Superellipsoid

Superellipsoid collection with exponent parameters, created using POV-Ray. Here, e = 2/r, and n = 2/t (equivalently, r = 2/e and t = 2/n).[1]

In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter , and whose vertical sections through the center are superellipses with the squareness parameter . It is a generalization of an ellipsoid, which is a special case when .[2]

Superellipsoids as computer graphics primitives were popularized by Alan H. Barr (who used the name "superquadrics" to refer to both superellipsoids and supertoroids).[2][3] In modern computer vision and robotics literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics.[4][5]

Superellipsoids have an rich shape vocabulary, including cuboids, cylinders, ellipsoids, octahedra and their intermediates.[6] It becomes an important geometric primitive widely used in computer vision,[6][5][7] robotics,[4] and physical simulation.[8] The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape.[6] Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available.[9] This makes it a desirable geometric primitive for robot grasping, collision detection, and motion planning.[4]

  1. ^ "POV-Ray: Documentation: 2.4.1.11 Superquadric Ellipsoid".
  2. ^ a b Barr (1981). "Superquadrics and Angle-Preserving Transformations". IEEE Computer Graphics and Applications. 1 (1): 11–23. doi:10.1109/MCG.1981.1673799. ISSN 1558-1756. S2CID 9389947.
  3. ^ Barr, A.H. (1992), Rigid Physically Based Superquadrics. Chapter III.8 of Graphics Gems III, edited by D. Kirk, pp. 137–159
  4. ^ a b c Ruan, Sipu; Wang, Xiaoli; Chirikjian, Gregory S. (2022). "Collision Detection for Unions of Convex Bodies With Smooth Boundaries Using Closed-Form Contact Space Parameterization". IEEE Robotics and Automation Letters. 7 (4): 9485–9492. doi:10.1109/LRA.2022.3190629. ISSN 2377-3766. S2CID 250543506.
  5. ^ a b Paschalidou, Despoina; Van Gool, Luc; Geiger, Andreas (2020). "Learning Unsupervised Hierarchical Part Decomposition of 3D Objects from a Single RGB Image". 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 1057–1067. doi:10.1109/CVPR42600.2020.00114. ISBN 978-1-7281-7168-5. S2CID 214634317.
  6. ^ a b c Liu, Weixiao; Wu, Yuwei; Ruan, Sipu; Chirikjian, Gregory S. (2022). "Robust and Accurate Superquadric Recovery: A Probabilistic Approach". 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 2666–2675. arXiv:2111.14517. doi:10.1109/CVPR52688.2022.00270. ISBN 978-1-6654-6946-3. S2CID 244715106.
  7. ^ Paschalidou, Despoina; Ulusoy, Ali Osman; Geiger, Andreas (2019). "Superquadrics Revisited: Learning 3D Shape Parsing Beyond Cuboids". 2019 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). pp. 10336–10345. arXiv:1904.09970. doi:10.1109/CVPR.2019.01059. ISBN 978-1-7281-3293-8. S2CID 128265641.
  8. ^ Lu, G.; Third, J. R.; Müller, C. R. (2012-08-20). "Critical assessment of two approaches for evaluating contacts between super-quadric shaped particles in DEM simulations". Chemical Engineering Science. 78: 226–235. Bibcode:2012ChEnS..78..226L. doi:10.1016/j.ces.2012.05.041. ISSN 0009-2509.
  9. ^ Ruan, Sipu; Chirikjian, Gregory S. (2022-02-01). "Closed-form Minkowski sums of convex bodies with smooth positively curved boundaries". Computer-Aided Design. 143: 103133. arXiv:2012.15461. doi:10.1016/j.cad.2021.103133. ISSN 0010-4485. S2CID 229923980.