structural:expansion, linguistic:paraphrasing, quality:correction
Circle

A common parametrization of a circle of radius r is γ (t) = (r cos t, r sin t). The formula for the curvature gives

It follows, as expected, that the radius of curvature is the radius of the circle, and that the center of curvature is the center of the circle.

The circle is a rare case where the arc-length parametrization is easy to compute, as it is

It is an arc-length parametrization, since the norm of

is equal to one. This parametrization gives the same value for the curvature, as it amounts to division by r in both the numerator and the denominator in the preceding formula.

The same circle can also be defined by the implicit equation F (x, y) = 0 with F (x, y) = x + y – r. Then, the formula for the curvature in this case gives