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Spider and Fly

Spider and Fly
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2024-02-20 01:08
AngleOptimizationGeometric Construction
This example demonstrates the shortest path problem between a spider and a fly. Key mathematical concepts include similar triangles, angles, and optimization. Using GeoGebra commands such as Mirror, Polygon, PointIn, etc., complex geometric shapes can be constructed. Suitable for high school mathematics education and learning.

GeoGebra Commands & Parameters

  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 Drawing Prompt Reference

Draw a rectangular prism with a square base. Place a point on one side face and another point on the opposite side face, representing a spider and a fly. Unfold the prism into a net, showing all possible paths from the spider to the fly, including polyline routes crossing different edges. In the net, rotate and mirror the spider and fly positions onto various faces, and draw multiple segments connecting them. Finally, highlight the shortest path and mark the intersection points where the path crosses edges.
The prompt above was compiled from the original drawing commands and is for reference only.