# parameterisation of arc of circle

A question that catches my curiosity is :

how many ways one can parameterize an arc of the circle of center $$(x_0, y_0)$$ and of radius $$r$$

1- parameterization by arc length $$\alpha(t)=(x_0+\cos(\frac{t}{r}),y_0+\sin(\frac{t}{r}))$$ 2-parametrisation by the support function $$h(t)=x_0 \cos(t)+y_0 \sin(t)+r$$ what else ?

PS : the case of parameterization like $$\alpha(t)=(x_0+r\cos(f(t)),y_0+r\sin(f(t)))$$ give an infinite way to parameterize arcs of circle but we can go beyond this case.

At first sight it seems there are plenty of creative ways to parameterize the circle, for instance things like $$(t,\pm\sqrt{1-t^2})$$ or $$(\cos t,\sin t)$$, but I think a better and perhaps disappointing way to look at all of them (which helps to count them) is considering re-paramterizations.
Let $$\gamma:I\rightarrow\mathbb R^n$$ be any smooth curve generating $$\Gamma=\gamma(I)$$. For simplicity, let's assume like in your case it is simple, meaning $$\Gamma$$ doesn't self intersect. As you said in your postscript, for any real monotonic function $$\sigma : J\rightarrow I$$, we can generate a new way of expressing $$\Gamma$$ in the $$J$$ domain as $$\gamma\circ \sigma$$. There are of course infinite ways to choose $$\sigma$$, but can we "go beyond this case"? Well, as long as we want the curve to be traced in the same way (for example not making the curve trace out the circle twice), the answer is no. The reason is that for any other parameterization $$\tilde{\gamma}:J\rightarrow \Gamma$$, we have that $$\sigma = \gamma^{-1}\circ\tilde{\gamma}$$ (which exists and is smooth) is a change of parameter satisfying $$\tilde{\gamma} = \gamma \circ \sigma$$, as required.
This means that $$(x_0 + r \cos f(t), y_0 + r\sin f(t))$$ is the most general form possible. In the case $$(x_0,y_0)=(0,0)$$ and $$r=1$$, how do we get $$(t,\sqrt{1-t^2})$$? we apply $$f(t)=\arccos t$$.