# Curvature of a plane curve.

Definition 6.13.1   Let be a function and be a point on the graph of . Denote by the circle (if it exists) verifying the following properties:
(i)
The circle has the same tangent at as the graph of ;
(ii)
It lies on the same side of the tangent as the graph does;
This circle is called the circle of curvature of the graph at . Its radius is called the radius of curvature at , and its center is called the center of curvature at .

Proposition 6.13.2   Suppose that is differentiable at least twice at and denote by the point whose coordinates are . Then:
1. The curvature at is given by the formula

2. The radius of curvature is given by

Example 6.13.3   Let , for . Let be the point of intersection of the graph of with the axis, i.e. .

The function is differentiable more tna twice over . We have:

Thus, the curvature at is equal to

The radius of curvature is equal to . See Figure 16.

The tangent to the graph at has equation , and the normal at has equation . The center of curvature is the point on at a distance of from and inside'' the curve; thus the coordinates of are .

Proposition 6.13.4   Suppose that the curve is given by a parametric representation of the form

where the functions are differentiable at least twice at . Denote by the point whose coordinates are .

The curvature at is given by the formula:

Physicists use to denote instead of and instead of ; the above formula becomes:

Example 6.13.5   The curve is given by the parametric equations

Find the curvature and the radius of curvature at the right end of the major axis.

The curve is an ellipse, whose cartesian equation is

The major axis is the axis and the point we have to consider is . This point corresponds to .

We use the formula in Proposition 13.4. We have:

We compute these derivatives for ; we have:

Thus, the curvature at is

 (6.2)

Noah Dana-Picard 2007-12-28