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牟合方蓋和球體的體積

牟合方蓋和球體的體積
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2016-05-27 16:47
立体几何几何作图
所属专题 Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes in Ancient Chinese Mathematics 中国古算中的体积 Volumes of Pyramid and Sphere in Ancient Chinese Mathematics Volume of Spheres in Ancient Chinese Mathematics
①这个范例演示了牟合方蓋与球體的几何关系和体积计算；②涉及的核心数学概念包括立体几何、体积计算、几何作图；③作图要点使用了Point, Sequence, CircleSector, Polygon, CircleArc, Segment, Prism, OrthogonalPlane, If, Intersect, Surface, CurveCartesian, Translate等GeoGebra命令和对象；④适合高中及以上学段，用于学习和探索立体几何体积问题。

GeoGebra 命令及参数

  1. Point(Segment[I, I + (0, 0, 1)]) → P
  2. Sequence(Polygon[((-sin(θ)), (-sin(θ)), 1 - cos(θ)), (sin(θ), (-sin(θ)), 1 - cos(θ)), (sin(θ), sin(θ), 1 - cos(θ)), ((-sin(θ)), sin(θ), 1 - cos(θ))], θ, 0, π, π / n) → list4
  3. CircleSector((0, 0, 0), (0, 0, 1), (-1, 0, 0)) → d
  4. CircleSector((0, 0, 0), (0, 1, 0), (0, 0, 1)) → g
  5. Polygon((0, 0), (-1, 0), (-1, 1), (0, 1)) → poly1, i, j, k, l
  6. CircleArc((0, 0, 0), (0, 1, 0), (0, 0, 1)) → h
  7. Segment((0, 0, 0), (0, 0, 1)) → m
  8. Prism(Polygon[A, B, C, D], F) → a, G, edgeBI, edgeCJ, edgeDK, edgeFI, edgeIJ, edgeJK, edgeFK, H, E, faceABIF, faceBCJI, faceCDKJ, faceADKF, faceFIJK, edgeAF
  9. Segment(B, F) → p
  10. Segment(B, E) → q
  11. Segment(B, H) → s
  12. Segment(I, I + (0, 0, 1)) → t
  13. OrthogonalPlane(P, t) → u
  14. Prism(Polygon[(0, 0, 0), (-1, 0, 0), (-1, 1, 0), (0, 1, 0)], -1) → v, M, edge1, edge2, edge3, edge4, edge5, edge6, edge7, edge8, N, O, Q, face1, face2, face3, face4, face5
  15. If(z(P) ≠ 1, (0, sqrt(1 - z(P)^(2)), z(P)), (0, 0, 1)) → K
  16. If(z(P) ≠ 1, ((-sqrt(1 - z(P)^(2))), 0, z(P)), (0, 0, 1)) → L
  17. Intersect(q, u) → U
  18. Intersect(p, u) → V
  19. Intersect(edgeBI, u) → W
  20. Intersect(s, u) → Z
  21. Surface((s * sqrt(1 - t^(2))), sqrt(1 - t^(2)), t, s, -1, 0, t, 0, 1) → c
  22. Surface((-sqrt(1 - t^(2))), (s * sqrt(1 - t^(2))), t, s, 0, 1, t, 0, 1) → b
  23. CircleArc((0, 0, 0), (0, 0, 1), (-1, 0, 0)) → e
  24. CurveCartesian((-sqrt(1 - t^(2))), sqrt(1 - t^(2)), t, t, 0, 1) → f
  25. Intersect(edge4, u) → R
  26. Intersect(edge3, u) → S
  27. If(z(P) ≠ 1, ((-sqrt(1 - z(P)^(2))), sqrt(1 - z(P)^(2)), z(P)), (0, 0, 1)) → J
  28. Intersect(edge2, u) → T
  29. Translate(Polygon[K, R, S, T, L, J], Vector[(0, 0)]) → poly2
  30. Segment((0, 0, 0), K) → f_1
  31. Segment(A_1, K) → g_1
  32. Segment((0, 0, 0), A_1) → h_1
  33. Segment(B, W) → j_1
  34. Translate(Polygon[W, Z, U, V], Vector[(0, 0)]) → poly3
  35. Segment(W, Z) → i_1
  36. Translate(Polygon[K, R, S, T, L, J], Vector[(0, 0)]) → poly4
  37. Translate(Polygon[W, Z, U, V], Vector[(0, 0)]) → poly5
  38. Segment(G, W) → k_1
  39. Segment(A_1, (0, 0, 1)) → l_1
  40. Segment(L, A_1) → m_1
  41. Segment(V, W) → n_1
  42. Segment(A, B) → p_1
  43. Segment(B, C) → q_1
  44. Segment(C, D) → r_1
  45. Segment(D, A) → s_1
  46. Segment((0, 0), (0, 1)) → t_1
  47. Segment((0, 0), (-1, 0)) → b_1
  48. Segment((-1, 1), (0, 1)) → c_1
  49. Segment((-1, 0), (-1, 1)) → d_1