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Mapping a Triangle from a Plane to a Sphere

Mapping a Triangle from a Plane to a Sphere
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2022-12-24 17:31
VectorSolid GeometryTransformation
This example demonstrates how to map a triangle from a plane onto a sphere through specific transformations. Key math concepts include spherical geometry, transformations, vectors, and rotations. GeoGebra commands used in construction are Intersect, Vector, OrthogonalPlane, Point, Sphere, Circle, Plane, Line, Rotate, Polygon, PointIn, Segment, CircleArc, and CircleSector. Suitable for high school and above students to learn spatial geometry and transformations.

GeoGebra Commands & Parameters

  1. Intersect(xAxis, zAxis) → B_{1}
  2. Vector(B_{1}, P) → u
  3. OrthogonalPlane(P, u) → p
  4. Point(zAxis) → A_{1}
  5. Sphere(B_{1}, A_{1}) → a
  6. Circle(P, 5, u) → c
  7. Plane(P, A_{1}, B_{1}) → q
  8. Point(c) → C_{1}
  9. Line(B_{1}, P) → f
  10. Rotate(C_{1}, 45°, f) → C'_{1}
  11. Rotate(C'_{1}, 90°, f) → C''
  12. Rotate(C'', 90°, f) → C'''
  13. Rotate(C''', 90°, f) → D
  14. Polygon(C'_{1}, C'', C''', D) → q1, c', c'', c''', d
  15. PointIn(q1) → A
  16. PointIn(q1) → C
  17. PointIn(q1) → B
  18. Segment(A, B) → g
  19. Segment(C, B) → h
  20. Segment(A, C) → i
  21. Segment(A, B_{1}) → j
  22. Segment(B_{1}, C) → k
  23. Segment(B, B_{1}) → l
  24. Intersect(j, a) → B',
  25. Intersect(l, a) → A',
  26. Intersect(k, a) → C',
  27. CircleArc(B_{1}, B', C') → e
  28. CircleArc(B_{1}, B', A') → r
  29. CircleArc(B_{1}, A', C') → s
  30. CircleSector(B_{1}, B', C') → t
  31. CircleSector(B_{1}, A', C') → c_1
  32. CircleSector(B_{1}, B', A') → d_1

AI Drawing Prompt Reference

In 3D, use two perpendicular coordinate axes and their intersection to set a reference point related to the sphere’s center. Mark another point on one axis to define the sphere and construct a sphere using the sphere-center reference point and the axis point. Through a plane determined by the reference point and two axis-related points, define a circle whose plane contains the reference point and has a specific direction. Take the circle’s key intersection point, then draw lines from the reference point along the chosen directions. Rotate the circle point(s) stepwise around the same axis line by 45° and then by 90° three times to obtain several rotated points. Connect four of these rotated points to form a spherical quadrilateral, and draw circle arcs and circle sectors on the sphere to illustrate the mapping.
The prompt above was compiled from the original drawing commands and is for reference only.