The fundamental theorems of curves and hypersurfaces in centro-affine geometry

Centro-Equiaffine Differential Invariants of Curve Families

“…In this equality, except of the last determinant, the others is generated by the set (2.2) and in according to induction hypothesis. In Lemma 2.1, if we take 2) and eliminate the zero terms it is obtained that 2) ] generated by the set (2.2).…”

“….., y n = x 1 (n−2) and eliminate the zero terms, it is obtained that 2) x 1 (n) ] generated by the set (2.2). By the induction hypothesis, the set (2.2) is generator set of R < U > G .…”

Creation of a Degraded Layer on the Surface of Photoresist by Radical Irradiation

“…They are invariant under reparameterizations and the action of SL(2, R) and uniquely determine the shape of the curve, up to left multiplication by an element of SL(2, R) (cf [8]). The trajectory is independent of the parameterization if ϑ = 1.…”

“…If x is parameterized by the centro-affine arclength, then (8) reduces to the Hill equation x + kx = 0. As a consequence, many interesting classes of star-shaped curves can be described, for suitable choices of k, in terms of special functions such as confluent hypergeometric functions, Weber-Hermite functions, Mathieu functions and Lamé functions [17,18].…”

“…defined on star-shaped curves x : R →Ṙ 2 in the punctured plane, where s is the centroaffine arclength and k is the centro-affine curvature (see [8] for a comprehensive treatment of curves in centro-affine geometry). Motivation is provided by integrable equations arising from motions of plane curves in centro-affine geometry [2-5, 9, 16] and by the strong similarities with certain particle models on null curves in Minkowski and anti-de Sitter 3-spaces [12,13,15].…”