Coordinates of a segment formed by the sum of two angles

Question

If I add two arbitrary angles A + D, how do I find the coordinates of the arc formed by GH?

Corners C and B can be swapped.
Corners E and F can be swapped.
As long as A and D get added together.

Let
r = 1
A = 22.5 degrees
D = 45 degrees

Answer 1

In general, say your angles are $\alpha$ and $\beta$. In polar coordinates your red dots will be $$(r,\theta) = (1,0),(1,\alpha+\beta).$$ Convert to Cartesian: $$ (x,y) = (1,0),(\cos(\alpha+\beta),\sin(\alpha+\beta)). $$ If you want the equation of the arc itself, just parametrize the curve. In particular, the parametrization is $$ (x,y) = (\cos t,\sin t)$$ Where $t$ varies in $[0,\alpha+\beta]$.