Mathematics and Statistics Vol. 9(5), pp. 749 - 759
DOI: 10.13189/ms.2021.090515
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Triangle Conics, Cubics and Possible Applications in Cryptography

Veronika Starodub ^1,*, Ruslan V. Skuratovskii ², Sergii S. Podpriatov ^3
1 American University in Bulgaria, Bulgaria
² Interregional Academy of Personnel Management, Kiev and The National Technical University of Ukraine, Igor Sikorsky Kyiv Polytechnic Institute, Ukraine
³ MD, Kyiv, Ukraine

ABSTRACT

We research triangle cubics and conics in classical geometry with elements of projective geometry. In recent years, N.J. Wildberger has actively dealt with this topic using an algebraic perspective. Triangle conics were also studied in detail by H.M. Cundy and C.F. Parry recently. The main task of the article is development of a method for creating curves that pass through triangle centers. During the research, it was noticed that some different triangle centers in distinct triangles coincide. The simplest example: an incenter in a base triangle is an orthocenter in an excentral triangle. This is the key for creating an algorithm. Indeed, we can match points belonging to one curve (base curve) with other points of another triangle. Therefore, we get a new fascinating geometrical object. During the research number of new triangle conics and cubics are derived, their properties in Euclidian space are considered. In addition, it is discussed corollaries of the obtained theorems in projective geometry, which proves that all of the discovered results could be transferred to the projective plane. It is well known that many modern cryptosystems can be naturally transformed into elliptic curves. We investigate the class of curves applicable in cryptography.

KEYWORDS
Triangle Cubics, Conics, Curves, Projective Geometry, Euclidian Space

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Veronika Starodub , Ruslan V. Skuratovskii , Sergii S. Podpriatov , "Triangle Conics, Cubics and Possible Applications in Cryptography," Mathematics and Statistics, Vol. 9, No. 5, pp. 749 - 759, 2021. DOI: 10.13189/ms.2021.090515.

(b). APA Format:
Veronika Starodub , Ruslan V. Skuratovskii , Sergii S. Podpriatov (2021). Triangle Conics, Cubics and Possible Applications in Cryptography. Mathematics and Statistics, 9(5), 749 - 759. DOI: 10.13189/ms.2021.090515.