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基于八面体的形状构造

基于八面体的形状构造
几何画板及绘图内容加载中，请稍等……
2017-01-21 18:33
多边形向量立体几何几何作图
所属专题 My 3D Collection
①本范例演示了如何基于八面体构造一个复杂的几何形状；②涉及的核心数学概念包括立体几何、多边形构造、向量运算和几何变换；③作图要点——使用了GeoGebra命令如Angle（角度）、Rotate（旋转）、Ray（射线）、Sequence（序列）、Point（点）、IndexOf（索引）、Element（元素）、Flatten（展平）、Intersect（相交）、AffineRatio（仿射比）、Midpoint（中点）、Polygon（多边形）、Vector（向量）；④适合高中及以上学生学习和探索立体几何构造方法。

GeoGebra 命令及参数

  1. Angle(A, O, B) → α
  2. Rotate(FF, α / 2, O_1) → F'
  3. Ray(O_1, F') → f
  4. Ray(O_1, FF) → g
  5. Ray(M, Vector[O_1, FF]) → h
  6. Ray(F', Vector[O_1, F']) → i
  7. Sequence(O_1 + (i * Vector[O_1, M]), i, 1.5, 5, 0.5) → B_1, {(0, 1.0606601717798214), (0, 1.4142135623730951), (0, 1.7677669529663689), (0, 2.121320343559643), (0, 2.4748737341529163), (0, 2.8284271247461903), (0, 3.1819805153394642), (0, 3.5355339059327378)}
  8. Point(g) → A_1
  9. Sequence(Distance[A_1, Element[B_1, i]], i, 1, Length[B_1]) → C_1, {1.414213562373095, 1.0606601717798212, 0.7071067811865475, 0.3535533905932735, 0, 0.35355339059327395, 0.7071067811865479, 1.0606601717798214}
  10. IndexOf(Min[C_1], C_1) → D_1
  11. Element(B_1, D_1) → E_1
  12. Sequence(O_1 + (i * Vector[O_1, F']), i, 1.5, 5, 0.5) → B_6, {(-1.0606601717798212, 1.0606601717798214), (-1.414213562373095, 1.4142135623730951), (-1.7677669529663687, 1.7677669529663689), (-2.1213203435596424, 2.121320343559643), (-2.474873734152916, 2.4748737341529163), (-2.82842712474619, 2.8284271247461903), (-3.181980515339464, 3.1819805153394642), (-3.5355339059327373, 3.5355339059327378)}
  13. Point(f) → A_6
  14. Sequence(Distance[A_6, Element[B_6, i]], i, 1, Length[B_6]) → C_6, {3.4999999999999996, 3, 2.5, 1.9999999999999998, 1.5000000000000002, 0.9999999999999999, 0.4999999999999996, 0}
  15. IndexOf(Min[C_6], C_6) → D_6
  16. Element(B_6, D_6) → E_6
  17. Flatten(Unique[Sequence[Sequence[O_1 + (ii * Vector[O_1, M]) + (jj * Vector[O_1, F']), ii, 1, 5, 0.5], jj, 0.5, 5, 0.5]]) → LL
  18. Sequence(Distance[A_2, Element[LL, i]], i, 1, Length[LL]) → C_2, {0.49999999999999994, 0.35355339059327373, 0.49999999999999994, 0.790569415042095, 1.1180339887498951, 1.4577379737113252, 1.8027756377319948, 2.150581316760657, 2.5000000000000004, 0, 0.35355339059327373, 0.7071067811865477, 1.0606601717798212, 1.4142135623730951, 1.7677669529663687, 2.121320343559643, 2.474873734152917, 2.8284271247461907, 0.49999999999999994, 0.790569415042095, 1.1180339887498951, 1.4577379737113252, 1.8027756377319948, 2.150581316760657, 2.5000000000000004, 2.8504385627478457, 3.2015621187164243, 1, 1.2747548783981963, 1.5811388300841898, 1.903943276465977, 2.23606797749979, 2.5739075352467506, 2.915475947422651, 3.259601202601325, 3.6055512754639896, 1.4999999999999998, 1.7677669529663689, 2.06155281280883, 2.3717082451262845, 2.6925824035672523, 3.0207614933986435, 3.3541019662496847, 3.691205765058351, 4.031128874149275, 2, 2.2638462845343543, 2.5495097567963922, 2.8504385627478452, 3.16227766016838, 3.48209706929603, 3.807886552931955, 4.138236339311712, 4.47213595499958, 2.4999999999999996, 2.7613402542968153, 3.0413812651491097, 3.3354160160315836, 3.640054944640259, 3.952847075210474, 4.272001872658765, 4.59619407771256, 4.924428900898053, 3, 3.2596012026013246, 3.535533905932738, 3.824264635194589, 4.1231056256176615, 4.430011286667337, 4.743416490252569, 5.062114182829148, 5.385164807134504, 3.5000000000000004, 3.7583240945932275, 4.0311288741492755, 4.315669125408018, 4.6097722286464435, 4.911720676097126, 5.220153254455275, 5.5339859052946645, 5.852349955359814, 4, 4.257346591481601, 4.527692569068709, 4.808846015417836, 5.0990195135927845, 5.396758286230726, 5.70087712549569, 6.0104076400856545, 6.324555320336759}
