Center for Image in Science and Art _ UL
Calculation of the minimum distance between ellipsoids and superellipsoids
Author: Daniel Simões Lopes.
Date: March 2010.
Description: The calculation of the minimum distance between surfaces plays an important role in computational mechanics, namely, in the study of mechanisms composed by rigid parts modeled as (super)ellipsoidal surfaces. Computing the minimum distance between convex surfaces is crucial to determine if the rigid objects of a mechanism are in contact among themselves or with the surrounding environment. The exhibited equations formulate the geometric conditions that characterize the pair of points which declare the minimum distance between surfaces. Such points share a common normal vector. The proposed mathematical framework to solve the minimum distance problem relies on simple algebraic and differential geometry, vector calculus, and on the C2 continuous implicit representations of the surfaces. The employed methodology establishes a set of collinear and orthogonal constraints between vectors defining the contacting surfaces that, allied with loci constraints. Several examples are represented for both ellipsoids (central image, top row) and superellipsoids (central image, bottom row).
Technique: Computer generated image.
Source: D.S. Lopes, M.T. Silva, J.A. Ambrósio, and P. Flores, A mathematical framework for contact detection between quadric and superquadric surfaces, Multibody System Dynamics, 24(3): 255-280, 2010. (DOI: 10.1007/s11044-010-9220-0)
Image and caption provided by: Daniel Simões Lopes, IDMEC/IST – TULisbon
Calculation of the minimum distance between ellipsoids and superellipsoids
Author: Daniel Simões Lopes.
Date: March 2010.
Description: The calculation of the minimum distance between surfaces plays an important role in computational mechanics, namely, in the study of mechanisms composed by rigid parts modeled as (super)ellipsoidal surfaces. Computing the minimum distance between convex surfaces is crucial to determine if the rigid objects of a mechanism are in contact among themselves or with the surrounding environment. The exhibited equations formulate the geometric conditions that characterize the pair of points which declare the minimum distance between surfaces. Such points share a common normal vector. The proposed mathematical framework to solve the minimum distance problem relies on simple algebraic and differential geometry, vector calculus, and on the C2 continuous implicit representations of the surfaces. The employed methodology establishes a set of collinear and orthogonal constraints between vectors defining the contacting surfaces that, allied with loci constraints. Several examples are represented for both ellipsoids (central image, top row) and superellipsoids (central image, bottom row).
Technique: Computer generated image.
Source: D.S. Lopes, M.T. Silva, J.A. Ambrósio, and P. Flores, A mathematical framework for contact detection between quadric and superquadric surfaces, Multibody System Dynamics, 24(3): 255-280, 2010. (DOI: 10.1007/s11044-010-9220-0)
Image and caption provided by: Daniel Simões Lopes, IDMEC/IST – TULisbon