Characteristics of conic segments in Bézier form

The rational Bézier form has become a standard in CAD-CAM packages and data exchange formats, because it encompasses both conic segments (in the quadratic case) and general free-form geometry. We present several results on the relationship between the quadratic rational Bézier form and the classical definition of conics in terms of their characteristics, such as foci, centre, axis and eccentricity. First, we recall a simple geometric procedure to compute arbitrary conic segments of given focus in Bézier form. Second, from this procedure we derive the geometric characteristics of a given Bézier conic in a straightforward manner, by employing complex arithmetic. For a central conic, a simple quadratic equation defines the foci location, and its solution furnishes not only an explicit formula for the foci, but also for the centre, axis direction and linear eccentricity.