The λ-Point Map between Two Legendre Plane Curves

Abstract

The λ-point map between two Legendre plane curves, which is a map from the plane into the plane, is introduced. The singularity of this map is studied through this paper and many known plane map singularities are realized as special cases of this construction. Precisely, the corank one and corank two singularities of the λ-point map between two Legendre plane curves are investigated and the geometric conditions for this map to have corank one singularities, such as fold, cusp, swallowtail, lips, and beaks are obtained. Additionally, the geometric conditions for the λ-point map to have a sharksfin singularity, which is a corank two singularity, are obtained.

Singular Surfaces of Osculating Circles in Three-Dimensional Euclidean Space

“…For a map germ f : (U ⊆ R m , 0) → (R n , 0), p ∈ U is a singular point of f . We say p is of co-rank α if and only if min(m, n) − rank(d f p ) = α [19]. For the purpose of discussing the types of singular points, it is necessary to calculate the co-rank of the singular points that appear on the OFB surfaces.…”