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卡瓦利里原理 - 体积

卡瓦利里原理 - 体积
几何画板及绘图内容加载中，请稍等……
2019-01-13 00:43
解析几何立体几何几何作图
这个范例演示了卡瓦利里原理，即两个立体图形如果在每个平行截面上的面积相等，则这两个立体图形的体积相等。核心概念包括立体几何和面积相等。通过使用多边形、棱柱、高度等 GeoGebra 命令，可以直观地展示这一原理。适合初中及以上学生学习立体几何。

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'
  26. Vector(R0, B) → v
  27. Length(list2) → ll
  28. Sequence(Vector[(f((i * δ)) + x1, y1, (i * δ)), A + (0, 0, (i * δ))], i, 0, w - 1) → list3
  29. Translate(Element[list2, floor(u)], Vector[((u - floor(u)) * Element[list3, floor(u)])]) → h_1
  30. Sequence(Prism[Polygon[A + (0, 0, (i * δ)), B + (0, 0, (i * δ)), C + (0, 0, (i * δ))], δ], i, 0, floor(u) - 2) → list6
  31. 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, floor(u), w - 1) → list5