The geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space

The aim of this paper is to give the geometrical and algebraic interpretations of Euler–Rodrigues formula in Minkowski 3-space. First, for the given non-lightlike axis of a unit length in [Formula: see text] and angle, the spatial displacement is represented by a [Formula: see text] semi-orthogonal rotation matrix using orthogonal projection. Second, we obtain the classifications of Euler–Rodrigues formula in terms of semi-skew-symmetric matrix corresponds to spacelike, timelike or lightlike axis and rotation angle with the help of exponential map. Finally, an alternative method is given to find rotation axis and the Euler–Rodrigues formula is expressed via split quaternions in Minkowski 3-space.

MARGINALLY TRAPPED MERIDIAN SURFACES OF PARABOLIC TYPE IN THE FOUR-DIMENSIONAL MINKOWSKI SPACE

A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their basic invariants. We find all marginally trapped meridian surfaces of parabolic type and give a geometric construction of these surfaces.

On the Gauss Map of Surfaces of Revolution with Lightlike Axis in Minkowski 3-Space

By studying the Gauss mapGand Laplace operatorΔhof the second fundamental formh, we will classify surfaces of revolution with a lightlike axis in 3-dimensional Minkowski space and also obtain the surface of Enneper of the 2nd kind, the surface of Enneper of the 3rd kind, the de Sitter pseudosphere, and the hyperbolic pseudosphere that satisfy conditionΔhG=ΛG, Λbeing a3×3real matrix.