Tangentials in cubic structures

Abstract

In this paper we study geometric concepts in a general cubic structure. The well-known relationships on the cubic curve motivate us to introduce new concepts into a general cubic structure. We will define the concept of the tangential of a point in a general cubic structure and we will study tangentials of higher-order. The characterization of this concept will be also given by means of the associated totally symmetric quasigroup. We will introduce the concept of associated and corresponding points in a cubic structure, and discuss the number of mutually different corresponding points. The properties of the introduced geometric concepts will be investigated in a general cubic structure.

Inflection Points in Cubic Structures

“…Various concepts, which appear in any cubic structure, and relations between them, are introduced and studied in [3] and in this paper. In the future, the authors intend to use cubic structures to study the properties of some types of configurations (see [4][5][6][7]) among which are, for example, Steiner's triplets.…”

“…In [3] (Th. 4.3), we proved the following: If a 1 , a 2 , a 3 , and a 4 are associated points with the common tangential a , then points p, q, and r exist such that [a 1 , a 2 , p], [a 3 , a 4 , p], [a 1 , a 3 , q], [a 2 , a 4 , q], [a 1 , a 4 , r] and [a 2 , a 3 , r], and points a , p, q, and r are associated.…”

“…Two concepts in cubic structures are defined in [3]. The point a is the tangential of point a if the statement [a, a, a ] holds.…”

On the Noteworthy Properties of Tangentials in Cubic Structures

“…The concept of tangentials of points was introduced in [5]. The point a ′ is said to be the tangential of the point a if [a, a, a ′ ] holds true.…”

“…In the case of a cubic structure where collinear triples of non-singular points are observed on a cubic curve in the complex plane, ranks 0, 1, or 2 appear, depending on whether the cubic has a spike, an ordinary double point, or is without singular points. We will mention here some of the results from [5] in the form of several lemmas. Lemma 3.…”