A NEW REPRESENTATION FOR QUARTIC CURVES AND COMPLETE SETS OF GEOMETRIC INVARIANTS
Mustafa Ünel, WILLIAM A. WOLOVICH · International Journal of Pattern Recognition and Artificial Intelligence · 1999
Many free-form object boundaries can be modeled by quartics with bounded zero sets. The fact that any nondegenerate closed-bounded algebraic curve of even degree n=2p can be expressed as the product of p conics, which are real ellipses, plus a remaining polynomial of degree n-2,12 can be utilized to express a nondegenerate quartic as the product of two leading ellipses plus a third conic which might be either a closed curve (an ellipse) or an open curve (a hyperbola). However, it can be shown that the leading ellipses can be modified with appropriate constants by constraining the third conic to be a circle, thus implying a 2-ellipse and 1-circle; i.e. an elliptical-circular(E2C)representation of the quartic. The use of such representations is to simplify the analysis of quartics by exploiting the well-known properties of conics and to develop a set of functionally independent geometric invariants for recognition purposes. Also, it is shown that the underlying Euclidean transformation between two configurations of the same quartic can be determined using the centers of the three conics.