Arc Length Parametrization for Combined Function

Question

The arc length parameterization for the circle is easily written as

$$x(t), y(t) = r\cos(t), r\sin(t)$$

But how can we do this if our geometry is composed of multiple arcs of circles with different radii?

$$Arc=\sqrt{\left(\dfrac{dx(t)}{dt}\right)^2+\left(\dfrac{dy(t)}{dt}\right)^2}$$

how we can calculate/define $dt$ along the curve?

$$Arc=\sqrt{\left(\dfrac{dx(n+1)-dx(n-1)}{2dt(n)}\right)^2+\left(\dfrac{dy(n+1)-dy(n-1)}{2dt(n)}\right)^2}$$

Answer 1

There are 2 problems with what you are trying to do.

1. You assume $r$ is the same everywhere, but it looks like you have circles of different radii.
2. Your parameterization works for the unit circle on $t \in [0,1)$. You may need to be in different intervals.

I would do the following.

1. Write down a general parameterization of the circle with radius $r$ for $t \in [a,b)$ for any $a<b \in \mathbb{R}$ (not just for $[a,b) = [0,1)$ as you have).
2. Cut your curve into different arcs, and parameterize different arcs with different ranges of $t$, making $r = r(t)$ change as well, being constant on different intervals.
3. Verify that the transition points between arcs exactly correspond to the interval boundaries.

Now you will have a range for $t$ over which the parameterization will trace out your entire curve.