  We derive a simple bijection between geometric plane perfect matchings on
$2n$ points in convex position and triangulations on $n+2$ points in convex
position. We then extend this bijection to monochromatic plane perfect
matchings on periodically $k$-colored vertices and $(k+2)$-gonal tilings of
convex point sets. These structures are related to a generalization of
Temperley-Lieb algebras and our bijections provide explicit one-to-one
relations between matchings and tilings. Moreover, for a given element of one
class, the corresponding element of the other class can be computed in linear
time.
