Boolean Combination of Circular Arcs using Orthogonal Spheres

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.