积极 AI Logo 积极 AI

二十面体球体构造

二十面体球体构造
几何画板及绘图内容加载中，请稍等……
2017-11-11 05:29
三角函数立体几何微积分导数几何作图
①本范例演示了如何通过 GeoGebra 构造一个二十面体球体及其相关几何元素；②涉及的核心数学概念包括立体几何、导数、中值定理、平行四边形和微分方程；③作图要点提炼自构造摘要，使用了点、二十面体、中点、球体、正交线、交点、角度、平面、序列、旋转、镜像等 GeoGebra 命令和对象；④适合高中及以上数学学习者，用于理解复杂几何体的构造和数学应用。

GeoGebra 命令及参数

  1. Point(Circle[Midpoint[A, B], (Distance[A, B] * sqrt(3) / 2), Segment[A, B]]) → C
  2. Icosahedron(A, B, C) → a, D, faceABC, faceACF, faceAFG, faceADG, faceABD, faceBDH, faceBEH, faceBCE, faceCEI, faceCFI, E, faceEHK, faceEIK, faceIKL, faceFIL, faceHJK, faceJKL, faceGJL, faceFGL, faceDHJ, faceDGJ, F, edgeAC, edgeAB, edgeBC, edgeAF, edgeCF, edgeAG, edgeFG, edgeAD, edgeDG, edgeBD, G, edgeBH, edgeDH, edgeBE, edgeEH, edgeCE, edgeCI, edgeEI, edgeFI, edgeEK, edgeHK, H, edgeIK, edgeIL, edgeKL, edgeFL, edgeJK, edgeHJ, edgeJL, edgeGL, edgeGJ, edgeDJ, I, J, K, L
  3. Midpoint(D, I) → M
  4. Sphere(M, A) → b
  5. OrthogonalLine(M, xOyPlane) → f
  6. Intersect(f, b) → N, O
  7. Angle(J, M, O) → α
  8. Plane(J, f) → c
  9. OrthogonalLine(M, c) → g
  10. Sequence(Rotate[Rotate[J, (-a), g], (6 * a), f], a, 0, α, α / 36) → L_1
  11. Rotate(L_1, (2 * π / 3), f) → L_2
  12. Sequence(CircleArc[M, Element[L_1, i], Element[L_2, i]], i, 1, Length[L_1]) → L_3
  13. Rotate(L_3, (2 * π / 3), f) → L_4
  14. Rotate(L_3, (-2 * π / 3), f) → L_5
  15. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, J]], a, 1, 4) → L_6
  16. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, K]], a, 2, 4) → L_7
  17. Mirror(L_6, M) → L_8
  18. Sequence(Rotate[{L_3, L_4, L_5}, ((a * 2) * π / 5), Line[M, L]], a, 2, 3) → L_9
  19. Mirror(L_9, M) → L_{10}
  20. Rotate({L_3, L_4, L_5}, (4 * π / 5), Line[M, J]) → L_{11}
  21. Rotate(L_{11}, (4 * π / 5), Line[M, D]) → L_{12}
  22. Rotate(L_{11}, (2 * π / 5), Line[M, D]) → L_{13}
  23. Rotate(L_{11}, (6 * π / 5), Line[M, G]) → L_{14}
  24. Rotate(L_{12}, (-2 * π / 5), Line[M, A]) → L_{15}