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圆锥曲面面积计算 by Anthony OR 柯志明

圆锥曲面面积计算 by Anthony OR 柯志明
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2014-09-28 12:21
圆立体几何几何作图
所属专题 Surface Area and Volume Surface Area and Volume Surface Area and Volume
①本范例演示圆锥曲面面积的计算方法；②涉及核心数学概念包括圆锥几何性质、圆周率π、曲面面积公式；③作图要点——使用Point, If, Sequence, Segment, Min, Max, CircleArc, Midpoint, CircleSector, Circle等GeoGebra命令/对象构建圆锥模型并计算曲面面积；④适合高中数学教学和几何学习，帮助学生理解曲面面积计算原理。

GeoGebra 命令及参数

  1. Point(CircleArc[(0, 1), (1, 1), (0, 0)]) → A
  2. If(y(A) > 0, Angle[(0, 0), (0, 1), A], 6.283185307179586) → α
  3. Sequence((sin(i * θ), 1 - cos(i * θ)), i, 0, 2 * n) → list1
  4. Segment(E, (x(E) + 1, y(E))) → a
  5. Point(a) → F
  6. Min(2 * (x(F) - x(E)), 1) → k
  7. Max(2 * (x(F) - x(E)) - 1, 0) → k'
  8. Sequence(If[k < 1, (x(O) + l * sin(i * φ) * cos(i * φ) / sin(φ), y(O) + l * sin(i * φ) * sin(i * φ) / sin(φ)), (x(O) + l * i, y(O))], i, 0, 2 * n) → list2
  9. Sequence(Sector[Rotate[Translate[Circle[(x(O) + 0, y(O) + 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
  10. Sequence(Sector[Rotate[Rotate[Translate[Circle[(x(O) + 0, y(O) + 1), Element[list1, i]], Vector[Element[list1, i], Element[list2, i]]], -(2 * i - 1) * k * θ / 2, Element[list2, i]], 180° * k', (x(O) + n * l - sin(θ / 2) / 2, y(O) + 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) + n * l - sin(θ / 2) / 2, y(O) + 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) + n * l - sin(θ / 2) / 2, y(O) + cos(θ / 2) / 2)]], i, n + 1, 2 * n) → list4
  11. If(y(A) > 0, CircleArc[(0, 1), (0, 0), A], 6.283185307179586) → s
  12. CircleArc((0, 1 / (1 - m)), O, L) → b
  13. Segment((0, 0), (s, 0)) → d
  14. Midpoint(d) → B'
  15. CircleArc((0, 1), A, (0, 0)) → c
  16. Midpoint(a) → C
  17. Point(Segment[(0, 1), (s, 1)]) → D
  18. Segment(D, (x(D), y(D) - 1)) → e
  19. Midpoint(e) → G
  20. CircleSector((0, 1), (0, 0), A) → f
  21. Circle((0, 1), 1) → g
  22. Midpoint(A, D) → H