Geometry of lightlike locus on mixed type surfaces in Lorentz-Minkowski 3-space from a contact viewpoint

Singularities of Slant Focal Surfaces along Lightlike Locus on Mixed Type Surfaces

“…If θ = 0, F S 0 γ coincides with the osculating lightlike surface f L of f along L( f ) (cf. [1]). Moreover, if θ = π 2 , we call F S π/2 γ the focal surfaces of the lightlike locus γ.…”

“…By taking the derivative of both the sides as follows: e(u) =x θ (u)(−α L e(u) − α G (u)L(u)) + (x θ ) (u)L(u) + (y θ ) (u)(cos θe(u) + sin θN(u)) We can see that F S 0 γ is a tangent flat approximation of f along L( f ) (cf. [1]). Then we have the F S 0 γ (green surface) in the following picture.…”

“…It can be a singular point of the induced metric. When p is the lightlike point of the first kind [1,2], the lightlike locus f | L( f ) may be a spacelike regular curve in [2]. Then we have tangent vector e of f | L( f ) and two lightlike vectors L, N along L( f ).…”

“…It is the envelope of family of normal planes which are spanned by two lightlike vectors L and N satisfying a symmetric (or, dual) condition along L( f ). Moreover, the osculating lightlike surface of lightlike locus is considered in [1]. It is the envelope of family of osculating planes which are also limiting tangent plane spanned by e and L of f along f | L( f ) .…”

Pseudo‐spherical evolutes of lightlike loci on mixed type surfaces in Minkowski 3‐space

“…A brief description of the organization of this paper is as follows. In Section 2, we recall some fundamental conceptions in Minkowski 3‐space and introduce a frame along the lightlike locus in Honda et al 16 In Section 3, we consider pseudo‐spherical height functions on the lightlike locus. By analyzing these functions, we obtain an invariant $$$ \sigma $$$.…”