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球面二次曲线构造

球面二次曲线构造
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2017-01-27 23:38
三角函数立体几何复数几何作图函数图像
①本范例演示球面二次曲线的构造与性质；②涉及复数、自然数、台形、三角法、放物线等核心数学概念；③通过GeoGebra命令Intersect（交集）、Sphere（球体）、Segment（线段）、Point（点）、Rotate（旋转）、RemoveUndefined（移除未定义对象）、PolyLine（多段线）、CircleArc（圆弧）、If（条件判断）等实现；④适合高中及以上数学学习者和教师用于教学演示与探索。

GeoGebra 命令及参数

  1. Intersect(xAxis, zAxis) → A
  2. Sphere(A, 1) → a
  3. Segment((1.5, -1.5, 0), (1.5, -0.5, 0)) → f
  4. Segment((1.5, 1.5, 0), (1.5, 0.5, 0)) → g
  5. Point(f) → B
  6. Point(g) → C
  7. Rotate((1, 0), θ) → D
  8. Rotate((1, 0), (-θ)) → E
  9. RemoveUndefined(If[(2 * θ) + ϕ < (2 * π), Sequence[If[r > 0, Intersect[Circle[(cos(abs(r)) * D), abs(sin(abs(r))), Vector[D]], Circle[(cos(ϕ - abs(r)) * E), abs(sin(ϕ - abs(r))), Vector[E]], 1], Intersect[Circle[(cos(abs(r)) * D), abs(sin(abs(r))), Vector[D]], Circle[(cos(ϕ - abs(r)) * E), abs(sin(ϕ - abs(r))), Vector[E]], 2]], r, (-ϕ), ϕ, π / 360], Sequence[If[r > 0, Intersect[Circle[(cos(abs(r)) * D), abs(sin(abs(r))), Vector[D]], Circle[(cos(ϕ - abs(r)) * E), abs(sin(ϕ - abs(r))), Vector[E]], 1], Intersect[Circle[(cos(abs(r)) * D), abs(sin(abs(r))), Vector[D]], Circle[(cos(ϕ - abs(r)) * E), abs(sin(ϕ - abs(r))), Vector[E]], 2]], r, ((-ϕ)) / 2, ϕ / 2, π / 180]]) → list1
  10. PolyLine(list1) → h
  11. Point(h) → F
  12. CircleArc(A, D, F) → c
  13. If((2 * θ) + ϕ < (2 * π), CircleArc[A, E, F], CircumcircleArc[E, (-E), F]) → d
  14. RemoveUndefined(Sequence[If[r > 0, Intersect[Circle[(cos(ϕ - abs(r)) * D), abs(sin(ϕ - abs(r))), Vector[D]], Circle[(cos(abs(r)) * E), abs(sin(abs(r))), Vector[E]], 1], Intersect[Circle[(cos(ϕ - abs(r)) * D), abs(sin(ϕ - abs(r))), Vector[D]], Circle[(cos(abs(r)) * E), abs(sin(abs(r))), Vector[E]], 2]], r, ((-ϕ)) / 2, ϕ / 2, π / 180]) → list2
  15. PolyLine(list2) → i