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Non-Euclidean Geometry on a Sphere

Suppose instead of working on the Euclidean plane we study geometry on a sphere in (Euclidean) three space. We interpret point to mean any point on the sphere and we interpret line to mean any great circle on the sphere (that is any circumference of the sphere).
a. Is Euclidâ??s parallel postulate true in this setting?
b. How would you define the angle which two great circles make at a point where they
intersect on the sphere?
c. How would you define a triangle on the sphere? Give some examples.
d. Is the sum of the angles in a triangle greater than, less than, or equal to 180 degrees in this setting?
e. Does the angle sum depend on the area of the triangle? How?
f. How does part d relate to showing that the parallel postulate implies the triangle postulate?

Solution Preview

a. No. Any two great circles on the sphere intersect, so there are no parallel "lines" in this context.

b. The angle which two great circles make at an intersection point on the sphere is define as the angle which the tangent lines to each great circle make at that point.

c. A triangle on a sphere is any region bounded by three ...

Solution Summary

We answer several questions pertaining to the geometry of a sphere and how it differs from Euclidean geometry (on a plane).