I have a set of $n$ points in the plane. I have a set of distances given for pairs of points and a set of angles given for triplets of points. How do I know if the point set is defined apart from translation and rotation?
This sounds like something that is too general and/or solved >200 years ago, but I cannot find a solution online.
If only a set of distances is given, I know, that the minimal set of distances gives a Laman graph, so I can solve this multiple ways (Henneberg construction, pebble algorithms, rigidity percolation).
In some cases it is possible to convert angles into corresponding distances and trace back the problem to Laman graphs. But I'm still stuck, because there are some edge cases (e.g. Laman graphs without triangles, angles given for triangles where no distances can be deduced without using akin angles) where my ideas fail.
If only one angle is given (apart from some degenerated cases), the set is rigid if the three points defining the angle are not from the same rigid subset. But angles can be redundant more "trickily" than distances, so I think my best hope is if I can convert all angles to corresponding distances somehow.
Also I hope this can be solved in a graph-y way, e.g. by treating angles as directed hyper-edges, because I'm more familiar with discrete maths than classic geometry. (I may understand the solution anyways, but when I started working with circles and lines and I got much less effective.)
edit: by angle given by a triplet ( $A, B, C$ ) of points I mean the angle of $AB$ line with $BC$ line.
It sounds like you're looking for the notion of "direction-length rigidity". In this paper the authors give an inductive characterization of direction-length rigid "mixed graphs" (where there are two kinds of edges, corresponding to direction and length constraints) using the analogues of the Henneberg operations that you seem to be familiar with. This paper discusses the direction-length analogue of global rigidity and it also contains a nice summary of the results in the other paper.