A latin bitrade ◇(T◇, T◇⊗) is a pair of partial latin squares that define the difference between two arbitrary latin squares L ⊇ T and L⊗⊇ T⊗) of the same order. A 3-homogeneous bitrade (T◇, T⊗) has three entries in each row, three entries in each column, and each symbol appears three times in T◇. Cavenagh  showed that any 3-homogeneous bitrade may be partitioned into three transversals. In this paper we provide an independent proof of Cavenagh's result using geometric methods. In doing so we provide a framework for studying bitrades as tessellations in spherical, euclidean or hyperbolic space. Additionally, we show how latin bitrades are related to finite representations of certain triangle groups.