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鞍面光影演示

鞍面光影演示
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2020-04-22 04:00
向量函数立体几何微积分几何作图
①本范例演示鞍面（双曲抛物面）的光影效果与几何形态；②涉及核心数学概念包括曲面、不定积分、双曲线、整数、三角形、阶跃函数等；③作图要点运用了线段、直线、交点、垂线、点、向量、平移、轨迹、多边形等 GeoGebra 命令与对象；④适合高中及以上数学学习者，用于理解空间几何与微积分在三维曲面上的应用。

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) → R_1
  88. Line(R_1, A_s) → k_7
  89. Line(A_s, N_1) → l_7
  90. Intersect(c_2, k_7) → S_5
  91. Intersect(e_2, l_7) → T_1
  92. Segment(S_5, A_s) → m_7
  93. Segment(A_s, T_1) → n_7
  94. Line(S_5, a) → p_7
  95. Line(T_1, a) → q_7
  96. Intersect(q_7, k) → U_1
  97. Intersect(p_7, j) → G_2
  98. Segment(G_2, S_5) → r_7
  99. Segment(U_1, T_1) → s_7
  100. Line(U_1, j) → g_3
  101. Line(G_2, k) → h_3
  102. Intersect(g_3, f) → S_1'
  103. Intersect(g, h_3) → S_2'
  104. Segment(U_1, S_1') → i_3
  105. Segment(S_2', G_2) → k_3
  106. Segment(S_1', S_2') → n_s
  107. Line(S_1', n) → l_3
  108. Line(S_2', n) → m_3
  109. Intersect(b_1, l_3) → S_1
  110. Intersect(e_1, m_3) → S_2
  111. Segment(S_1, S_1') → n_3
  112. Segment(S_2', S_2) → p_3
  113. Point(n_s) → Q_1
  114. Line(Q_1, f) → q_3
  115. Intersect(g, q_3) → V_1
  116. Intersect(j, q_3) → W_1
  117. Line(V_1, n) → r_3
  118. Line(W_1, n) → s_3
  119. Intersect(r_1, s_3) → Z_1
  120. Intersect(r_3, e_1) → A_2
  121. Segment(A_2, Z_1) → t_3
  122. Line(Q_1, n) → a_3
  123. Intersect(t_3, a_3) → B_2
  124. Locus(B_2, Q_1) → ponthalmaz2
  125. Line(S_2', a) → b_3
  126. Line(S_1', a) → c_3
  127. Intersect(t_2, c_3) → C_2
  128. Intersect(b_3, f_3) → D_2
  129. Segment(S_1', C_2) → d_3
  130. Segment(S_2', D_2) → e_3
  131. Line(Q_1, a) → f_4
  132. Line(B_2, d) → g_4
  133. Intersect(g_4, f_4) → E_2
  134. Locus(E_2, Q_1) → ponthalmaz3
  135. Intersect(i_3, k_3) → A_p'
  136. Line(A_p', n) → h_4
  137. Intersect(d, h_4) → A_p
  138. Segment(A_p, A_p') → i_4
  139. Segment(S_1, A_p) → j_4
  140. Segment(A_p, S_2) → k_4
  141. Line(G_2, i_4) → l_4
  142. Line(U_1, i_4) → m_4
  143. Intersect(l_4, r_1) → I_2
  144. Intersect(m_4, s_1) → J_2
  145. Segment(I_2, G_2) → n_4
  146. Segment(U_1, J_2) → p_4
  147. Segment(A_p, I_2) → q_4
  148. Segment(A_p, J_2) → r_4
  149. Point(ponthalmaz3) → K_1
  150. Point(ponthalmaz3) → M_1
  151. Point(ponthalmaz3) → O_1
  152. Point(ponthalmaz3) → P_1
  153. Point(ponthalmaz3) → F_2
  154. Point(ponthalmaz3) → H_2
