Carrier method for the general evaluation and control of pose, molecular conformation, tracking, and the like
A general framework for processing and evaluating geometric object information is developed in terms of the conformal model of the three-dimensional Euclidean geometry. This is done using the conformal geometric algebra \(\mathcal{C}\ell_{4,1} \simeq \mathcal{C}\ell_{1,1}\otimes \mathcal{C}\ell_{3,0}\), where \(\mathcal{C}\ell_{p,q}\) is the Clifford algebra of the quadratic space \(\mathbb{R}^{p,q}\). The conformal model has two classes of geometric objects: (i) four spheres \(S\) of dimensions zero (points), one (point pairs), two (circles) and three (spheres), and (ii) four flat objects \(F\) of dimensions zero (finite-infinite point pair), one (line), two (plane) and three (the embedded three-dimensional euclidean space). The authors derive general expressions for the spheres \(S\) and the flat objects \(F\) that are distinguished only by the corresponding scalar, vector, bivector and trivector Euclidean carriers, and obtain formulas to compute these carriers and the positions and (for the spheres) the radius of all objects. There are several potential applications for this algebraic framework, and the authors discuss one application to computational biochemistry.
scientific article; zbMATH DE number 5593791