# Differential Geometry/Curvature and Osculating Circle

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

The magnitude of ${\displaystyle f''(s)}$ is called the curvature of the curve ${\displaystyle C}$ at the point ${\displaystyle f(s)}$. 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 ${\displaystyle f''(s)}$ is always 0, which proves that it is a straight line through elementary integrations.

We can also consider the normal vector ${\displaystyle f''(s)}$ to be the curvature vector.

The point that is away from ${\displaystyle f(s)}$ 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 ${\displaystyle f(s)}$ 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 ${\displaystyle f(s)}$. It is very obvious that the unit tangent vector at the point ${\displaystyle f(s)}$ is tangent to the osculating circle at ${\displaystyle f(s)}$.