Characteristic conic of rational bilinear map

Bilinear and rational bilinear maps defined by planar quadrilaterals play important roles in computer graphics and geometric design. We show that the inverse of a rational bilinear map has a close relationship with a conic, which is named as ”characteristic conic” in this paper. We show that the characteristic conic of a rational bilinear map can be expressed as the rational quadratic Bézier curve, with its control points and weights in a closed form. Furthermore, we show that the characteristic conic is an envelope of two one-parameter families of straight lines, and all characteristic conics of a fixed quadrilateral form a one-parameter family.