Rational Wedge Construction for Polycons and Polypols

Abstract

Each polycon wedge is of the form $\mathrm{W}_{\mathrm{i}}(\mathrm{x,y}) =\mathrm{ k}_{\mathrm{i}}\mathrm{P}^{\mathrm{i}}\mathrm{R}^{\mathrm{i}}/\mathrm{Q}$, where k_i normalizes the polycon wedge to unity at node (x_i, y_i) and the three polynomials are determined so that the properties enumerated in Sect. 1.5 are achieved. We first elaborate on the construction mentioned in Sect.3.1 and then prove that this construction yields the required properties. Polynomial Pⁱ in the numerator is called the opposite factor. It is the product of the linear and quadratic forms which vanish on the sides opposite node i. Polynomial Rⁱ, the other factor in the numerator, is called the adjacent factor. It is unity for all side nodes and for all vertex nodes at the intersection of two linear sides. For these nodes, Pⁱ is of degree m − 2 when the polycon is of order m. A vertex node at the intersection of a linear and a conic side has an opposite factor of degree m − 3. The adjacent factor is the linear form that vanishes on (A;B) in Fig. 5.1. If the EIP of the sides adjacent to node i is at infinity (that is, if point B is on the absolute line in the projective plane), then line (A;B) is parallel to the linear side at vertex i.