This paper presents a novel method of velocity field estimation for the points on moving contours in a 2-D image sequence. The method determines the corresponding point in a next image frame by considering the curvature change of a given point on the contour. In traditional methods, there are errors in optical flow estimation for the points which have low curvature variations since those methods compute solutions by approximating normal optical flow. The proposed method computes optical flow vectors of contour points minimizing the curvature changes. As a first step, snakes are used to locate smooth curves in 2-D imagery. Thereafter, the extracted curves are tracked continuously. Each point on a contour has a unique corresponding point on the contour in the next frame whenever the curvature distribution of the contour varies smoothly. The experimental results showed that the proposed method computes accurate optical flow vectors for various moving contours.