Superellipse

Examples of superellipses for

A superellipse, also known as a Lamé curve after Gabriel Lamé, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but defined by an equation that allows for various shapes between a rectangle and an ellipse.

In two dimensional Cartesian coordinate system, a superellipse is defined as the set of all points on the curve that satisfy the equationwhere and are positive numbers referred to as semi-diameters or semi-axes of the superellipse, and is a positive parameter that defines the shape. When , the superellipse is an ordinary ellipse. For , the shape is more rectangular with rounded corners, and for , it is more pointed.[1] [2][3]


In the polar coordinate system, the superellipse equation is (the set of all points on the curve satisfy the equation):

  1. ^ Shi, Pei-Jian; Huang, Jian-Guo; Hui, Cang; Grissino-Mayer, Henri D.; Tardif, Jacques C.; Zhai, Li-Hong; Wang, Fu-Sheng; Li, Bai-Lian (15 October 2015). "Capturing spiral radial growth of conifers using the superellipse to model tree-ring geometric shape". Frontiers in Plant Science. 6: 856. doi:10.3389/fpls.2015.00856. ISSN 1664-462X. PMC 4606055. PMID 26528316.
  2. ^ 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. ^ 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.