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Spherical Quadratic Curve Construction

Spherical Quadratic Curve Construction
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2017-01-27 23:38
TrigonometrySolid GeometryComplex NumberGeometric ConstructionGraphing
In collections surface 3D /3D-Oberflächen 1
This example demonstrates the construction and properties of spherical quadratic curves. It involves core mathematical concepts such as complex numbers, natural numbers, trapezoids, trigonometry, and parabolas. GeoGebra commands like Intersect, Sphere, Segment, Point, Rotate, RemoveUndefined, PolyLine, CircleArc, and If are used to create the visualization. Suitable for high school mathematics students and teachers for educational purposes.

GeoGebra Commands & Parameters

  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