, h_6, m_5, i_6, ö_k, j_6, k_6, l_6, o_3, m_6, n_6, p_6, q_6, n_5, r_6, u_3, v_3, p_5, q_5, r_5, v_2, w_2, z_2
  204. Point(ponthalmaz1) → W_3
  205. Point(ponthalmaz1) → Z_3
  206. Point(ponthalmaz1) → A_4
  207. Point(ponthalmaz1) → B_4
  208. Point(ponthalmaz1) → C_4
  209. Point(ponthalmaz1) → D_4
  210. Point(ponthalmaz1) → E_4
  211. Point(ponthalmaz1) → F_4
  212. Point(ponthalmaz1) → G_4
  213. Point(ponthalmaz1) → H_4
  214. Point(ponthalmaz1) → I_4
  215. Point(ponthalmaz1) → J_4
  216. Point(ponthalmaz1) → K_4
  217. Point(ponthalmaz1) → L_4
  218. Point(ponthalmaz1) → M_4
  219. Point(ponthalmaz1) → N_4
  220. Point(ponthalmaz1) → O_4
  221. Point(ponthalmaz1) → P_4
  222. Point(ponthalmaz1) → Q_4
  223. Point(ponthalmaz1) → R_4
  224. Point(ponthalmaz1) → S_4
  225. Point(ponthalmaz1) → T_4
  226. Point(ponthalmaz1) → U_4
  227. Point(ponthalmaz1) → V_4
  228. Point(ponthalmaz1) → W_4
  229. Point(r_1) → Z_4
  230. Point(ponthalmaz3) → A_5
  231. Point(ponthalmaz3) → B_5
  232. Point(ponthalmaz3) → C_5
  233. Point(ponthalmaz3) → D_5
  234. Point(ponthalmaz3) → E_5
  235. Point(ponthalmaz3) → F_5
  236. Point(ponthalmaz3) → G_5
  237. Point(ponthalmaz3) → H_5
  238. Point(ponthalmaz3) → I_5
  239. Point(ponthalmaz3) → J_5
  240. Point(ponthalmaz3) → K_5
  241. Point(ponthalmaz3) → L_5
  242. Intersect(e_1, t_2) → R_6
  243. Intersect(t_2, k_6) → S_6
  244. Point(ponthalmaz1) → T_6
  245. Point(ponthalmaz1) → U_6
  246. Point(ponthalmaz1) → V_6
  247. Point(ponthalmaz1) → W_6
  248. Point(ponthalmaz1) → Z_6
  249. Point(ponthalmaz1) → A_7
  250. Point(ponthalmaz1) → B_7
  251. Point(ponthalmaz1) → C_7
  252. Point(ponthalmaz1) → D_7
  253. Point(ponthalmaz1) → E_7
  254. Point(ponthalmaz1) → F_7
  255. Point(ponthalmaz1) → G_7
  256. Point(ponthalmaz1) → H_7
  257. Point(ponthalmaz1) → I_7
  258. Intersect(i_6, ö_k) → J_7
  259. Point(ponthalmaz1) → K_7
  260. Point(ponthalmaz1) → L_7
  261. Point(ponthalmaz1) → M_7
  262. Point(ponthalmaz1) → N_7
  263. Point(ponthalmaz1) → O_7
  264. Point(ponthalmaz1) → P_7
  265. Point(ponthalmaz1) → Q_7
  266. Point(ponthalmaz1) → R_7
  267. Point(ponthalmaz1) → S_7
  268. Point(ponthalmaz1) → T_7
  269. Polygon(S_1, Q_2, C_3, D_3, E_3, G_3, H_3, J_3, K_3, J_7, W_3, Z_3, R_2, A_4, B_4, C_4, D_4, E_4, F_4, G_4, P_4, T_6, U_6, S_2, V_6, W_6, Z_6, A_7, B_7, G_7, K_7, L_7, M_7, N_7, T_2, O_7, P_7, Q_7, R_7, S_7, T_7, I_7, Z_4, U, U_2, V_2, W_2, A_3, B_3) → poligon4, s_6, h_7, i_7, j_7, t_7, a_7, b_7, c_7, d_7, e_7, w_3, t_6, z_3, f_8, g_8, h_8, i_8, j_8, k_8, l_8, m_8, n_8, a_6, u_6, v_6, w_6, z_6, p_8, q_8, r_8, s_8, t_8, a_8, b_6, b_8, o_7, c_8, d_8, e_8, f_9, g_9, h_9, z_4, u_4, c_6, d_6, e_6, f_7, g_7
  270. Point(ponthalmaz3) → M_5
  271. Point(ponthalmaz3) → N_5
  272. Point(ponthalmaz3) → O_5
  273. Point(ponthalmaz3) → P_5
  274. Point(ponthalmaz3) → Q_5
  275. Point(ponthalmaz3) → R_5
  276. Point(ponthalmaz3) → S_5
  277. Point(ponthalmaz3) → T_5
  278. Point(ponthalmaz3) → U_5
  279. Point(ponthalmaz3) → V_5
  280. Point(ponthalmaz3) → W_5
  281. Point(ponthalmaz3) → Z_5
  282. Point(ponthalmaz3) → A_6
  283. Point(ponthalmaz3) → B_6
  284. Point(ponthalmaz3) → C_6
  285. Point(ponthalmaz3) → D_6
  286. Point(ponthalmaz3) → E_6
  287. Point(ponthalmaz3) → F_6
  288. Point(t_2) → G_6
  289. Point(ponthalmaz1) → H_6
  290. Point(ponthalmaz1) → I_6
  291. Point(ponthalmaz1) → J_6
  292. Point(ponthalmaz1) → K_6
  293. Point(ponthalmaz1) → L_6
  294. Point(ponthalmaz1) → M_6
  295. Point(s_1) → N_6
  296. Point(s_1) → O_6
  297. Intersect(e_1, t_2) → P_6
  298. Polygon(A_s, D_2, U_5, V_5, W_5, Z_5, A_6, B_6, C_6, D_6, E_6, F_6, M_5, E_2, C_s, N_1, G_6, H_6, I_6, J_6, K_6, L_6, M_6, N_5, N_6, O_6, K_1, P_6, O_5, P_5, Q_5, R_5, S_5, T_5) → poligon5, a_s, q_9, u_5, v_5, w_5, z_5, r_9, s_9, t_9, a_9, b_9, i_9, c_9, d_9, c_s, e_9, f_{10}, g_{10}, h_{10}, i_{10}, j_{10}, k_{10}, j_9, l_{10}, m_{10}, o_6, n_{10}, p_{10}, k_9, o_5, l_9, m_9, n_9, p_9