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圆锥曲面面积演示 by Anthony OR 柯志明

圆锥曲面面积演示 by Anthony OR 柯志明
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2014-12-05 15:25
圆三角函数立体几何几何作图
所属专题 Area, Surface Area, Volume, Cross Section, 3D 3D Geometry & Cross Sections Volumes and Surface Areas of Pyramids, Cones and Spheres GCSE Higher Ch7 Area and Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume Geometry: Surface Area & Volume GA Chapter 11: Surface Area and Volume Volumes and Surface Areas of Pyramids, Cones and Spheres More about Volumes and Surface Areas Connolly Geometry Volume Volumes and Surface Areas of Pyramids, Cones and Spheres Volumes and Surface Areas of Pyramids, Cones and Spheres
①本范例演示如何将圆锥的曲面展开成一个扇形，并进一步分解重组以验证其面积为πrl；②涉及核心数学概念包括曲面面积、圆锥几何、勾股定理、三角函数；③作图要点运用了Point, If, Sequence, CircleArc, Segment, Midpoint, Circle, Intersect, Rotate, Surface, CircumcircleArc, Vector, Angle等GeoGebra命令与对象；④适合高中数学教学及几何深入学习。

GeoGebra 命令及参数

  1. Point(CircleArc[(0, 1), (1, 1), (0, 0)]) → Q
  2. If(y(Q) > 0, CircleArc[(0, 1), (0, 0), Q], 6.283185307179586) → s
  3. If(y(Q) > 0, Angle[(0, 0), (0, 1), Q], 6.283185307179586) → α'
  4. Sequence((sin((i * θ)), 1 - cos((i * θ))), i, 0, (2 * n)) → list1
  5. If(t ≤ 1, t, 1) → k
  6. If(t ≥ 1, t - 1, 0) → k'
  7. Sequence(If[k < 1, (x(O_1) + ((l * sin((i * φ))) * cos((i * φ)) / sin(φ)), y(O_1) + ((l * sin((i * φ))) * sin((i * φ)) / sin(φ))), (x(O_1) + (l * i), y(O_1))], i, 0, (2 * n)) → list2
  8. Sequence(Sector[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]]], i, 1, n) → list3
  9. CircleArc((0, 1 / (1 - m)), O_1, L) → b
  10. Segment((0, 0), (s, 0)) → d
  11. Midpoint(d) → B'
  12. CircleArc((0, 1), Q, (0, 0)) → c
  13. Segment(O, (x(O), y(O) - 1)) → e
  14. Midpoint(e) → G
  15. Circle((0, 1), 1) → g
  16. Midpoint(Q, O) → H
  17. Segment(O, A) → a
  18. Circle(P, 1, yAxis) → p
  19. Intersect(xOyPlane, p) → D, C
  20. Rotate(O_1, α, a) → F
  21. Surface((u * cos(v)), (u * sin(v)) + 1, f(u), u, 0, abs(x(D)), v, -1.5708, (3 * π / 2) - (2 * γ)) → o
  22. Rotate(D_1, α, a) → D'
  23. CircumcircleArc(D_1, C, D') → q
  24. Circle(O, D_1, a) → c_1
  25. Segment(P, D') → i
  26. Segment(P, D_1) → a_1
  27. Point(b) → J
  28. Segment(O, D_1) → b_1
  29. Vector(A + (0, 0, 0.3), A + (0, 0, 0.06)) → u
  30. Angle(D_1, O, A) → β
  31. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O_1) + 0, y(O_1) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)], Rotate[Rotate[Translate[Element[list1, i + 1], Vector[Element[list1, i], Element[list2, i]]], (((-((2 * i) - 1)) * k) * θ / 2), Element[list2, i]], (180° * k'), (x(O_1) + (n * l) - sin(θ / 2) / 2, y(O_1) + cos(θ / 2) / 2)]], i, n + 1, (2 * n)) → list4

AI 绘图提示语参考

绘制一个圆锥的侧面展开图，首先画一个圆心在(0,1)的圆，然后在圆上取一点Q，通过Q画一段圆弧s，接着画出角α'。然后构造一个包含n个点的序列list1，再根据k和k'的值构造另一个序列list2。利用list1和list2，通过旋转和平移，构造一系列扇形，最终形成圆锥侧面展开的图形。
以上提示语为根据原始绘图命令整理而来，仅供参考。