Cubic curves in the triangle plane

Pinkernell, Guido M.

doi:10.1007/bf01223040

Cited by 10 publications

(2 citation statements)

References 18 publications

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“…Jerabek hyperbola is a curve that passes through vertices of triangle, circumcenter, orthocenter, Lemoine point, isogonal conjugate of the de Longchamps point [5].…”

Section: Definitionmentioning

confidence: 99%

Triangle Conics, Cubics and Possible Applications in Cryptography

Starodub¹,

Skuratovskii²,

Подпрятов³

2021

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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.

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“…Jerabek hyperbola is a curve that passes through vertices of triangle, circumcenter, orthocenter, Lemoine point, isogonal conjugate of the de Longchamps point [5].…”

Section: Definitionmentioning

confidence: 99%

Triangle Conics, Cubics and Possible Applications in Cryptography

Starodub¹,

Skuratovskii²,

Подпрятов³

2021

View full text Add to dashboard Cite

show abstract

“…Definition 1. Jerabek hyperbola is a curve which passes through verticies of trianlge, circumcenter, orthocenter, Lemoine point, isogonal conjugate of the de Longchamps point [6].…”

Section: Base Trianglementioning

confidence: 99%

Triangle conics and cubics

Skuratovskii

Strarodub

2021

RMM

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This is a paper about 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 was to develop an algorithm for creating curves, which 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 was 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 intersting geometrical object. During the research were derived number of new triangle conics and cubics, were considered their properties in Euclidian space. In addition, was discussed corollaries of the obtained theorems in projective geometry, what proves that all of the descovered results could be transfered to the projeticve plane.

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Locus properties of the Neuberg cubic

Čerin¹

1998

J Geom

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Cubic curves in the triangle plane

Cited by 10 publications

References 18 publications

Triangle Conics, Cubics and Possible Applications in Cryptography

Triangle Conics, Cubics and Possible Applications in Cryptography

Triangle conics and cubics

Locus properties of the Neuberg cubic

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