Abstract
In this paper we will address the problem of recovering covariant transformations between objects—specifically; lines, planes, circles, spheres and point pairs. Using the covariant language of conformal geometric algebra (CGA), we will derive such transformations in a very simple manner. In CGA, rotations, translations, dilations and inversions can be written as a single rotor, which is itself an element of the algebra. We will show that the rotor which takes a line to a line (or plane to a plane etc) can easily be formed and we will investigate the nature of the rotors formed in this way. If we can recover the rotor between one object and another of the same type, a useable metric which tells us how close one line (plane etc) is to another, can be a function of how close this rotor is to the identity. Using these ideas, we find that we can define metrics for a number of common problems, specifically recovering the transformation between sets of noisy objects.
Highlights
In this paper we will address the problem of recovering covariant transformations between objects—; lines, planes, circles, spheres and point pairs
We will use the standard extension of the 3D geometric algebra, where our 5D conformal geometric algebra (CGA) space is made up of the standard spatial basis vectors
Taking the positive or negative square root for β changes the sign of the rotor, which makes no difference to the transformation. These expressions hold for all CGA objects: lines, planes, circles, spheres, point pairs
Summary
Our primary aim in this paper is to simultaneously estimate the rotation and translation that takes one object (line to line/circle to circle/plane to plane/sphere to sphere/point-pair to point-pair) to another. There are many methods that estimate rigid body transformations with points [1,2,3,4]. In [5] the authors estimate a general rotor between arbitrary objects using the idea of carriers—while interesting, this method lacks simplicity and does not deal directly with the objects themselves. This article is part of the Topical Collection on Proceedings of AGACSE 2018, IMECCUNICAMP, Campinas, Brazil, edited by Sebastia Xambo-Descamps and Carlile Lavor. 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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