Exercises

Find the coordinates 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 $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$.

   $\alpha = x + 3x^{\frac{1}{3}} y^{\frac{2}{3}}$, $\beta = y + 3 x^{\frac{2}{3}} y^{\frac{1}{3}}$; 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 $y^3 = a^2x$.

   Ans. $\alpha = \frac{a^4 + 15y^4}{6a^2 y}$, $\beta = \frac{a^4y - 9y^5}{2a^4}$.

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

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.

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 $$\begin{cases}x = 3t^2, \\ y = 3t - t^3. \end{cases}$$

Ans. $$\begin{cases}\alpha = \frac{3}{2} ( 1 + 2t^2 - t^4 ), \\ \beta = -4 t^3. \end{cases}$$

8. The curve $$\begin{cases} x = a(\cos t + t \sin t), \\ y = a(\sin t - t \cos t). \end{cases}$$.

Ans. $$\begin{cases}\alpha = a \cos t, \\ \beta = a \sin t. \end{cases}$$.

9. The curve $$\begin{cases}x = 3t, \\ y = t^2 -6. \end{cases}$$.

Ans. $$\begin{cases}\alpha = -\frac{4}{3} t^3, \\ \beta = 3t^2 - \frac{3}{2}. \end{cases}$$.

10. The curve $$\begin{cases}x = 6 - t^2 \\ y = 2t. \end{cases}$$.

Ans. $$\begin{cases}\alpha = 4 - 3t^2, \\ \beta = -2t^3. \end{cases}$$.

11. The curve $$\begin{cases}x = 2t, \\ y = t^2 - 2. \end{cases}$$.

Ans. $$\begin{cases}\alpha = -2 t^3, \\ \beta = 3t^2. \end{cases}$$.

12. The curve $$\begin{cases}x = 4t, \\ y = 3 + t^2. \end{cases}$$.

Ans. $$\begin{cases}\alpha = -t^3, \\ \beta = 11 + 3t^2. \end{cases}$$.

13. The curve $$\begin{cases}x = 9 - t^2, \\ y = 2t. \end{cases}$$.

Ans. $$\begin{cases}\alpha = 7 - 3t^2, \\ \beta = -2t^3. \end{cases}$$.

14. The curve $$\begin{cases}x = 2t, \\ y = \frac{1}{3}t^3. \end{cases}$$.

Ans. $$\begin{cases}\alpha = \frac{4t - t^5}{4}. \\ \beta = \frac{12 + 5t^4}{6t}. \end{cases}$$.

15. The curve $$\begin{cases}x = \frac{1}{3} t^3, \\ y = t^2. \end{cases}$$.

Ans. $$\begin{cases}\alpha = \frac{4t^3 + 12t}{3} \\ \beta = -\frac{2t^2 + t^4}{2}. \end{cases}$$.

16. The curve $$\begin{cases}x = 2t, \\ y = \frac{3}{t}. \end{cases}$$.

Ans $$\begin{cases}\alpha = \frac{12t^4 + 9}{4t^3} \\ \beta = \frac{27 + 4t^4}{6t}. \end{cases}$$.

17. $x = 4 -t^2$, $y = 2t$.

18. $x = 2t$, $y = 16 -t^2$.

19. $x = t$, $y = \sin\, t$.

20. $x = \frac{4}{t}$, $y = 3t$.

21. $x = t^2$, $y = \frac{1}{6} t^3$.

22. $x = t$, $y = t^3$.

23. $x = \sin t$, $y = 3\cos t$.

24. $x = 1 -\cos t$, $y = t -\sin t$.

25. $x = \cos^4t$, $y = \sin^4t$.

26. $x = a\sec t$, $y = b\tan t$.