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The performance of automatic 3-D face recognition can be significantly improved by coping with the nonrigidity of the facial surface. In this paper, we propose a geodesic polar parameterization of the face surface. With this parameterization, the intrinsic surface attributes do not change under isometric deformations and, therefore, the proposed representation is appropriate for expression-invariant 3-D face recognition. We also consider the special case of an open mouth that violates the isometry assumption and propose a modified geodesic polar parameterization that also leads to invariant representation. Based on this parameterization, 3-D face recognition is reduced to the classification of expression-compensated 2-D images that can be classified with state-of-the-art algorithms. Experimental results verify theoretical assumptions and demonstrate the benefits of the geodesic polar parameterization on 3-D face recognition.