  155. Point(ponthalmaz3) → C_4
  156. Point(ponthalmaz3) → D_4
  157. Point(ponthalmaz3) → E_4
  158. Point(ponthalmaz3) → F_4
  159. Point(ponthalmaz3) → G_4
  160. Point(ponthalmaz3) → H_4
  161. Point(ponthalmaz3) → I_4
  162. Point(ponthalmaz3) → J_4
  163. Polygon(R_1, C_s, C_4, D_4, E_4, F_4, G_4, H_4, I_4, J_4, D_2, N_1, C_2, K_1, M_1, O_1, P_1, F_2, H_2) → poligon5, j_7, g_8, h_8, i_8, j_8, k_8, l_8, m_8, n_8, p_8, q_8, c_s, t_7, a_7, b_7, c_7, d_7, e_7, f_8
  164. Point(ponthalmaz2) → K_2
  165. Point(ponthalmaz2) → L_2
  166. Point(ponthalmaz2) → M_2
  167. Point(ponthalmaz2) → N_2
  168. Point(ponthalmaz2) → O_2
  169. Point(ponthalmaz2) → P_2
  170. Point(ponthalmaz2) → Q_2
  171. Point(ponthalmaz2) → R_2
  172. Point(ponthalmaz2) → S_4
  173. Point(ponthalmaz2) → T_2
  174. Point(ponthalmaz2) → U_2
  175. Point(ponthalmaz2) → V_2
  176. Point(ponthalmaz2) → W_2
  177. Point(ponthalmaz2) → Z_2
  178. Point(ponthalmaz2) → A_3
  179. Point(ponthalmaz2) → B_3
  180. Point(ponthalmaz2) → C_3
  181. Point(ponthalmaz2) → D_3
  182. Point(ponthalmaz2) → E_3
  183. Point(ponthalmaz2) → F_3
  184. Polygon(L_1, S_2, S_4, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, K_2, E_3, F_3, S_1, L_2, M_2, N_2, O_2, P_2, Q_2, R_2) → poligon2, s_4, g_5, h_5, i_5, j_5, v_2, w_2, z_2, k_5, l_5, m_5, t_4, n_5, p_5, q_5, o_1, a_4, b_4, c_4, d_4, o_2, e_4, f_5
  185. Polygon(A_p, S_2, S_4, T_2, U_2, V_2, W_2, Z_2, A_3, B_3, C_3, D_3, K_2, E_3, F_3, S_1, L_2, M_2, N_2, O_2, P_2, Q_2, R_2) → poligon3, r_5, g_6, h_6, i_6, j_6, k_6, l_6, m_6, n_6, p_6, q_6, s_5, r_6, s_6, t_6, a_6, t_5, a_5, b_5, c_5, d_5, e_5, f_6
  186. Point(ponthalmaz1) → G_3
  187. Point(ponthalmaz1) → H_3
  188. Point(ponthalmaz1) → I_3
  189. Point(ponthalmaz1) → J_3
  190. Point(ponthalmaz1) → K_3
  191. Point(r_4) → L_3
  192. Point(ponthalmaz1) → M_3
  193. Point(r_1) → N_3
  194. Polygon(U, G_3, H_3, I_3, J_3, K_3, L_3, M_3, N_3) → poligon4, u_3, b_6, c_6, d_6, e_6, f_7, g_7, h_7, i_7
  195. Point(ponthalmaz3) → O_3
  196. Point(ponthalmaz3) → P_3
  197. Point(ponthalmaz3) → Q_3
  198. Point(ponthalmaz3) → R_3
  199. Point(ponthalmaz3) → S_3
  200. Point(ponthalmaz3) → T_3
  201. Point(ponthalmaz3) → U_3
  202. Point(ponthalmaz3) → V_3
  203. Point(ponthalmaz3) → W_3
  204. Point(ponthalmaz3) → Z_3
  205. Point(ponthalmaz3) → A_4
  206. Point(ponthalmaz3) → B_4