In an optimization problem I have to parametrize a circular arc. Thus far, I have reduced a more general problem to the figure below:
The figure shows a symmetrical circular arc, with chord length L, and the internal angle beta at both end points. The coordinate system has its origin at the left end point. The x axis points along the chord, and the y-axis upward in the image (not pictured, sorry).
I wish to find points on this arc, preferably equally spaced on the arc, but since the angle beta is low (<10° in most cases), points equally spaced on the x-axis would do.
My problem is that the angle beta can be extremely small, zero, or negative. If I were to calculate a radius, it may be extremely large, infinite or complex. I want to avoid this if possible because it would cause numerical problems.
How can I parametrize this arc without calculating the radius, and using Cartesian coordinates with an origin as described above?
I can filter the negative case out with a control structure, if necessary.