  Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension
two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale
dual of $A$. In this paper we compute the Euclidean distance degree and polar
degrees of $X_A$ (along with other associated invariants) combinatorially
working from the matrix $B$. Our approach allows for the consideration of
examples that would be impractical using algebraic or geometric methods. It
also yields considerably simpler computational formulas for these invariants,
allowing much larger examples to be computed much more quickly than the
analogous combinatorial methods using the matrix $A$ in the codimension two
case.
