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蜘蛛与苍蝇

蜘蛛与苍蝇
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2024-02-20 01:08
角最值优化几何作图
这个范例演示了蜘蛛和苍蝇之间的最短路径问题。涉及的核心数学概念包括相似三角形、角度和最值优化。通过使用GeoGebra命令如Mirror、Polygon、PointIn等，可以构造出复杂的几何图形。适合高中数学教学和学习。

GeoGebra 命令及参数

  1. Mirror(A, xAxis) → D
  2. Mirror(A, (0, 0, 0)) → C
  3. Mirror(A, yAxis) → B
  4. Polygon(A, B, C, D) → v1, a, b, c, d
  5. Prism(v1, E) → quad, F, KanteBF, KanteCG, KanteDH, KanteEF, KanteFG, KanteGH, KanteEH, G, H, FlächeABFE, FlächeBCGF, FlächeCDHG, FlächeADHE, Deck, KanteAE
  6. Net(quad, l_{13}) → net1, I, R, S, T, U, V, Fläche7, Fläche8, Fläche9, Fläche10, Fläche11, J, Fläche12, Kante19, Kante20, Kante21, Kante22, Kante23, Kante24, Kante25, Kante26, Kante27, K, Kante28, Kante29, Kante30, Kante31, Kante32, Kante33, Kante34, Kante35, Kante36, Kante37, L, M, N, O, P, Q
  7. PointIn(FlächeADHE) → spider
  8. PointIn(FlächeBCGF) → fly
  9. Distance(A, B) → lang
  10. Rotate(spider_B, 90°, A) → spider2
  11. Rotate(spider2, 90°, M) → spider3
  12. Rotate(spider_B, -90°, D) → spider1
  13. Segment(spider3, fly_B) → l_{10}
  14. Segment(spider2, fly_B) → l_8
  15. Segment(spider1, fly_B) → l_2
  16. Segment(spider_B, fly_B) → l_5
  17. Rotate(fly_B, -90°, B) → fly2
  18. Rotate(fly2, -90°, N) → fly3
  19. Segment(spider_B, fly3) → l_{11}
  20. Segment(spider2, fly2) → l_9
  21. Segment(spider1, fly2) → l_3
  22. Segment(spider_B, fly2) → l_6
  23. Rotate(fly_B, 90°, C) → fly1
  24. Segment(spider2, fly1) → l_7
  25. Segment(spider1, fly1) → l_1
  26. Segment(spider_B, fly1) → l_4
  27. OrthogonalLine(A, xOyPlane) → f
  28. Distance(A, E) → hoch
  29. Segment(spider3, fly3) → l_{12}
  30. Min(lStr) → minim
  31. Distance(B, C) → breit
  32. Net(quad, g) → net2, W, I_1, J_1, K_1, L_1, M_1, Fläche1, Fläche2, Fläche3, Fläche4, Fläche5, Z, Fläche6, Kante1, Kante2, Kante3, Kante4, Kante5, Kante6, Kante7, Kante8, Kante9, A_1, Kante10, Kante11, Kante12, Kante13, Kante14, Kante15, Kante16, Kante17, Kante18, Kante38, B_1, C_1, D_1, F_1, G_1, H_1
  33. Intersect(l_1, Kante31) → S11
  34. Intersect(l_1, Kante29) → S12
  35. Rotate(S11, ((g - 1) * 90°), c) → S11g
  36. Rotate(S12, ((g - 1) * 90°), c) → S12g
  37. Rotate(spider_B, ((g - 1) * 90°), d) → spiderg
  38. Rotate(fly_B, ((g - 1) * 90°), b) → flyg
  39. Segment(spiderg, S11g) → s11
  40. Segment(S11g, S12g) → s12
  41. Segment(S12g, flyg) → s13
  42. Intersect(l_2, Kante31) → S21
  43. Intersect(c, l_2) → S22
  44. Intersect(b, l_2) → S23
  45. Segment(S22, S23) → s23
  46. Rotate(S21, ((g - 1) * 90°), c) → S21g
  47. Segment(spiderg, S21g) → s21
  48. Segment(S21g, S22) → s22
  49. Segment(S23, flyg) → s24
  50. Intersect(d, l_5) → S51
  51. Intersect(b, l_5) → S52
  52. Segment(spiderg, S51) → s51
  53. Segment(S51, S52) → s52
  54. Segment(S52, flyg) → s53
  55. Intersect(l_8, Kante23) → S91
  56. Intersect(a, l_8) → S82
  57. Intersect(b, l_8) → S83
  58. Segment(S82, S83) → s83
  59. Segment(S83, flyg) → s84
  60. Rotate(S91, ((g - 1) * 90°), a) → S81g
  61. Segment(spiderg, S81g) → s81
  62. Segment(S81g, S82) → s82
  63. Intersect(l_9, Kante25) → S92
  64. Rotate(S91, ((g - 1) * 90°), a) → S91g
  65. Rotate(S92, ((g - 1) * 90°), a) → S92g
  66. Segment(spiderg, S91g) → s91
  67. Segment(S91g, S92g) → s92
  68. Segment(S92g, flyg) → s93
  69. Intersect(l_3, Kante31) → S31
  70. Intersect(a, l_3) → S33
  71. Rotate(S31, ((g - 1) * 90°), c) → S31g
  72. Intersect(c, l_3) → S32
  73. Segment(spiderg, S31g) → s31
  74. Segment(S31g, S32) → s32
  75. Segment(S32, S33) → s33
  76. Intersect(d, l_6) → S61
  77. Intersect(a, l_6) → S62
  78. Intersect(Kante25, l_6) → S63
  79. Rotate(S63, ((g - 1) * 90°), a) → S63g
  80. Segment(spiderg, S61) → s61
