Superellipsoid

Abstract: A distance volume is a volume dataset where the value stored at each voxel is the shortest distance to the surface of the object being represented by the volume. Distance volumes are a useful representation in a number of computer graphics applications. We present a technique for generating a distance volume with sub-voxel accuracy from one type of geometric model, a constructive solid geometry (CSG) model consisting of superellipsoid primitives. The distance volume is generated in a two step process. The first step calculates the shortest distance to the CSG model at a set of points within a narrow band around the evaluated surface. Additionally, a second set of points, labeled the zero set, which lies on the CSG model's surface are computed. A point in the zero set is associated with each point in the narrow band. Once the narrow band and zero set are calculated, a fast marching method is employed to propagate the shortest distance and closest point information out to the remaining voxels in the volume. Our technique has been used to scan convert a number of CSG models, producing distance volumes which have been utilized in a variety of computer graphics applications, e.g. CSG surface evaluation, offset surface generation, and 3D model morphing.

Abstract: Funding information National Natural Science Foundation of China, Grant/Award Number: 51679207; Research Grants Council of Hong Kong, Grant/Award Number: GRF Project No. 16205418, CRF Project No. C6012-15G and TBRS Project No. T22-603/15N; the Hong Kong Scholars Program, Grant/Award Number: XJ2018048 Summary Particle morphology plays a key role in affecting physical and mechanical behaviors of granular media. While various mathematical approaches and shape descriptors have been proposed to describe the morphological properties of granular particles, it remains a challenge to effectively incorporate them for efficient discrete modeling of granular materials. This study presents a new poly-superellipsoid-based approach for three-dimensional discrete element method (DEM) modeling of non-spherical convex particles. A uniform mathematical description of 3D poly-superellipsoidal surface is employed to represent a realistic granular particle, which is shown to be versatile and effective in reproducing a wide range of shape features (including elongation, flatness, angularity, and asymmetry) for real particles in nature. A novel optimization approach based on hybrid Levenberg-Marquardt (LM) and Gilbert-Johnson-Keerthi (GJK) algorithms is further developed for efficient and robust contact detection in DEM simulation of poly-superellipsoidal assemblies. Simulations of granular packing and triaxial compression tests show that the proposed approach is generally robust and efficient for both dynamic and quasistatic modeling of granular media.

Abstract: A new model for representing an unorganised 3D data points set is proposed. Based on superquadrics, this model allows to describe the points set with a union of superellipsoids. Two different segmentation and modeling methods are developed in order to determine the whole model: a region growing approach and a split and merge one. This second method leads to a low sensitive model compared to the one obtained by the region growing. The model is simple and compact: only 11 parameters are needed per superellipsoid. It seems promising for 3D object compression and 3D object indexing and retrieval. As the topological relations of the superellipsoids are known, the model can be associated to a graph. The graph theory can thus be used in order to compare and to measure the similarity between 3D objects.