Conics in rational cubic Bézier form made simple

We revisit the rational cubic Bézier representation of conics, simplifying and expanding previous works, elucidating their connection, and making them more accessible. The key ingredient is the concept of conic associated with a given (planar) cubic Bézier polygon, resulting from an intuitive geometric construction: Take a cubic semicircle, whose control polygon forms a square, and apply the perspective that maps this square to the given polygon. Since cubic conics come from a quadratic version by inserting a base point, this conic admitting the polygon turns out to be unique. Therefore, detecting whether a cubic is a conic boils down to checking out whether it coincides with the conic associated with its control polygon. These two curves coincide if they have the same shape factors (aka, shape invariants) or, equivalently, the same oriented curvatures at the endpoints. Our results hold for any cubic polygon (with no three points collinear), irrespective of its convexity. However, only polygons forming a strictly convex quadrilateral define conics whose cubic form admits positive weights. Also, we provide a geometric interpretation for the added expressive power (over quadratics) that such cubics with positive weights offer. In addition to semiellipses, they encompass elliptical segments with rho-values over the negative unit interval.