Tzitzeica Curves and Surfaces

Abstract

Tzitzeica curves and surfaces represent early examples of affineinvariant geometrical objects. At the time Gheorghe Tzitzeica was studying these objects, affine differential geometry (ADG) was in its infancy. ADG was motivated by Felix Kleinʼs influential Erlangen program, where a geometry was defined by its set of invariants under a group of symmetries. We find that the issue lends itself well to a relatively elementary discussion suitable for upper-division undergraduates and nonspecialists, while still providing the basic thrust of this elegant subject. Moreover, the topic is an excellent one to illustrate the utility of Mathematicaʼs symbolic manipulation and graphics capabilities. For this reason, the article nicely complements the existing literature on the uses of software in differential geometry (such as [1]), and it provides material that would be useful for inclusion in a differential geometry course either as an application or a project.

Tzitzeica-Type centro-affine invariants in Minkowski spaces

“…As an example, we can choose a Minkowski circle which is a Tzitzeica curve in the Minkowski space; its equivalent in the Euclidian space it is a equilateral hyperbola which is also a Tzitzeica curve, but in the Euclidian space. Some other examples of Tzitzeica curves can be seen in [3]. The geometric meaning of these objects depends on the geometry but they both fullfiled the Tzitzeica condition.…”

“…Theorem 2.5. Let h : I ⊂ R −→ R 3 1 be a centro-affine transformation of a curve c : I ⊂ R −→ R 3 1 , c(t) = (x(t), y(t), z(t)). Then the Tzitzeica function I c is a centro-affine invariant of the curve c, and furthermore it satisfies the relation…”

“…As an example let us consider the Minkowski sphere −x 2 +y 2 +z 2 = 1. It is easy to see that this surface is a Tzitzeica surface in the Minkowski 3-dimensional space R 3 1 , but the equation of this Minkowski sphere in the Euclidian 3-dimensional space represents an hyperboloid which is different to the Euclidian sphere. The Tzitzeica surface induced by the equation of the Euler's line of a triangle as in [2] is an example of Tzitzeica surface in a Minkowski space according to our theory, too.…”

“…All the results are presented in a 3-dimensional Minkowski space. DOI: 10.2478/v10309-012-0037-0 2 The Tzitzeica function for curves in R 3 1 The space R 3 1 is defined as a space to be the usual three-dimensional R-vector space consisting of vectors…”

“…Definition 2.2. We call the function I c from lemma 2.1 the Tzitzeica function for the curve c : I ⊂ R −→ R 3 1 in the Minkowski space R 3 1 .…”