| Spherical Geometry |
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| Spherical Geometry |
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Spherical geometry is based on the surface of a sphere, not a plane. The first four basic axioms are still true in spherical geometry. However, the parallel postulate is negated, and now there are NO parallel lines possible! |
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AXIOMS
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EXISTENCE
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| 1. A straight line can be drawn between any two points | Choose any two points on the surface of the sphere, and draw a "straight" line between them. However, in spherical geometry, a straight line is a great circle. The shortest distance between two points always lies on a great circle. Longitude lines are great circles. |
| 2. A finite line can be infinitely extended in both directions | A straight line will go around the sphere and connect to itself. A line of infinite length would go around the sphere an infinite amount of times. |
| 3. A circle can be drawn with any center or radius | Choose a center and find all the points a given distance (radius) from the center. In spherical geometry, a great circle is both a line and a circle! Latitude lines are actually not lines, they are circles in spherical geometry. |
| 4. All right angles are equal to each other | Right angles can be found on the surface of the sphere -- think of the intersection of longitude and latitude -- they are always perpendicular. |
| 5. There are NO parallel lines | Parallel lines are defined as never intersecting. Then there are no parallel lines in spherical geometry -- any two lines (great circles) drawn on the sphere will intersect in two places. In geography, these are known as antipodal points. |
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