Lamarle Formula in 3-Dimensional Lorentz Space

TL;DR

This paper extends the Lamarle Formula to 3D Lorentz space, establishing relationships between Gaussian curvature and distribution parameters for various types of ruled surfaces, supported by examples.

Contribution

It introduces Lorentzian Lamarle formulas for different ruled surface classes in 3D Lorentz space, expanding classical surface theory.

Findings

Derived Lorentzian Lamarle formulas for spacelike and timelike ruled surfaces.

Established relationships between Gaussian curvature and distribution parameters.

Provided examples illustrating these relationships.

Abstract

The Lamarle Formula, given by Kruppa in \cite{Kr}, is known as a relationship between the Gaussian curvature and the distribution parameter of a ruled surface in the surface theory. The ruled surfaces were investigated in 3 different classes with respect to the character of base curves and rulings, \cite{Tu1},\cite{Tu2}. In this paper on account of these studies, the relationships between the Gaussian curvatures and distribution parameters of spacelike ruled surface, timelike ruled surface with spacelike ruling and timelike ruled surface with timelike ruling are obtained, respectively. These relationships are called as Lorentzian Lamarle formulas. Finally some examples concerning with these relations are given.