Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/212

 wrapped around the ruler (or curve). It is clear from the results of the last section that when the string is unwound and kept taut, the free end will describe the curve $$P_1 P_9$$. Hence the name evolute.

The curve $$P_1 P_9$$ is said to be an involute of $$C_1 C_9$$. Obviously any point on the string will describe an involute, so that a given curve has an infinite number of involutes but only one evolute.

The involutes $$P_1 P_9, P_1' P_9', P_1 P_9$$ are called parallel curves since the distance between any two of them measured along their common normals is constant.

The student should observe how the parabola and ellipse in §119 may be constructed in this way from their evolutes.

EXAMPLES

Find the coördinates of the center of curvature and the equation of the evolute of each of the following curves. Draw the curve and its evolute, and draw at least one circle of curvature.

4. Show that in the parabola $$x^{\frac{1}{2}} + y^{\frac{1}{2}} = a^{\frac{1}{2}}$$ we have the relation $$\alpha + \beta = 3(x + y).$$

5. Given the equation of the equilateral hyperbola $$2 xy = a^2$$ show that


 * $$\alpha + \beta = \frac{(y + x)^3}{a^2}, \alpha - \beta = \frac{(y - x)^3}{a^2}.$$

From this derive the equation of the evolute $$(\alpha + \beta)^{\frac{2}{3}} - (\alpha - \beta)^{\frac{2}{3}} = 2 a^{\frac{2}{3}}.$$

Find the parametric equations of the evolutes of the following curves in terms of the parameter t. Draw the curve and its evolute, and draw at least one circle of curvature.