This paper presents a high-capacity data hiding method for 3D polygonal meshes. By slightly modifying the distance from a vertex to its traversed neighbors based on quantization, a watermark (i.e., a string of binary numbers) can be embedded into a polygonal mesh during a mesh traversal process. The impact of embedding can be tuned by appropriately choosing the quantization step. The embedded data is robust against those content-preserving manipulations, such as rotation, uniformly scaling and translation, as well as mantissa truncation of vertex coordinate to a certain degree, but sensitive to malicious manipulations. Therefore, it can be used for authentication and content annotation of polygonal meshes. Compared with the previous work, the capacity of the proposed method is relatively high, tending to 1 bit/vertex. Besides to define the embedding primitive over a neighborhood so as to achieve resistance to substitution attacks, the security is also improved by making it hard to estimate the quantization step from the modified distances. A secret key is used to order the process of mesh traversal so that it is even harder to construct a counterfeit mesh with the same watermark. The numerical results show the efficacy of the proposed method.