Evolutes

The locus of the centers of curvature of a given curve is called
the *evolute* of that curve.
Consider the circle of curvature corresponding to a point on a curve.
If moves along the given curve, we may suppose the corresponding
circle of curvature to roll along the curve with it, its radius
varying so as to be always equal to the radius of curvature
of the curve at the point . The curve in Figure 14.5
described by the center of the circle is the evolute of .

It is instructive to make an approximate construction of the evolute of a curve by estimating (from the shape of the curve) the lengths of the radii of curvature at different points on the curve and then drawing them in and drawing the locus of the centers of curvature.

Formula (14.7) gives the coordinates of any point on the evolute expressed in terms of the coordinates of the corresponding point of the given curve. But is a function of ; therefore

To find the ordinary rectangular equation of the evolute we eliminate between the two expressions. No general process of elimination can be given that will apply in all cases, the method to be adopted depending on the form of the given equation. In a large number of cases, however, the student can find the rectangular equation of the evolute by taking the following steps:

General directions for finding the equation of the evolute in rectangular coordinates.

- FIRST STEP. Find
from (14.9).
- SECOND STEP. Solve the two resulting equations for and
in terms of and .
- THIRD STEP. Substitute these values of and in the given equation. This gives a relation between the variables and which is the equation of the evolute.

*Solution.
.
*

*First step.
,
.
*

*Second step.
,
.
*

*Third step
;
or,
.
*

*Remembering that denotes the abscissa and the
ordinate of a rectangular system of coordinates,
we see that the evolute of the parabola is the semi-cubical
parabola ; the centers of curvature for , , ,
being at , , , respectively.*

Solution. , .

First step. , .

Second step. , .

Third step. , the equation of the evolute of the ellipse , , , , are the centers of curvature corresponding to the points , , , , on the curve, and , , correspond to the points , , .

When the equations of the curve are given in parametric form, we proceed to find and , as in §12.5, from

and then substitute the results in formulas (14.9). This gives the parametric equations of the evolute in terms of the same parameter that occurs in the given equations.

Find the equation of the evolute in parametric form, plot the curve and the evolute, find the radius of curvature at the point where , and draw the corresponding circle of curvature.

Solution. , , , . Substituting in above formulas (14.12) and then in (14.9), gives

the parametric equations of the evolute. Assuming values of the parameter , we calculate , ; from (14.13) and (14.14). Now plot the curve and its evolute.

The point is common to the given curve and its evolute. The given curve (a semi-cubical parabola) lies entirely to the right and the evolute entirely to the left of .

The circle of curvature at , where , will have its center at on the evolute and radius . To verify our work, find radius of curvature at . From (12.5), we get

Solution. As in Example 12.5.2, we get

Since (14.17) and (14.14) are identical in form, we have:

david joyner 2008-11-22