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卡瓦列里原理演示：棱柱体积相等性

卡瓦列里原理演示：棱柱体积相等性
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2016-04-10 07:27
多边形立体几何变换几何作图
所属专题 Area, Surface Area, Volume, Cross Section, 3D
① 该范例演示卡瓦列里原理，通过两个高度相同、横截面积相等的棱柱展示体积相等的现象。② 涉及核心数学概念：卡瓦列里原理、立体几何体积计算、横截面积比较、棱柱与多面体的性质。③ 作图要点：使用 Polygon 创建底面多边形，Prism 生成棱柱，CurveCartesian 与 Surface 绘制截面曲线与曲面，Point、OrthogonalPlane、Intersect、PointIn 等命令构建顶点、正交平面、交线与内部点，形成完整的三维几何模型。④ 适合高中立体几何教学、大学微积分体积积分引入，也可作为几何作图与动画演示的素材。

GeoGebra 命令及参数

  1. Polygon((1, 0, -2), (2, -3, -2), (3, 2, -2)) → poly1, a, b, c
  2. Prism(poly1, 6) → d, A, edge3, edgeAB, edgeBC, edgeAC, B, C, face1, face3, face2, faceABC, edge1, edge2
  3. Polygon((-4, 0, -2), (-3, -3, -2), (-2, 2, -2)) → poly2, e, f, g
  4. CurveCartesian(sin((π / 3 * (t + 2))) - 4, 0, t, t, -2, 4) → i
  5. CurveCartesian(sin((π / 3 * (t + 2))) - 3, -3, t, t, -2, 4) → j
  6. Polygon((-4, 0, 4), (-3, -3, 4), (-2, 2, 4)) → poly3, h, l, m
  7. CurveCartesian(sin((π / 3 * (t + 2))) - 2, 2, t, t, -2, 4) → k
  8. Surface((u * (sin((π / 3 * (t + 2))) - 4)) + ((1 - u) * (sin((π / 3 * (t + 2))) - 3)), (u * 0) + ((1 - u) * (-3)), t, t, -2, 4, u, 0, 1) → n
  9. Surface((u * (sin((π / 3 * (t + 2))) - 4)) + ((1 - u) * (sin((π / 3 * (t + 2))) - 2)), (u * 0) + ((1 - u) * 2), t, t, -2, 4, u, 0, 1) → o
  10. Surface((u * (sin((π / 3 * (t + 2))) - 2)) + ((1 - u) * (sin((π / 3 * (t + 2))) - 3)), (u * 2) + ((1 - u) * (-3)), t, t, -2, 4, u, 0, 1) → p
  11. CurveCartesian(-6, 0, t, t, -10, 10) → q
  12. Point(q) → E
  13. OrthogonalPlane(E, zAxis) → r
  14. Intersect(edge2, r) → F
  15. Intersect(edge3, r) → G
  16. Intersect(edge1, r) → H
  17. Polygon(H, G, F) → poly4, f_1, h_1, g_1
  18. CurveCartesian((u * (sin((π / 3 * (z(E) + 2))) - 4)) + ((1 - u) * (sin((π / 3 * (z(E) + 2))) - 3)), (u * 0) + ((1 - u) * (-3)), z(E), u, 0, 1) → s
  19. Polygon(I, J, K) → poly5, k_1, i_1, j_1
  20. PointIn(poly4) → Pointpoly4
  21. PointIn(poly5) → Pointpoly5