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Stereographic Projection: Circles to Circles

Stereographic Projection: Circles to Circles
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2019-01-29 18:19
TrigonometryCoordinate GeometrySolid GeometryTransformationGeometric Construction
This example demonstrates how stereographic projection maps circles on a sphere to circles on a plane. The core mathematical concepts involved include spherical geometry, projection transformations, and analytical geometry. The construction utilizes GeoGebra commands such as Sphere, OrthogonalPlane, PointIn, Segment, Point, IntersectPath, Ray, Intersect, OrthogonalLine, Polygon, Locus, and Angle, along with objects like button1, center, radius, sph, P, topPlane, Psph, s, AuxP, TangPAux, TangPsph, circleSph, Pcirc, Proj, TangPcirc, ProjXY, AuxPerp, segaux, PAux, t1, segaux2, PAux2, t2, T1, pcirc, p, paux, T2, f, g, h, Prj, angleT1, angleT2, a, b, angleT3, angleT4, c. It is suitable for high school and above students studying spherical projection and analytical geometry.

GeoGebra Commands & Parameters

  1. Sphere((0, 0, center), radius) → sph
  2. OrthogonalPlane(P, zAxis) → topPlane
  3. PointIn(sph) → Psph
  4. Segment((0, 0, center), Psph) → s
  5. Point(s) → AuxP
  6. OrthogonalPlane(AuxP, s) → TangPAux
  7. OrthogonalPlane(Psph, s) → TangPsph
  8. IntersectPath(TangPAux, sph) → circleSph
  9. Point(circleSph) → Pcirc
  10. Ray(P, Pcirc) → Proj
  11. OrthogonalPlane(Pcirc, Segment[(0, 0, center), Pcirc]) → TangPcirc
  12. Intersect(Proj, xOyPlane) → ProjXY
  13. OrthogonalLine(ProjXY, xOyPlane) → AuxPerp
  14. Ray(P, Intersect[AuxPerp, topPlane]) → segaux
  15. Intersect(segaux, TangPcirc) → PAux
  16. Segment(PAux, Pcirc) → t1
  17. Ray((0, 0, 0), ProjXY) → segaux2
  18. Intersect(segaux2, TangPcirc) → PAux2
  19. Segment(PAux2, Pcirc) → t2
  20. Polygon(P, PAux, Pcirc) → T1, pcirc, p, paux
  21. Polygon(ProjXY, PAux2, Pcirc) → T2, f, g, h
  22. Locus(ProjXY, Pcirc) → Prj
  23. Angle(Pcirc, PAux, P) → angleT1
  24. Angle(Pcirc, PAux2, ProjXY) → angleT2
  25. Angle(Pcirc, P, PAux) → angleT3
  26. Angle(PAux2, ProjXY, Pcirc) → angleT4