Circular segment from central angle

The parameterized solutions come from solving the following

The goal is to get rid of r. We replace it with c/2/sin(α)

1/sin(α) diverges as the central angle and thus α goes to 0. We'd like to be able to divide it through and have the terms simplify. We'll return to x in a bit, for now we'll focus on y. Factoring a sin(α) out of the difference in cosines might be easier it they're replaced with a product

That looks promising. Replacing the sin(u±v) with sin(u)cos(v)±cos(u)sin(v) would make a lot of sin terms to factor out

We can divide both of those sums by a sin(α) if we multiply by a second one. This makes it clear why y shouldn't diverge.

If there was a trigonometric identity for dividing sines we would be done with both x and y. But there isn't so we'll introduce the "boxySine" function which I named after what it looks like.

This boxy sine, which is really a patched ratio of two sines lets us write the x and y coordinates of the point on the arc without referring to the radius or diverging in any way

Adding x₀ turns out to be superfluous with the definition of boxySine