  Sherman-Zelevinsky and Cerulli constructed canonically positive bases in
cluster algebras associated to affine quivers having at most three vertices.
Both constructions involve cluster monomials and normalized Chebyshev
polynomials of the first kind evaluated at a certain "imaginary" element in the
cluster algebra. Using this combinatorial description, it is possible to define
for any affine quiver $Q$ a set $\mathcal B(Q)$ which is conjectured to be the
canonically positive basis of the acyclic cluster algebra $\mathcal A(Q)$.
  In this article, we provide a geometric realization of the elements in
$\mathcal B(Q)$ in terms of the representation theory of $Q$. This is done by
introducing an analogue of the Caldero-Chapoton cluster character where the
usual quiver Grassmannian is replaced by a constructible subset called
transverse quiver Grassmannian.
