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Regular Quadrilateral on a Sphere

Regular Quadrilateral on a Sphere
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2014-07-24 02:56
PolygonSolid GeometryGeometric Construction
In collections Spherical Geometry Ideas
This example illustrates a regular quadrilateral on a sphere. The angles range from 90 degrees to 180 degrees. Using GeoGebra commands such as Sphere, PointIn, Line, Rotate, Plane, OrthogonalLine, Intersect, and CurveCartesian, this geometric figure can be constructed. Suitable for high school geometry education.

GeoGebra Commands & Parameters

  1. Sphere(O, 1) → a
  2. PointIn(a) → A
  3. PointIn(a) → E
  4. Line(O, E) → b
  5. Rotate(A, 90*°, b) → B
  6. Rotate(B, 90*°, b) → C
  7. Rotate(C, 90*°, b) → D
  8. Plane(A, O, B) → c
  9. OrthogonalLine(O, c) → d
  10. Intersect(d, a, 1) → F
  11. Plane(A, d) → e
  12. OrthogonalLine(O, e) → f
  13. Intersect(f, a, 1) → G
  14. CurveCartesian((x(A) * cos(t)) + (x(G) * sin(t)), (y(A) * cos(t)) + (y(G) * sin(t)), (z(A) * cos(t)) + (z(G) * sin(t)), t, 0, Angle[A, O, B]) → g
  15. Rotate(g, 90*°, b) → g'
  16. Rotate(g', 90*°, b) → g''
  17. Rotate(g'', 90*°, b) → g'''

AI Drawing Prompt Reference

Construct a regular quadrilateral on a unit sphere. Start by picking a point on the sphere as a vertex, and choose another point on the sphere along with the sphere's center to define an axis of rotation. Rotate the first vertex around this axis by 90 degrees three times to obtain four vertices, forming a spherical square. Then, through the first vertex, the center, and the second vertex, define a plane; from the center, draw a line perpendicular to this plane and intersect it with the sphere to get a point. Next, through the first vertex and this new point, define another plane; from the center, draw a perpendicular to this plane and intersect with the sphere to get another point. Using these two points and the first vertex, trace an arc on the sphere, then rotate this arc around the same axis by 90 degrees three times to complete the four edges of the spherical quadrilateral.
The prompt above was compiled from the original drawing commands and is for reference only.