A locus problem

This wasn't received well at Mathoverflow. So here goes:
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Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB is a right angle, then the locus of points is the circle with the diameter AB. Ref: https://mathpages.com/home/kmath173/kmath173.htm#:~:text=Loci%20of%20Equi%2Dangular%20Points&text=Given%20a%20line%20segment%20AB,circle%20with%20the%20diameter%20AB.

Question: Given two line segments AB and CD lying on same plane, what is the locus of points P such that the sum of the angles APB and CPD is a constant? What are the qualitative features of this locus and how does this locus depend upon the relative position, length and orientation of the two segments and the value of the angle sum?

Note: Numerical calculations indicate the following: when AB and CD are kept fixed near to each other or intersecting and the angle sum varied, (1) if the angle sum is small, the locus is a connected curve that lies far away from the two line segments and surrounding them and shows concavities; (2) for larger values of angle sum, the locus lies closer to the two segments and appears convex. And in some cases, for still larger values of the angle sum, the locus probably breaks into two closed curves. Basically, we seem to have a one-parameter family of curves that go from convex to non-convexr>

An analogous question in 3D (with 2 line segments - that could be either skew or coplanar - and an angle sum) could give surfaces as loci of P.
Picture below contours of the angle sum measured with respect to a pair of short and intersecting line segments near the origin