Abstract
In this paper, we analyze involutes of pseudo-null curves in Lorentz–Minkowski 3-space. Pseudo-null curves are spacelike curves with null principal normals, and their involutes can be defined analogously as for the Euclidean curves, but they exhibit properties that cannot occur in Euclidean space. The first result of the paper is that the involutes of pseudo-null curves are null curves, more precisely, null straight lines. Furthermore, a method of reconstruction of a pseudo-null curve from a given null straight line as its involute is provided. Such a reconstruction process in Euclidean plane generates an evolute of a curve, however it cannot be applied to a straight line. In the case presented, the process is additionally affected by a choice of different null frames that every null curve allows (in this case, a null straight line). Nevertheless, we proved that for different null frames, the obtained pseudo-null curves are congruent. Examples that verify presented results are also given.
Highlights
An involute of a curve in Euclidean plane E2 is a curve to which all tangent lines of the initial curve at corresponding points are orthogonal
Once we have proved that the involutes of a pseudo-null curve are null straight lines, we address with the inverse process that can be expressed with the Question
In the present paper we develop the related theory for the case of pseudo-null curves, which has not been previously considered in the literature and which shows notable and interesting differences
Summary
An involute of a curve in Euclidean plane E2 is a curve to which all tangent lines of the initial curve at corresponding points are orthogonal. When a curve c parametrized by arc length is given, its evolute is sought-after in the form e(u) = c(u) + μ(u)n(u), where n is the unit normal field of c. If the curve c is timelike or it is spacelike with non-null normal, involutes have been studied in [4,5,6], where parametrizations of associated evolutes, as adaptations of (2) to R31 , were obtained too. In this paper we reexamine that result proving that involutes of such curves are straight lines Such a situation does not exist in Euclidean case, straight lines do not have evolutes.
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