Evolute of Circle is its Center

Theorem
The evolute of a circle is a single point: its center.

Proof
By definition, the evolute of $C$ is the locus of the centers of curvature of each point on $C$

, take the circle $C$ of radius $a$ whose center is positioned at the origin of a cartesian plane.

From Equation of Circle, $C$ has the equation:
 * $x^2 + y^2 = a^2$

From the definition of curvature in cartesian form:
 * $k = \dfrac {y''} {\paren {1 + y'^2}^{3/2} }$

Here we have:

So:

Thus the curvature of $C$ is constant.

The radius of curvature of $C$ is likewise constant:
 * $\rho = a$

From Radius at Right Angle to Tangent, the normal to $C$ at all points on $C$ passes through the center of $C$.

We have that $a$ is the distance from $C$ to the center of $C$.

Thus it follows that the center of curvature of $C$ is the center of $C$ at all points.

Hence the result by definition of evolute.