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On the Geometry of Point-Plane Constraints on Rigid-Body Displacements

On the Geometry of Point-Plane Constraints on Rigid-Body Displacements In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Applicandae Mathematicae Springer Journals

On the Geometry of Point-Plane Constraints on Rigid-Body Displacements

Acta Applicandae Mathematicae , Volume 116 (2) – Jul 23, 2011

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References (20)

Publisher
Springer Journals
Copyright
Copyright © 2011 by Springer Science+Business Media B.V.
Subject
Mathematics; Theoretical, Mathematical and Computational Physics; Computer Science, general; Mechanics; Mathematics, general; Statistical Physics, Dynamical Systems and Complexity
ISSN
0167-8019
eISSN
1572-9036
DOI
10.1007/s10440-011-9634-6
Publisher site
See Article on Publisher Site

Abstract

In this paper the rigid-body displacements that transform a point in such a way that it remains on a particular plane are studied. These sets of rigid displacements are referred to as point-plane constraints and are given by the intersection of the Study quadric of all rigid displacements with another quadric in 7-dimensional projective space. The set of all possible point-plane constraints comprise a Segre variety. Two different classes of problems are investigated. First instantaneous kinematics, for a given rigid motion there are points in space which, at some instant, have no torsion or have no curvature to some order. The dimension and degrees of these varieties are found by very simple computations. The corresponding problems for point-sphere constraints are also found. The second class of problems concern the intersections of several given constraints. Again the characteristics of these varieties for different numbers of constraints are found using very simple techniques.

Journal

Acta Applicandae MathematicaeSpringer Journals

Published: Jul 23, 2011

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