Rotation minimizing frames and spherical curves in simply isotropic and pseudo-isotropic 3-spaces

Abstract

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Euclidean and Lorentzian geometries, here we show how to build RM frames in isotropic geometries and apply them in the study of isotropic spherical curves. Indeed, through a convenient manipulation of osculating spheres described in terms of RM frames, we show that it is possible to characterize spherical curves via a linear equation involving the curvatures that dictate the RM frame motion. For the case of pseudo-isotropic space, we also discuss on the distinct choices for the absolute figure in the framework of a Cayley-Klein geometry and prove that they are all equivalent approaches through the use of Lorentz numbers (a complex-like system where the square of the imaginary unit is $+1$). Finally, we also show the possibility of obtaining an isotropic RM frame by rotation of the Frenet frame through the use of Galilean trigonometric functions and dual numbers (a complex-like system where the square of the imaginary unit vanishes).

The geometry of Gauss map and shape operator in simply isotropic and pseudo-isotropic spaces

“…Simply and pseudo isotropic spaces are examples of Cayley-Klein geometries [8], i.e., the study of properties in projective space P 3 invariant by the action of the projectivities that fix the so-called absolute figure, which for our isotropic geometries are given by a plane at infinity, identified with ω : x 0 = 0, and a degenerate quadric of index zero or one, identified with Q : x 2 0 + x 2 1 + δ x 2 2 = 0: δ = 1 for the I 3 absolute figure [14,17] and δ = −1 for the I 3 p one [6]. (Euclidean and Lorentzian geometries stand for the choice of ω : x 0 = 0 and Q : x 2 0 + x 2 1 + x 2 2 + δ x 2 3 = 0, δ = ±1.)…”

“…Simply and pseudo isotropic three-dimensional spaces. In the Cayley-Klein framework, the simply isotropic I 3 and pseudo-isotropic I 3 p geometries are the study of those properties in R 3 invariant by the action of the 6-parameter groups B 6 [14,17] and B p 6 [6] given by On the xy plane I 3 and I 2 p look just like the Euclidean E 2 and Lorentzian E 2 1 plane geometries. The projection of a vector u = (u 1 , u 2 , u 3 ) on the xy plane is called the top view of u and it is denoted by u = (u 1 , u 2 , 0).…”

“…where σ ∈ {−1, +1}: σ = +1 in I 3 [14] and σ = −1 in I 3 p [6]. Up to a rigid motion, we can express an isotropic sphere in one of the two normal forms below:…”

“…In addition, every admissible surface in I 3 p is timelike, i.e., g ij is non-degenerated with index 1 [3,7]. Regular admissible curves may be either spacelike, α ′ , α ′ pz > 0, or timelike, α ′ , α ′ pz < 0, while lightlike curves, α ′ , α ′ pz = 0, are non-admissible [6].…”

“…The study of I 3 has been initiated by the Austrian geometer Karl Strubecker in the 1930's [17,18,19,20,21] (see also [14] and references therein), while that of I 3 p began only recently [3,7]. Besides its mathematical interest [1,3,9,23,24], see also the recent contributions by this Author [6,7], isotropic geometry finds applications in economics [4,5], image processing [10], and shape interrogation [12]. In addition, this theory may prove useful in understanding the geometry of surfaces with zero mean curvature in semi-Riemannian spaces [15,16].…”