Curvature: The curvature at a point on a curve is the reciprocal of the radius of the osculating circle, which has a second order tangency at the point. The curvature (more specifically the Gaussian curvature) at a point of a surface in three-space can be defined by looking at the curves of intersection with planes that are perpendicular to the surface at the point. Comparing the curvatures of all these curves, the product of the largest and the smallest (the principal curvatures) is defined to be the curvature of the surface at that point. If these curve in opposite directions, as on a saddle surface, the curvature is negative.
It's intrinsic: The curvature is intrinsic in the sense that it can be defined in terms of measurements within the surface. So an isometry of one surface to another, a map that preserves the length of curves, preserves the curvature at each point.
Constant curvature: A sphere is an example of a surface of constant positive curvature. It is the only such surface that is complete, i.e. it is closed, has no singularities, and every geodesic (shortest distance curve, in this case an arc of a great circle) extends without end. Because the curvature is constant, the geometry of the surface is the same around each point, and there are analogs of the formulas of Euclidean geometry. For example, there is an analog of the Pythagorean theorem saying that on a surface of constant curvature 1, the cosine of the hypotenuse of any right triangle is the product of the cosines of the legs.
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