  19. Sequence(Distance[A_3, Element[LL, i]], i, 1, Length[LL]) → C_3, {1.5, 1.2747548783981961, 1.118033988749895, 1.0606601717798212, 1.118033988749895, 1.2747548783981961, 1.5, 1.767766952966369, 2.0615528128088303, 1, 0.7905694150420949, 0.7071067811865475, 0.7905694150420947, 0.9999999999999999, 1.274754878398196, 1.5811388300841895, 1.9039432764659772, 2.23606797749979, 0.5000000000000001, 0.35355339059327373, 0.5000000000000001, 0.7905694150420948, 1.118033988749895, 1.457737973711325, 1.8027756377319948, 2.150581316760657, 2.4999999999999996, 0, 0.35355339059327395, 0.7071067811865475, 1.060660171779821, 1.414213562373095, 1.7677669529663689, 2.121320343559643, 2.474873734152917, 2.82842712474619, 0.4999999999999998, 0.7905694150420948, 1.1180339887498945, 1.457737973711325, 1.8027756377319948, 2.150581316760657, 2.4999999999999996, 2.8504385627478457, 3.2015621187164243, 0.9999999999999999, 1.2747548783981961, 1.5811388300841895, 1.9039432764659772, 2.23606797749979, 2.5739075352467498, 2.915475947422651, 3.2596012026013246, 3.6055512754639896, 1.4999999999999996, 1.7677669529663687, 2.0615528128088303, 2.3717082451262845, 2.6925824035672514, 3.0207614933986426, 3.3541019662496847, 3.6912057650583505, 4.031128874149275, 1.9999999999999998, 2.2638462845343543, 2.5495097567963922, 2.850438562747845, 3.16227766016838, 3.48209706929603, 3.8078865529319543, 4.138236339311712, 4.472135954999579, 2.5, 2.7613402542968157, 3.04138126514911, 3.335416016031584, 3.640054944640259, 3.9528470752104745, 4.272001872658766, 4.59619407771256, 4.924428900898053, 3, 3.2596012026013237, 3.5355339059327373, 3.824264635194589, 4.123105625617661, 4.430011286667337, 4.743416490252569, 5.062114182829148, 5.385164807134504}
  20. Sequence(Distance[A_4, Element[LL, i]], i, 1, Length[LL]) → C_4, {2.1505813167606567, 1.8027756377319946, 1.457737973711325, 1.1180339887498945, 0.7905694150420943, 0.4999999999999998, 0.35355339059327373, 0.5000000000000004, 0.7905694150420952, 1.7677669529663687, 1.414213562373095, 1.060660171779821, 0.7071067811865475, 0.3535533905932735, 0, 0.35355339059327395, 0.7071067811865479, 1.0606601717798219, 1.457737973711325, 1.1180339887498945, 0.7905694150420943, 0.4999999999999998, 0.35355339059327373, 0.5000000000000001, 0.7905694150420952, 1.1180339887498953, 1.457737973711325, 1.274754878398196, 0.9999999999999996, 0.7905694150420947, 0.7071067811865475, 0.7905694150420949, 1.0000000000000002, 1.2747548783981966, 1.5811388300841904, 1.9039432764659772, 1.2747548783981961, 1.1180339887498947, 1.0606601717798212, 1.118033988749895, 1.2747548783981963, 1.5000000000000002, 1.7677669529663687, 2.061552812808831, 2.3717082451262845, 1.457737973711325, 1.414213562373095, 1.457737973711325, 1.5811388300841898, 1.767766952966369, 1.9999999999999998, 