  Construction of the diagrammatic version of the affine Temperley-Lieb algebra
of type $\widetilde{A_N}$ as a subring of matrices over the Laurent polynomials
is given. We move towards geometrical understanding of cellular structure of
the Temperley-Lieb algebra. We represent its center as a coordinate ring of the
certain affine algebraic variety and describe this variety constructing its
desingularization.
