Computer Science – Computational Geometry
Scientific paper
2011-12-27
Computer Science
Computational Geometry
Scientific paper
The Cayley configuration space of a 1-dof linkage in 2D is the set of realizable distance-values for an independent non-edge f. We study (a) the Cayley size, i.e., the number of intervals, (b) the Cayley computational complexity of computing the interval endpoints, as a function of the number of intervals, (c) the Cayley algebraic complexity of describing the interval end points. In both parts of the paper, we restrict ourselves to 1-dof linkages obtained by dropping a bar from a minimally rigid, tree-decomposable linkage. These linkages are widely used in engineering and CAD, because they are quadratically-radical solvable (QRS): for rational bar lengths, the point coordinates of all standard Cartesian realizations belong to an extension field over the rationals obtained by nested square-roots. Additionally, if a relative local orientation is specified for each point, there is a linear time algorithm to compute the point coordinates of the unique cartesian realization satisfying the specified orientations (if one exists). In Part I of this paper, we formally characterize the interval endpoints of the Cayley configuration space by their corresponding realizations, and give an algorithmic description of the Cayley configuration space. We then consider (b) and (c) above for linkages with low Cayley complexity. It follows that for such a linkage, we can find a path of continuous motion (provided one exists) between two given realizations in time linear in a natural measure of the length of the path, and the number of such paths is at most two. In addition, we give a natural, minimal solution type, i.e. a minimal set of local orientations, whose specification guarantees Cayley size of 1 and O(|V|^2) Cayley computational complexity. Specifying fewer local orientations results in a superpolynomial blow-up provided P is different from NP.
Gao Heping
Sitharam Meera
Wang Menghan
No associations
LandOfFree
Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part I: Structure and Extreme Points does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part I: Structure and Extreme Points, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cayley Configuration Spaces of 1-dof Tree-decomposable Linkages, Part I: Structure and Extreme Points will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-37610