2.2638462845343548, 2.5495097567963922, 2.8504385627478452, 1.7677669529663684, 1.8027756377319943, 1.903943276465977, 2.0615528128088303, 2.263846284534354, 2.5, 2.7613402542968153, 3.0413812651491097, 3.3354160160315836, 2.1505813167606567, 2.23606797749979, 2.3717082451262845, 2.5495097567963922, 2.7613402542968157, 3, 3.2596012026013246, 3.535533905932738, 3.824264635194589, 2.57390753524675, 2.6925824035672523, 2.8504385627478452, 3.0413812651491106, 3.2596012026013246, 3.5000000000000004, 3.7583240945932275, 4.0311288741492755, 4.315669125408018, 3.020761493398643, 3.162277660168379, 3.335416016031583, 3.5355339059327378, 3.758324094593227, 4, 4.257346591481601, 4.5276925690687095, 4.808846015417836}
  21. Sequence(Distance[A_5, Element[LL, i]], i, 1, Length[LL]) → C_5, {2.5, 2.2638462845343543, 2.0615528128088303, 1.903943276465977, 1.8027756377319946, 1.7677669529663687, 1.8027756377319946, 1.9039432764659772, 2.0615528128088303, 2, 1.7677669529663689, 1.5811388300841895, 1.457737973711325, 1.414213562373095, 1.457737973711325, 1.5811388300841895, 1.7677669529663689, 2, 1.5, 1.2747548783981961, 1.1180339887498947, 1.0606601717798212, 1.118033988749895, 1.2747548783981961, 1.5, 1.767766952966369, 2.06155281280883, 0.9999999999999999, 0.7905694150420947, 0.7071067811865475, 0.7905694150420947, 0.9999999999999999, 1.2747548783981961, 1.58113883008419, 1.9039432764659776, 2.2360679774997894, 0.5000000000000001, 0.35355339059327373, 0.4999999999999998, 0.7905694150420948, 1.118033988749895, 1.4577379737113254, 1.8027756377319943, 2.150581316760657, 2.5, 0, 0.35355339059327395, 0.7071067811865475, 1.0606601717798214, 1.4142135623730954, 1.7677669529663684, 2.1213203435596433, 2.4748737341529163, 2.8284271247461903, 0.4999999999999996, 0.7905694150420947, 1.118033988749895, 1.4577379737113254, 1.8027756377319943, 2.1505813167606567, 2.5, 2.8504385627478452, 3.2015621187164247, 0.9999999999999999, 1.2747548783981961, 1.58113883008419, 1.9039432764659767, 2.2360679774997902, 2.57390753524675, 2.9154759474226504, 3.2596012026013246, 3.605551275463989, 1.5000000000000002, 1.767766952966369, 2.0615528128088307, 2.371708245126285, 2.692582403567252, 3.0207614933986435, 3.3541019662496847, 3.6912057650583505, 4.0311288741492755, 2, 2.263846284534354, 2.5495097567963922, 2.850438562747845, 3.1622776601683795, 3.4820970692960302, 3.807886552931954, 4.138236339311713, 4.47213595499958}
  22. IndexOf(Min[C_2], C_2) → D_2
  23. IndexOf(Min[C_3], C_3) → D_3
  24. IndexOf(Min[C_4], C_4) → D_4
  25. IndexOf(Min[C_5], C_5) → D_5
  26. Element(LL, D_2) → E_2
  27. Element(LL, D_3) → E_3
  28. Element(LL, D_4) → E_4
  29. Element(LL, D_5) → E_5
  30. Intersect(g, Line[A_2, f]) → G_2
  31. Intersect(g, Line[A_3, f]) → G_3
  32. Intersect(g, Line[A_4, f]) → G_4
  33. Intersect(g, Line[A_5, f]) → G_5
  34. Intersect(f, Line[A_2, g]) → H_2
  35. Intersect(f, Line[A_3, g]) → H_3
  36. Intersect(f, Line[A_4, g]) → H_4
  37. Intersect(f, Line[A_5, g]) → H_5
  38. AffineRatio(O_1, M, A_1) → I_1
  39. AffineRatio(O_1, M, G_2) → I_2
  40. AffineRatio(O_1, M, G_3) → I_3
  41. AffineRatio(O_1, M, G_4) → I_4
  42. AffineRatio(O_1, M, G_5) → I_5
  43. AffineRatio(O_1, F', A_6) → J_6
  44. AffineRatio(O_1, F', H_5) → J_5
