Question

A plane curve is given by γ(θ)=(rcosθ,rsinθ)

γ(θ)=(rcos⁡θ,rsin⁡θ), where r is a smooth function of θ

θ (so that (r,θ)

(r,θ) are the polar coordinates of γ(θ)

γ(θ)). Under what conditions is γ

γ regular? Find all functions r(θ)

r(θ) for which γ

γ is unit-speed. Show that, if γ

γ is unit-speed, the image of γ

γ is a circle; what is its radius?

Answer 1

Curve is regular if $\gamma' (\theta)\ne0$ everywhere.

$\gamma' (\theta)=(-rsin\theta,rcos\theta)$

$||\gamma' (\theta)||=r$

So curve is regular if $r\ne 0$

$s(\theta)=\int^{\theta}_0rd\alpha=r\theta\implies \theta=s/r$

Unit speed curve:

$\gamma(s)=(rcos(s/r),rsin(s/r))$

Image of $\gamma(s)$ :

$||\gamma(s)||=r$