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.

Involute of flexible ruler.

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.

1.	The hyperbola ${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{1}}}=1.$	Ans.	$\alpha ={\frac {(a^{2}+b^{2})x^{3}}{a^{4}}},\beta =-{\frac {(a^{2}+b^{2})y^{3}}{b^{4}}};$
			evolute $(a\alpha )^{\frac {2}{3}}-(b\beta )^{\frac {2}{3}}=(a^{2}+b^{2})^{\frac {2}{3}}.$
2.	The hypocycloid $x^{\frac {2}{3}}+y^{\frac {2}{3}}=a^{\frac {2}{3}}.$	Ans.	$\alpha =x+3x^{\frac {1}{3}}y^{\frac {2}{3}},\beta =y+3x^{\frac {2}{3}}y^{\frac {1}{3}};$
			evolute $(\alpha +\beta )^{\frac {2}{3}}+(\alpha -\beta )^{\frac {2}{3}}=2a^{\frac {2}{3}}.$
3.	Find the coordinates of the center of curvature of the cubical parabola $y^{3}=a^{2}x.$
		Ans.	$\alpha ={\frac {a^{4}+15y^{4}}{6a^{2}y}},\beta ={\frac {a^{4}y-9y^{5}}{2a^{4}}}.$

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 $2xy=a^{2}$ 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 $(\alpha +\beta )^{\frac {2}{3}}-(\alpha -\beta )^{\frac {2}{3}}=2a^{\frac {2}{3}}.$

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.

6. The hypocycloid	${\begin{cases}x=a\cos ^{3}t,\\y=a\sin ^{3}t.\end{cases}}$	Ans.	${\begin{cases}\alpha =a\cos ^{3}t+3a\cos t\sin ^{2}t,\\\beta =3a\cos ^{2}t\sin t+a\sin ^{3}t.\end{cases}}$
7. The curve	${\begin{cases}x=3t^{2},\\y=3t-t^{3}.\end{cases}}$		${\begin{cases}\alpha ={\frac {3}{2}}(1+2t^{2}-t^{4}),\\\beta =-4t^{3}.\end{cases}}$