![]() Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions. This is more than an analogy spherical and plane geometry and others can all be unified under the umbrella of geometry built from distance measurement, where "lines" are defined to mean shortest paths (geodesics). Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles. This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic relationships between great circles and the also-undefined "points". In the intrinsic approach, a great circle is a geodesic a shortest path between any two of its points provided they are close enough. ![]() In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in Elliptic geometry. In spherical geometry, the basic concepts are point and great circle. In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. It can also be analyzed by "intrinsic" methods that only involve the surface itself, and do not refer to, or even assume the existence of, any surrounding space outside or inside the sphere. The sphere has for the most part been studied as a part of 3-dimensional Euclidean geometry (often called solid geometry), the surface thought of as placed inside an ambient 3-d space. ![]() Long studied for its practical applications – spherical trigonometry – to navigation, spherical geometry bears many similarities and relationships to, and important differences from, Euclidean plane geometry. Spherical geometry is the geometry of the two- dimensional surface of a sphere. A sphere with a spherical triangle on it. ![]()
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