Leaf image identification is a significant and challenging application of computer vision and image processing. A central issue associated with this task is how to effectively and efficiently describe the leaf images and measure their similarities. In this paper, a novel shape descriptor termed R-angle is proposed. R-angle describes the curvature of the contour by measuring the angle between the intersections of the shape contour with a circle of radius R centered at points sampled around the contour. It is intrinsically invariant to group transforms including scaling, rotation and translation. Varying the parameter R of the proposed R-angle naturally introduces the notation of scale, which we leverage to provide a coarse-to-fine description of the local curvature. A local scale arrangement is proposed by taking the distance between each contour point and the center of the shape to be the maximum scale for a given contour point. Two matching schemes, including L1-norm matching and dynamic programming based matching, are applied to measure the similarities of the leaf shapes. The retrieval experiments conducted on two challenging leaf image datasets indicate that the proposed method significantly outperforms the state-of-the-art methods for leaf identification. An additional experiment on an animal dataset also indicates its potential for general shape recognition.