A public fragile watermarking scheme based on the sensitivity of vertex geometry is proposed for 3D model authentication. In the 3D fragile watermarking embedding, slightly perturbing the positions of a subset of vertices is usually needed to keep them in some predefined relationship with their neighboring vertices. Two problems frequently arise in the embedding stage: the causality problem and the convergence problem. The causality problem arises while the neighboring relationship of a former processed vertex is influenced by the perturbing of its latter processed neighboring vertices. The convergence problem means that the original model has been heavily distorted before some vertices reach the predefined relationship. In this paper, we propose a multi-function vertex embedding method and an adjusting-vertex method to overcome these two problems. The proposed method does not need the original model and watermarks for authentication; moreover, the key for extracting watermarks is relatively smaller than that of previous works. For some artistic or technical models, sometimes it is very important to control the distortion ratio caused by watermark embedding. Our method can control the average distortion by the keys used in watermark embedding. Experimental and analytic results on various kinds of 3D models show the effectiveness of the scheme.