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The 3D facial geometry contains ample information about human facial expressions. Such information is invariant to pose and lighting conditions, which have imposed serious hurdles on many 2D facial analysis problems. In this paper, we perform person and gender independent facial expression recognition based on properties of the line segments connecting certain 3D facial feature points. The normalized distances and slopes of these line segments comprise a set of 96 distinguishing features for recognizing six universal facial expressions, namely anger, disgust, fear, happiness, sadness, and surprise. Using a multi-class support vector machine (SVM) classifier, an 87.1% average recognition rate is achieved on the publicly available 3D facial expression database BU-3DFE. The highest average recognition rate obtained in our experiments is 99.2% for the recognition of surprise. Our result outperforms the result reported in the prior work, which uses elaborately extracted primitive facial surface features and an LDA classifier and which yields an average recognition rate of 83.6% on the same database.