The relation between $p$ and $r$ where $p$ is the length of the perpendicular from the fixed point $O$ on the tangent to the curve at any point $P$ is called pedal equation of the curve.
I want to find the radius of curvature of the pedal curves.
Let $p=f(r)$ be the pedal equation.
$\psi=\theta+\phi……..(1)$
and $p=r\sin\phi……….(2)$
Differentiating $(1)$ with respect to $s$,
$\frac{d\psi}{ds}=\frac{d\theta}{ds}+\frac{d\phi}{ds}………….(3)$
Differentiating $(2)$ with respect to $r$,
$\frac{dp}{dr}=\sin\phi+r\cos\phi\frac{d\phi}{dr}$
$\frac{dp}{dr}=\sin\phi+r\cos\phi\frac{d\phi}{ds}\frac{ds}{dr}$
$\frac{dp}{dr}=r\frac{d\theta}{ds}+r\frac{dr}{ds}\frac{ds}{dr}\frac{d\phi}{ds}$
$\because \sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}$
$\frac{dp}{dr}=r(\frac{d\theta}{ds}+\frac{d\phi}{ds})=r\frac{d\psi}{ds}$….from $(3)$
$\frac{dp}{dr}=r\frac{1}{\rho}$
We know that radius of curvature$=\rho=\frac{ds}{d\phi}$
$\therefore \rho=r\frac{dr}{dp}$
In this derivation i do not understand why does $\sin\phi=r\frac{d\theta}{ds}$ and $\cos\phi=\frac{dr}{ds}?$.Please help me.
Best Answer
Move along the curve from $P$ to a nearby point $P'$ with polar coordinates $(r+\delta r, \theta +\delta\theta)$. We denote the change in arc length $\delta s$. For $P$ and $P'$ sufficiently close together, we can assume that the curve between them approximates to a straight line.
We therefore have a triangle $OPP'$ with $OP=r,\; PP'=\delta s,\;OP'=r+\delta r,\; \angle POP'=\delta \theta\; \text{and}\; \angle OP'P=\phi.$
(Apologies for the lack of diagram — you may want to draw one!)
Now using the cosine rule, $$ \begin {align} r^2&=(r+\delta r)^2+\delta s^2-2(r+\delta r)\delta s \cos \phi \\ 2(r+\delta r)\delta s\cos\phi&=2r\delta r+\delta r^2+\delta s^2 \\ 2r\delta s \cos \phi&\approx2r\delta r \\ \cos\phi&\approx\frac{\delta r}{\delta s} \end {align} $$ And the sine rule, $$ \begin {align} \frac {\sin\phi}{r}&=\frac{\sin\delta\theta}{\delta s} \\ \sin\phi&=r\frac{\sin\delta\theta}{\delta \theta}\frac{\delta \theta}{\delta s} \\ \sin\phi&\approx r\frac{\delta \theta}{\delta s} \end {align} $$