  45. AffineRatio(O_1, F', H_4) → J_4
  46. AffineRatio(O_1, F', H_3) → J_3
  47. AffineRatio(O_1, F', H_2) → J_2
  48. Midpoint(A1, B1) → C1
  49. Polygon(F', M, A_1, A_2, A_3, A_4, A_5, A_6) → wielokąt1, f', m, a_1, a_2, a_3, a_4, a_5, a_6
  50. Vector(O, A1) → D1
  51. Vector(O, B1) → E1
  52. Vector(O, C1) → F1
  53. Polygon(B1, A1, O1, P1, Q1, G1, H1, I1, J1, K1, L1, M1, N1) → R1, S1, AB1, AC1, AD1, AE1, T1, U1, V1, W1, X1, Y1, Z1, AA1
  54. Midpoint(A2, B2) → C2
  55. Midpoint(A3, B3) → C3
  56. Midpoint(A4, B4) → C4
  57. Midpoint(A5, B5) → C5
  58. Midpoint(A6, B6) → C6
  59. Midpoint(A7, B7) → C7
  60. Midpoint(A8, B8) → C8
  61. Midpoint(A9, B9) → C9
  62. Midpoint(A10, B10) → C10
  63. Midpoint(A11, B11) → C11
  64. Midpoint(A12, B12) → C12
  65. Vector(O, A2) → D2
  66. Vector(O, B2) → E2
  67. Vector(O, C2) → F2
  68. Vector(O, A3) → D3
  69. Vector(O, B3) → E3
  70. Vector(O, C3) → F3
  71. Vector(O, A4) → D4
  72. Vector(O, B4) → E4
  73. Vector(O, C4) → F4
  74. Vector(O, A5) → D5
  75. Vector(O, B5) → E5
  76. Vector(O, C5) → F5
  77. Vector(O, A6) → D6
  78. Vector(O, B6) → E6
  79. Vector(O, C6) → F6
  80. Vector(O, A7) → D7
  81. Vector(O, B7) → E7
  82. Vector(O, C7) → F7
  83. Vector(O, A8) → D8
  84. Vector(O, B8) → E8
  85. Vector(O, C8) → F8
  86. Vector(O, A9) → D9
  87. Vector(O, B9) → E9
  88. Vector(O, C9) → F9
  89. Vector(O, A10) → D10
  90. Vector(O, B10) → E10
  91. Vector(O, C10) → F10
  92. Vector(O, A11) → D11
  93. Vector(O, B11) → E11
  94. Vector(O, C11) → F11
  95. Vector(O, A12) → D12
  96. Vector(O, B12) → E12
  97. Vector(O, C12) → F12
  98. Polygon(B2, A2, O2, P2, Q2, G2, H2, I2, J2, K2, L2, M2, N2) → R2, S2, AB2, AC2, AD2, AE2, T2, U2, V2, W2, X2, Y2, Z2, AA2
  99. Polygon(B3, A3, O3, P3, Q3, G3, H3, I3, J3, K3, L3, M3, N3) → R3, S3, AB3, AC3, AD3, AE3, T3, U3, V3, W3, X3, Y3, Z3, AA3
  100. Polygon(B4, A4, O4, P4, Q4, G4, H4, I4, J4, K4, L4, M4, N4) → R4, S4, AB4, AC4, AD4, AE4, T4, U4, V4, W4, X4, Y4, Z4, AA4
  101. Polygon(B5, A5, O5, P5, Q5, G5, H5, I5, J5, K5, L5, M5, N5) → R5, S5, AB5, AC5, AD5, AE5, T5, U5, V5, W5, X5, Y5, Z5, AA5
  102. Polygon(B6, A6, O6, P6, Q6, G6, H6, I6, J6, K6, L6, M6, N6) → R6, S6, AB6, AC6, AD6, AE6, T6, U6, V6, W6, X6, Y6, Z6, AA6
  103. Polygon(B7, A7, O7, P7, Q7, G7, H7, I7, J7, K7, L7, M7, N7) → R7, S7, AB7, AC7, AD7, AE7, T7, U7, V7, W7, X7, Y7, Z7, AA7
  104. Polygon(B8, A8, O8, P8, Q8, G8, H8, I8, J8, K8, L8, M8, N8) → R8, S8, AB8, AC8, AD8, AE8, T8, U8, V8, W8, X8, Y8, Z8, AA8
  105. Polygon(B9, A9, O9, P9, Q9, G9, H9, I9, J9, K9, L9, M9, N9) → R9, S9, AB9, AC9, AD9, AE9, T9, U9, V9, W9, X9, Y9, Z9, AA9
  106. Polygon(B10, A10, O10, P10, Q10, G10, H10, I10, J10, K10, L10, M10, N10) → R10, S10, AB10, AC10, AD10, AE10, T10, U10, V10, W10, X10, Y10, Z10, AA10
  107. Polygon(B11, A11, O11, P11, Q11, G11, H11, I11, J11, K11, L11, M11, N11) → R11, S11, AB11, AC11, AD11, AE11, T11, U11, V11, W11, X11, Y11, Z11, AA11
  108. Polygon(B12, A12, O12, P12, Q12, G12, H12, I12, J12, K12, L12, M12, N12) → R12, S12, AB12, AC12, AD12, AE12, T12, U12, V12, W12, X12, Y12, Z12, AA12

AI 绘图提示语参考

画一个基于八面体的复杂几何形状，包含多个点、线、射线和面。首先确定一个中心点和一个八面体的基本结构，然后通过旋转和向量操作生成多个点集。接着，利用这些点集和几何变换构建形状的边界，并计算关键点之间的距离和比例关系。最后，将这些元素组合起来，形成一个具有八面体特征的复杂几何体。
以上提示语为根据原始绘图命令整理而来，仅供参考。