Could any one tell me what are the fundamental contrasts with postulates of Euclidean Geometry and Spherical Geometry?
I myself see these things, please tell me if there are more:
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Lines in EG are great Circles in SG, we can extend infinitely a line in EG but there is no great circle in a sphere which has arbitrary radius.
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Parallel Line exists in EG but not in SG as any two great circle meets in two points also a contrast that in EG any two non parallel line meets only at one pt.
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Sum of angles exceeds $180$ degree in SG.
along with this I would like to ask: is there any spherical triangle whose angles add up to exacly $180$ degree, which euclidean postulates is violating and why please tell me
Best Answer
A consequence of the parallel lines postulate in Euclidean geometry implies that the interior angles of a triangle always add up to 180 degrees (see any book on EG). Thus modifying the parallel postulate modifies the possibilities for the sum of the interior angles of a triangle. In particular if you postulate that parallel lines don't exist then this implies that the sum of interior angles of a triangle are strictly greater than 180 dergrees, thus there are no triangles on a sphere whose angles add up to 180. Like many other structures in math, there is a trichotomy (let $S$ be the sum of the interior angles of a triangle in a given geometry) :
Hyperbolic geometry: $S<180$,
Euclidean geometry: $S=180$,
Spherical (or parabolic geometry): $S>180$.
This really comes from the trichotomy of real numbers:
$r\in \mathbb{R}$ is either negative, zero, or positive.
This is because the trichotomy mentioned earlier about triangles really comes from the curvature of the associated geometries. Hyperbolic geometry is a negatively curved geometry, euclidean geometry has zero curvature (i.e., it's flat), and parabolic geometry is positively curved.