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 σ -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.
Highlights
The caustic, called evolute, is an important study object in physics and nonlinear sciences
The caustics have wide applications in many other research fields, such as optics, mechanics, and electromagnetism; they have drawn the attention of many scientists
In [1], in order to confirm the location of caustic regression points, the researchers gave a new geometric variational criterion
Summary
The caustic, called evolute, is an important study object in physics and nonlinear sciences. From the geometric point of view, the evolute of a curve in the Euclidean plane is usually defined by the locus of centres of osculating circles of the base curve or the envelope of the normal lines of the base curve. As it is an important object of studying in classical differential geometry, it is widely considered by many scholars (see [5, 6]). We consider the evolutoids of the regular mixed-type curves in R21 and study their propositions. All maps and manifolds in this paper are infinitely differentiable
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