Page:Elements of the Differential and Integral Calculus - Granville - Revised.djvu/209

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 * the parametric equations of the evolute. Assuming values of the parameter $$t$$, we calculate $$x, y; \alpha, \beta$$ from (B) and (C); and tabulate the results as follows:


 * Now plot the curve and its evolute.


 * {| class="wikitable" style="float: right; text-align: center;"


 * t||x||y||&aplha;||β
 * 3
 * $$\tfrac{5}{2}$$
 * $$\tfrac{9}{2}$$
 * 2
 * $$\tfrac{5}{4}$$
 * $$\tfrac{4}{3}$$
 * $$-\tfrac{35}{4}$$
 * $$\tfrac{19}{3}$$
 * $$\tfrac{3}{2}$$
 * $$\tfrac{13}{16}$$
 * $$\tfrac{9}{16}$$
 * $$-\tfrac{91}{32}$$
 * $$3$$
 * 1
 * $$\tfrac{1}{2}$$
 * $$\tfrac{1}{6}$$
 * $$-\tfrac{1}{2}$$
 * $$\tfrac{7}{6}$$
 * 0
 * $$\tfrac{1}{4}$$
 * 0
 * $$\tfrac{1}{4}$$
 * 0
 * -1
 * $$\tfrac{1}{2}$$
 * $$-\tfrac{1}{6}$$
 * $$-\tfrac{1}{2}$$
 * $$-\tfrac{7}{6}$$
 * $$-\tfrac{3}{2}$$
 * $$\tfrac{13}{16}$$
 * $$-\tfrac{9}{16}$$
 * $$-\tfrac{91}{32}$$
 * $$-3$$
 * -2
 * $$\tfrac{5}{4}$$
 * $$-\tfrac{4}{3}$$
 * $$-\tfrac{35}{4}$$
 * $$-\tfrac{19}{3}$$
 * -3
 * $$\tfrac{5}{2}$$
 * $$-\tfrac{9}{2}$$
 * }
 * -2
 * $$\tfrac{5}{4}$$
 * $$-\tfrac{4}{3}$$
 * $$-\tfrac{35}{4}$$
 * $$-\tfrac{19}{3}$$
 * -3
 * $$\tfrac{5}{2}$$
 * $$-\tfrac{9}{2}$$
 * }
 * $$-\tfrac{9}{2}$$
 * }
 * }
 * }


 * The point $$(\tfrac{1}{4}, 0)$$ is common to the given curve and its evolute. The given curve (semi cubical parabola) lies entirely to the right and the evolute entirely to the left of $$x = \tfrac{1}{4}$$.


 * The circle of curvature at $$A (\tfrac{1}{2}, \tfrac{1}{6})$$, where $$t = 1$$, will have its center at $$A' (-\tfrac{1}{2}, -\tfrac{7}{6})$$ on the evolute and radius = $$AA'$$. To verify our work find radius of curvature at A. From (42), §103, we get

$$R = \frac{t(1 + t^2)^{\frac{3}{2}}}{2} = \sqrt{2}, \text{when} t 1.$$


 * This should equal the distance

$$AA' = \sqrt{(\frac{1}{2} + \frac{1}{2})^2 +(\frac{1}{6} - \frac{7}{6})^2} = \sqrt{2}.$$


 * [[Image:Wag 119-4 evolute example_3.jpg|center|371px|Evolute of the curve.]]

Find the parametric equations of the evolute of the cycloid,


 * {| style="width:100%;"


 * style="width:3%;"|
 * align="center"|$$\begin{cases} x = a(t - \sin t) \\ y = a(1 - \cos t).\end{cases}$$
 * }


 * {| style="width:100%;"


 * colspan="2"|Solution. As in, §103, we get
 * align="center"|$$\frac{dy}{dx} = \frac{\sin t}{1 - \cos t}, \frac{d^2 y}{dx^2} = -\frac{1}{a(1 - \cos t)^2}.$$
 * colspan="2"|Substituting these results in formulas (50), §117, we get
 * style="width:3%;"|
 * align="center"|$$\begin{cases} \alpha = a(t + \sin t), \\ \beta = -a(1 - \cos t).\end{cases}$$ Ans.
 * }
 * style="width:3%;"|
 * align="center"|$$\begin{cases} \alpha = a(t + \sin t), \\ \beta = -a(1 - \cos t).\end{cases}$$ Ans.
 * }
 * }