  81. Segment(S61, S62) → s62
  82. Segment(S62, S63g) → s63
  83. Segment(S63g, flyg) → s64
  84. Intersect(l_4, d) → S41
  85. Intersect(l_4, c) → S42
  86. Intersect(l_4, Kante29) → S43
  87. Rotate(S43, ((g - 1) * 90°), c) → S43g
  88. Segment(spiderg, S41) → s41
  89. Segment(S41, S42) → s42
  90. Segment(S42, S43g) → s43
  91. Intersect(l_7, Kante23) → S71
  92. Intersect(a, l_7) → S72
  93. Intersect(c, l_7) → S73
  94. Intersect(l_7, Kante29) → S74
  95. Rotate(S71, ((g - 1) * 90°), a) → S71g
  96. Rotate(S74, ((g - 1) * 90°), c) → S74g
  97. Segment(S74g, flyg) → s75
  98. Segment(S72, S73) → s73
  99. Segment(S73, S74g) → s74
  100. Segment(spiderg, S71g) → s71
  101. Segment(S71g, S72) → s72
  102. Polygon(E_1, S, D, R) → v2, e_1, s, d_1, r
  103. Polygon(Q, C, P, E_4) → v3, q, c_1, p, e_4
  104. Polygon(T, E_2, M, A) → v4, t, e_2, m, a_1
  105. Polygon(B, N, E_3, O) → v5, b_1, n, e_3, o
  106. Segment(spider3, fly2) → h
  107. Segment(spider2, fly3) → i
  108. Polygon(E_2, E_5, V, M) → v6, j, e_5, v, m_1
  109. Polygon(N, U, E_6, E_3) → v7, n_1, u, e_6, k
  110. Intersect(Kante38, Kante37) → N_1
  111. Intersect(Kante38, Kante37) → O_1
  112. Intersect(Kante38, Kante37) → P_1
  113. Intersect(Kante6, Kante24) → R_1
  114. Intersect(Kante6, Kante24) → S_1
  115. Intersect(Kante6, Kante24) → T_1
  116. Intersect(a, Kante1) → V_1
  117. Intersect(a, Kante1) → W_1
  118. Intersect(l_{10}, a) → S103
  119. Intersect(b, Kante2) → A_2
  120. Intersect(l_{10}, b) → S104
  121. Intersect(l_{10}, Kante24) → S102
  122. Intersect(l_{10}, Kante37) → S101
  123. Rotate(Rotate[S101, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6) → S101g
  124. Rotate(S102, ((g - 1) * 90°), a) → S102g
  125. Segment(spiderg, S101g) → s101
  126. Segment(S101g, S102g) → s102
  127. Segment(S102g, S103) → s103
  128. Segment(S103, S104) → s104
  129. Segment(S104, flyg) → s105
  130. Mirror(spiderg, xzEbene) → spiderg'
  131. Mirror(S101g, xzEbene) → S101g'
  132. Mirror(S102g, xzEbene) → S102g'
  133. Mirror(S103, xzEbene) → S103'
  134. Mirror(S104, xzEbene) → S104'
  135. Mirror(flyg, xzEbene) → flyg'
  136. PolyLine(spiderg', S101g', S102g', S103', S104', flyg') → p10'
  137. Intersect(l_{11}, a) → S112
  138. Intersect(l_{11}, Kante24) → S113
  139. Intersect(l_{11}, Kante35) → S114
  140. Intersect(l_{11}, d) → S111
  141. Rotate(S113, ((g - 1) * 90°), a) → S113g
  142. Rotate(Rotate[S114, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6) → S114g
  143. PolyLine(spiderg, S111, S112, S113g, S114g, flyg) → p11
  144. Mirror(S111, xzEbene) → S111'
  145. Mirror(S112, xzEbene) → S112'
  146. Mirror(S113g, xzEbene) → S113g'
  147. Mirror(S114g, xzEbene) → S114g'
  148. PolyLine(spiderg', S111', S112', S113g', S114g', flyg') → p11'
  149. PolyLine(spiderg, S101g, S102g, S103, S104, flyg) → p10
  150. Segment(S43g, flyg) → s44
  151. Intersect(Kante37, l_{12}) → S121
  152. Intersect(Kante35, n_1) → U_1
  153. Intersect(l_{12}, Kante35) → S122
  154. Rotate(Rotate[S121, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6) → S121g
  155. Rotate(Rotate[S122, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6) → S122g
  156. Rotate(Rotate[Rotate[spider3, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6], ((-(g - 1)) * 90°), Kante38) → spider3g
  157. Rotate(Rotate[Rotate[fly3, ((g - 1) * 90°), a], ((-(g - 1)) * 90°), Kante6], ((-(g - 1)) * 90°), Kante17) → fly3g
  158. Segment(spider3g, S121g) → s121
  159. Segment(S121g, S122g) → s122
  160. Segment(S122g, fly3g) → s123

AI 绘图提示语参考

绘制一个长方体，其底面为正方形，四个顶点依次连接。在长方体的两个相对侧面上分别放置一个点和另一个点，代表蜘蛛和苍蝇。将长方体展开成平面展开图，并展示从蜘蛛到苍蝇的所有可能路径，包括经过不同棱边的折线。在展开图中，通过旋转和镜像操作，将蜘蛛和苍蝇的位置映射到展开图的各个面上，并画出多条连接它们的线段。最后，标出最短路径，并显示路径经过的棱边交点。
以上提示语为根据原始绘图命令整理而来，仅供参考。