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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-convex.

contour plots for angle sum when the two reference line segments are short and intersecting

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.

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COMMENT.-The nine variables at play (eight coordinates and the angle sum) can vary in many ways. For example, if $AB$ and $CD$ are defined by $(0,0),(1,1)$ and $(2,0),(3,1)$ respectively, the curves change much of shape as the attached figure shows.Towards $137^{\circ}$ the curve is reduced to a small arch and for greater values of the angles sum the curve simply disappear which suggests that there is not solution.

I tell you this because it seems that this example don't agree entirely with the Note in your post. The equation of the locus for this example is $$\tan^{-1}\left(\left|\frac{y-x}{x^2+y^2-x-y}\right|\right)+\tan^{-1}\left|\left(\frac{y-x+2}{x^2+y^2-5x+y+6}\right|\right)=\text { angle sum }$$

enter image description here

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  • $\begingroup$ thanks. i couldn't quite reproduce the curves you show - my method was to plot the angle sum at closely spaced points and to draw contour curves. the degree 1 curve above, i could get but for other values of angle sum the contour lines i got show more symmetry than curves above (after all, the two line segments are parallel). the region where the angle sum has large values is quite small. $\endgroup$ Dec 3, 2023 at 10:55
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This not an answer, but too long for a comment. There are two versions of you question depending on whether one uses directed angles or not. This is seen in your simpler example of a single segment where one gets the full circle or a lens shape accordingly. You are concerned with the latter case but I will begin with the former since it is simpler. I assumed that the first segment is the unit interval in the $x$-axis while second joins $(p,q)$ with $(p_1,q_1)$. Your curve is then the zero set of an algebraic function which can be computed explicitly--it is a quotient of a cubic by the square root of a polynomial of degree $8$. The corresponding zero sets can easily be visualised in concrete cases (I used Mathematica). For some special values of the angle, one gets simpler curves (explicitly, a cubic).

In the case you are interested in, the curves can have corners at the endpoints of the segments but I have not looked into that in detail.

By the way, this is an interesting question (which, given the nature of the internet and this site in particular, might explain the downvote).

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  • $\begingroup$ thank you. i am not trained in algebraic geometry so i am quite surprised by the numerical evidence that shows a curve going from closed convex to non-convex and even disconnected when a single parameter changes continuously (here, that parameter is the angle sum - the two line segments kept fixed of course). as for the downvote, it can be undone by an upvote - or two. $\endgroup$ Dec 3, 2023 at 10:53

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