Geodesic-spin-image (GSI), a new three-dimensional(3D) local shape descriptor, was proposed to solve the problems of ambiguous Euclidean measure and insufficient shape information description when utilizing the Spin Image (SI) descriptor. The local support region of GSI was defined by Geodesic measure instead of Euclidean measure. Besides, three new features were also introduced to compensate the azimuth information loss while these features were long axis length and short axis length of the support region's projection onto the tangent plane of the given point, and the distance from the centroid of the support region to the tangent plane. Based on the GSI, matching strategy of SI was modified through a combination of coarse matching utilizing the three new features and fine matching utilizing SI defined by geodesic measure, so as to improve the matching efficiency. Simulation experiment results show that compared with the original SI method, the proposed GSI descriptor is proved to provide more shape information and be more discriminative, and its matching strategy is more efficient.