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Designing a point cloud upsampler, which aims to generate a clean and dense point cloud given a sparse point representation, is a fundamental and challenging problem in computer vision. A line of attempts achieves this goal by establishing a point-to-point mapping function via deep neural networks. However, these approaches are prone to produce outlier points due to the lack of explicit surface-level constraints. To solve this problem, we introduce a novel surface regularizer into the upsampler network by forcing the neural network to learn the underlying parametric surface represented by bicubic functions and rotation functions, where the new generated points are then constrained on the underlying surface. These designs are integrated into two different networks for two tasks that take advantages of upsampling layers -- point cloud upsampling and point cloud completion for evaluation. The state-of-the-art experimental results on both tasks demonstrate the effectiveness of the proposed method. The implementation code will be available at https://github.com/corecai163/PSCU.