Evolutoids of the Mixed-Type Curves

Abstract

The evolutoid of a regular curve in the Lorentz-Minkowski plane ${ℝ}_1^2$ is the envelope of the lines between tangents and normals of the curve. It is regarded as the generalized caustic (evolute) of the curve. The evolutoid of a mixed-type curve has not been considered since the definition of the evolutoid at lightlike point can not be given naturally. In this paper, we devote ourselves to consider the evolutoids of the regular mixed-type curves in ${ℝ}_1^2$ . As the angle of lightlike vector and nonlightlike vector can not be defined, we introduce the evolutoids of the nonlightlike regular curves in ${ℝ}_1^2$ and give the conception of the $\sigma$ -transform first. On this basis, we define the evolutoids of the regular mixed-type curves by using a lightcone frame. Then, we study when does the evolutoid of a mixed-type curve have singular points and discuss the relationship of the type of the points of the mixed-type curve and the type of the points of its evolutoid.