Degree formulae for offset curves
The problem of relating the degree of the offset of a real irreducible affine plane curve given implicitly to the degree of the original variety is approached from three different points of view. The first method of computation is based on the number of intersection points between the offset curve and a generic line. Since the implicit equation of the offset curve is not known, the number of intersection points is computed by counting the points of the original variety that, in an one to one correspondence, generate the intersections between the line and the offset curve. An auxiliary curve is used to make difference between the fake and the non-fake intersection points and, as consequence, to derive a formula for the degree of the offset curve. A deterministic algorithm for computing the offset degree is a consequence of this formula. The second method of computation refers to projective algebraic plane curves represented by homogeneous polynomials. It is based on the resultant of the defining polynomial of the given curve and the polynomial defining generically the previously used auxiliary curve. Finally, it is shown that the hodograph to the given curve leads to an improved formula for the offset degree, which depends on less parameters. Many practical examples are discussed from the point of view of the relationship between their degree and the offset degree: the lemniscata, the ellipses and hyperbolas, the parabola, the circle, few polynomial curves, few rational curves as, for example, the cardioid.
scientific article; zbMATH DE number 2136959
scientific article