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Q  is a quiver of typeif its graph is of affine typeand if its arrows have a certain orientation. We develop a bijection between the set of indecomposable  kQ -modules whose dimension vectors are positive real roots of the root system associated to  Q  and a certain set of planar curves. We prove that the number of self-intersections of the curve which corresponds to the module  M  is equal to the dimension of. We also prove that, for many pairs of modules ( M ,  N ), the number of intersections between the corresponding two curves is equal to the dimension of, where  C  is the cluster category of  kQ -mod.

Computation of extension spaces for the path algebra of type Ã ( n - 1,1) using planar curves