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祖暅原理演示

祖暅原理演示
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2016-04-11 15:19
立体几何微积分积分几何作图
所属专题 Area, Surface Area, Volume, Cross Section, 3D Primary School Mathematics Collection Volumes and Surface Areas of Pyramids, Cones and Spheres Mensuration Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes in Ancient Chinese Mathematics 中国古算中的体积 More about Volumes and Surface Areas Volumes of Pyramid and Sphere in Ancient Chinese Mathematics Connolly Geometry Volume
①本范例通过动态演示两个几何体截面积相等来展示祖暅原理；②涉及核心数学概念包括体积、截面积、积分、立体几何；③作图要点使用了Polygon（多边形）、Prism（棱柱）、Height（高度）、CurveCartesian（笛卡尔曲线）、Surface（曲面）、Segment（线段）、Point（点）、Sequence（数列）等GeoGebra命令和对象；④适合高中及以上学生及数学爱好者学习立体几何与积分概念。

GeoGebra 命令及参数

  1. Polygon(A, B, C) → poly1, c, a_1, b
  2. Prism(poly1, 3.245704115496874) → d, D, edgeCF, edgeDE, edgeEF, edgeDF, E, F, faceABED, faceBCFE, faceACFD, faceDEF, edgeAD, edgeBE
  3. Height(d) → h
  4. If(t < 0.5, 0, (2 * (t - 0.5))) → t2
  5. If(t < 0.5, (2 * t), 1) → t1
  6. Polygon(C', A', B') → poly2, b', c', a'
  7. CurveCartesian(f(u) + x1, y1, u, u, 0, h) → e
  8. CurveCartesian(f(u) + x2, y2, u, u, 0, h) → g
  9. CurveCartesian(f(u) + x3, y3, u, u, 0, h) → i
  10. Surface((u * (f(v) + x1)) + ((1 - u) * (f(v) + x2)), (u * y1) + ((1 - u) * y2), v, u, 0, 1, v, 0, h) → j
  11. Surface((u * (f(v) + x1)) + ((1 - u) * (f(v) + x3)), (u * y1) + ((1 - u) * y3), v, u, 0, 1, v, 0, h) → a
  12. Surface((u * (f(v) + x2)) + ((1 - u) * (f(v) + x3)), (u * y2) + ((1 - u) * y3), v, u, 0, 1, v, 0, h) → k
  13. Segment(G, G + (0, 0, h)) → l
  14. Point(l) → H
  15. Polygon(I, J, K) → poly3, k_1, i_1, j_1
  16. Polygon(L, M, N) → poly4, n, l_1, m
  17. Polygon(O, P, Q) → poly5, q, o, p
  18. Polygon((-7, -7, h'), (7, -7, h'), (7, 7, h'), (-7, 7, h')) → poly6, r, s, b_1, c_1
  19. Segment(A, B) → d_1
  20. Segment(B, C) → e_1
  21. Segment(C, A) → f_1
  22. Sequence(Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ))], δ], i, 0, w - 1) → list2
  23. Sequence(Prism[Polygon[(f((i * δ)) + x1, y1, (i * δ)), (f((i * δ)) + x2, y2, (i * δ)), (f((i * δ)) + x3, y3, (i * δ))], δ], i, 0, w - 1) → list2'
  24. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, w - 1) → list1
  25. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, w - 1) → list1'