Rosen, Zvi

In this dissertation, we develop analogues to various notion in rigidity theory in the case of angular constraints. Initially, we were interested in angle dependencies. However, by defining a chromatic graph determined by the point-line incidence or angle graph of a given angularity, we were able to determine an assortment of properties. We provide a conjectured angular rigidity matroid involving count matroids, tranversal matroids, and traditional rigidity matroids. We consider realizations of chromatic graphs in R2 as well as C similar to the work in [3]. We extend the notions of pure conditions and infinitesimal motions using the chromatic rigidity matrix by applying techniques from algebra geometric as well as classical geometric results, such as Thales’ theorem. Some realizations I computed inspired curiosity in the space of realizations of angle-constrained graphs. We generate uniformly random sets of angle constraints to consider the space of realizations given these angle sets. We provide some results for the maximum number of possible realizations for some chromatic graphs on four vertices. We conclude with some directions for further research to develop our notions of angle-rigid graphs and their properties.