Sampling Superquadric Point Clouds with Normals

Abstract: Superquadrics provide a compact representation of common shapes and have been used both for object/surface modelling in computer graphics and as object-part representation in computer vision and robotics. Superquadrics refer to a family of shapes: here we deal with the superellipsoids and superparaboloids. Due to the strong non-linearities involved in the equations, uniform or close-to-uniform sampling is not attainable through a naive approach of direct sampling from the parametric formulation. This is specially true for more `cubic' superquadrics (with shape parameters close to $0.1$). We extend a previous solution of 2D close-to-uniform uniform sampling of superellipses to the superellipsoid (3D) case and derive our own for the superparaboloid. Additionally, we are able to provide normals for each sampled point. To the best of our knowledge, this is the first complete approach for close-to-uniform sampling of superellipsoids and superparaboloids in one single framework. We present derivations, pseudocode and qualitative and quantitative results using our code, which is available online.