Curvature of a plane curve.

- (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;

- The curvature
at
is given by the formula

- The radius of curvature is given by

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 .

*The curvature at
is given by the formula:
*

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

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