Differential Geometry/Curvature and Osculating Circle

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Consider a curve C of class of at least 2, parametrized by the arc length parameter, f(s).

The magnitude of f''(s) is called the curvature of the curve C at the point f(s). The multiplicative inverse of the curvature is called the radius of curvature.

The curvature is 0 at every point if and only if the curve is a straight line. Suppose that the curvature is always 0. Then f''(s) is always 0, which proves that it is a straight line through elementary integrations.

We can also consider the normal vector f''(s) to be the curvature vector.

The point that is away from f(s) by a distance of the radius of curvature in the direction of the principal normal unit vector is called the center of curvature of the point f(s) and the circle with the center on the center of curvature and with the radius as the radius of curvature is called the osculating circle at the point f(s). It is very obvious that the unit tangent vector at the point f(s) is tangent to the osculating circle at f(s).