Abstract
An orthogonal sphere representation of arcs on spatial circles can be used to compactly perform Boolean combinations of such arcs. We formulate this using conformal geometric algebra, of which the oriented nature allows both minor and major arcs to be treated. Easily computable quantities discriminate the cases of relative positions. An application in the first stages of a problem in Discretizable Molecular Distance Geometry is included. We give a suggestion on how to extend this characterization by orthogonal spheres to the manifolds of arcs in the subsequent stages, using probabilistic eigenspheres of the distributions.
Highlights
The central problem in distance geometry [10] is the determination of a spatial configuration of points based on their given mutual distances
In the Discretizable Molecular Distance Geometry Problem (DMDGP) [8], it is assumed that a few more distances between neighboring atoms along the chain are known as intervals [7]
We show that there is a way to represent arcs in conformal geometric algebra (CGA) by means of orthogonal spheres to the circle they reside on
Summary
The central problem in distance geometry [10] is the determination of a spatial configuration of points based on their given mutual distances. It was originated by Menger [11]. The field has practical applications for detecting molecular structure of proteins, since nuclear magnetic resonance techniques exist to determine distances between neighboring atoms [1]. As a start to an algorithmic solution to this puzzle, simplifications have been made. It is for instance assumed that bond lengths and angles along the protein carbon backbone are known exactly (unit lengths, and 120 degree angles). In the Discretizable Molecular Distance Geometry Problem (DMDGP) [8], it is assumed that a few more distances between neighboring atoms along the chain are known as intervals [7]. This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECCUNICAMP, Campinas, Brazil, edited by Sebastia Xambo-Descamps and Carlile Lavor. ∗Corresponding author
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