Consider a set S of points in the plane in convex position, where each point has an integer label from {0,1,…,n-1}. This naturally induces a labeling of the edges: each edge (i,j) is assigned label i+j, modulo n. We propose the algorithms for finding large non--crossing harmonic matchings or paths, i. e. the matchings or paths in which no two edges have the same label. When the point labels are chosen uniformly at random, and independently of each other, our matching algorithm with high probability (w.h.p.) delivers a nearly--perfect matching, a matching of size n/2 - O(n1/3lnn).