# Differential Geometry/Curvature and Osculating Circle

Consider a curve of class of at least 2, parametrized by the arc length parameter, .

The magnitude of is called the **curvature** of the curve at the point . 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 is always 0, which proves that it is a straight line through elementary integrations.

We can also consider the normal vector to be the curvature vector.

The point that is away from 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 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 . It is very obvious that the unit tangent vector at the point is tangent to the osculating circle at .