Abstract
Typically we do not add objects in conformal geometric algebra (CGA), rather we apply operations that preserve grade, usually via rotors, such as rotation, translation, dilation, or via reflection and inversion. However, here we show that direct linear interpolation of conformal geometric objects can be both intuitive and of practical use. We present a method that generates useful interpolations of point pairs, lines, circles, planes and spheres and describe algorithms and proofs of interest for computer vision applications that use this direct averaging of geometric objects.
Highlights
In this paper we will look at adding conformal geometric algebra (CGA) objects and adjusting the resulting multivectors to produce useful interpolations of the objects
We will use the standard extension of the 3D geometric algebra, where our 5D CGA space is made up of the standard spatial basis vectors {ei} i = 1, 2, 3, plus two additional basis vectors, e and ewith signatures, e2 = 1, e2 = −1
This paper has shown how we are able to add multiples of conformal objects by factoring the resulting multivector into a scalar plus 4-vector term and a valid geometric object
Summary
In this paper we will look at adding CGA objects and adjusting the resulting multivectors to produce useful interpolations of the objects. We will present a general technique that is valid for all geometric objects of grade 2 or above. This technique uses the decompositions presented in [6]. The objects we work with here will be CGA objects unless explicitly stated otherwise. We will use the standard extension of the 3D geometric algebra, where our 5D CGA space is made up of the standard spatial basis vectors {ei} i = 1, 2, 3, plus two additional basis vectors, e and ewith signatures, e2 = 1, e2 = −1.
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