Hitzer, Tachibana, Buchholz & Yu: Carrier method for pose


Carrier method for the general evaluation and control of pose, molecular conformation, tracking, etc.

by E. Hitzer, K. Tachibana, S. Buchholz, and I. Yu,

Submitted (01.01.2008) to:
Proceedings of Workshop on Clifford Analysis and Applications (WCAA 2007), “Het Pand”, Ghent, Belgium, September 03-05, 2007

Abstract: The four basic geometric objects of points, point pairs, circles and spheres correspond to outer product null-spaces of conformal points in conformal geometric algebra. All basic geometric objects have the same algebraic structure given by a minimal Euclidean blade scalar, vector, bivector, trivector) called carrier, by their scalar radius and center. Wedging with the infinity point we get four flat objects: finite-infinity point pairs, lines, planes and the whole 5D space R^{4,1}.

We state simple, but fully general, algebraic formulas to extract radii, and object specific (positioned and non-positioned) carriers. Quotients allow us to compute object position translators, relative translators, and relative rotors. The radii lead to rescaling operations. A very intuitive combination of these rotors, translators and rescaling finally yields fully general relative motors and matching operators which completely reveal the relative pose and extension of any two objects.

We thus develop a single general algebraic method to quantify and interpolate the relative pose of eight different classes of objects. We are absolutely free to begin with object points or higher order geometric algebra objects. Without extra effort the method can thus be adapted to whatever information is available.

Our method can be universally applied to the conformation of molecules, to human and robot pose problems, to the interpolation of motion, or to aerospace control and virtual reality problems, to name just a few. As an explicit example we apply our framework to the conformation of organic macromolecules: We model the conformation of a peptide which consists of 75 atoms.

Mathematics Subject Classification (2000): Primary 15A66; Secondary 62M45, 2H11, 92C40, 68T40.

Keywords: Geometric algebra, conformal geometric algebra, carrier, Euclidean objects, pose, molecular conformation, tracking.

Available upon request: Email hitzer_at_mech.fukui-u.ac.jp

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