Line–Cyclide Intersection and Colinear Point Quadruples in the Double Conformal Model

In this paper, we look at using the double conformal model for ray tracing and for tool positioning in Computer Numerically Controlled (CNC) machining. In particular, we explore the intersection of a line with a cyclide in the double conformal model, and how to extract the four points from the resulting colinear point quadruple. Further, we show how to directly construct a colinear point quadruple from four points, and we show how to find the line containing the points of a colinear point quadruple. We also briefly touch on barycentric coordinates and affine combinations in DCGA. Finally, we discuss applications of the double conformal model in ray tracing and in CNC machining.