  This paper proves the existence of an intersection-dimension formula for
preprojective modules over path algebras of type $\widetilde{D}_n$. Identical
intersection-dimension formulas have previously been provided for modules over
path algebras of type $A_n, D_n,$ and $\widetilde{A}_n$ due to Schiffler as
well as He, Zhou, and Zhu. These modules can be represented geometrically by
some set of curves on special surfaces. The intersection-dimension formula is
an equality of the intersection number between two curves and the dimensions of
the first extension spaces between the two modules they represent. This paper
takes a direct approach to proving the formula utilizing the known structure of
the Auslander-Reiten quiver of type $\widetilde{D}_n$. Future work will extend
the formula to the entire module category (not just the preprojective modules)
over path algebras of type $\widetilde{D}_n$.
