In this paper the plane Laguerre’s geometry in the augmented plane of dual numbers is presented. Basic integral and differential invariants of -curves in the plane are deduced, i.e. the -curve arc, -curvature, -minimal curves, -circle. Furthermore the contact of -curves, -osculating circle, -evolute of a curve and some special -motions are studied from the point of view of -Differential geometry.
We consider surfaces in hyperbolic 3-space and their duals. We study flat dual surfaces in hyperbolic 3-space by using extended Legendrian dualities between pseudo-hyperspheres in Lorentz-Minkowski 4-space. We define the flatness of a surface in hyperbolic 3-space by the degeneracy of its dual, which is similar to the case of the Gauss map of a surface in Euclidean 3-space. Such surfaces are a kind of ruled surfaces. Moreover, we investigate the singularities of these surfaces and the dualities...
Lorentzian surfaces in Lorentz three-space are studied using an indefinite version of the quaternions. A classification theorem for Bonnet pairs in Lorentz three-space is obtained.