  This work presents an initial analysis of using bijective mappings to extend
the Theory of Functional Connections to non-rectangular two-dimensional
domains. Specifically, this manuscript proposes three different mappings
techniques: a) complex mapping, b) projection mapping, and c) polynomial
mapping. In that respect, an accurate least-squares approximated inverse
mapping is also developed for those mappings having no closed-form inverse. The
advantages and disadvantages of using these mappings are highlighted and a few
examples are provided. Additionally, the paper shows how to replace boundary
constraints expressed in terms of a piecewise sequence of functions with a
single function, that is compatible and required by the Theory of Functional
Connections already developed by rectangular domains.
