Curvature (InstanceTopic, 3)

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Curvature is a geometric property of a space. It can generally be expressed as a real number.

If this number is zero, the space is said to be flat. In this case, the sum of the angles in any triangle adds up to 180 degrees, and two distinct parallel lines will never touch. If a flat space A is embedded within another space B, the shortest route between any two points in space A does not leave A. Flat space observes Euclidean geometry.

If the number is positive, the space is curved. In this case, the sum of the angles in a triangle may add up to more than 180 degrees. We cannot call two lines "parallel" any more in such a space, but if one constructs three coplanar lines E, L1, L2 such that E is perpendicular to both L1 and L2, then the lines L1 and L2 may intersect. On a sphere, they will intersect at two points and will actually appear as circles. On a paraboloid, they may intersect at zero points (as circles perpendicular to the central line of symmetry), exactly one point (as parabolas in a plane of symmetry), or (possibly? depending on how continity works at the pole - this may produce zero points instead) at infinitely many distinct points (in any other case).

If the number is negative, the space is hyperbolic. In this case, the sum of the angles in a triangle may add up to less than 180 degrees. This allows an infinite number of distinct regular infinite hyperbolic tilings which are not possible in non-hyperbolic space.

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