\(r = asin(n\theta) \)
\(\frac {dr}{d\theta} = ancos(n\theta)\)
\(\frac {d^2r}{d\theta^2} = -an^2sin (n \theta)\)
radius of curvature
\(S = \cfrac{\{r^2 + (\frac {dr}{d\theta})^2\}}{r^2 + 2 (\frac {dr}{d\theta})^2 - r\frac{d^2r}{d\theta^2}}\)
\(= \frac {(a^2sin^2n\theta + a^2 + n^2 cos^2n\theta)^{3/2}}{a^2 sin^2n\theta + 2a^2n^2 cos^2n\theta + a^2 n^2 sin^2 n\theta}\)
\(= \frac{a^3(sin^2n\theta + n^2cos^2n\theta)^{3/2}}{a^2 (sin^2\theta + n^2sin^2n\theta + 2n^2 cos^2n\theta)}\)
\(= \frac {a(sin^2n\theta + n^2cos^2n\theta)^{3/2}}{(n^2 + 1)sin^2n\theta + 2n^2 cos^2n\theta}\)
at \(\theta = 0\)
\(= \frac {a.n^3}{2n^2}\)
\(= \frac{an} 2\)
