Constant rations for inflection points of a cubic curve with a node or an acnode

L. N. Romakina

Čebyševskij sbornik, Tome 24 (2023) no. 3, pp. 122-138.

Résumé

In this paper, projective invariants of cubic curves with a node or an acnode are obtained. It is proved that on the projective plane every two inflection points of a cubic curve with a node (acnode) are in an equianharmonic ratio with the points of tangents of the given curve at its node (acnode) located on the line containing these inflection points. And every three inflection points of such a curve are in a quasi-anharmonic ratio with the point on tangent of this curve at its node (acnode) located on the line containing these inflection points. It is established that, on the projective plane, the family of all crunodal (acnodal) cubics defined up to a projective transformation, is two-parametric. It is proved that four lines containing the node (acnode) of a cubic, namely: the line of the inflection points, the tangent and pseudotangent to the curve at the inflection point, the tangent to the curve at the point conjugate to the inflection point, are in a constant cross ratio equal to $-3$. Based on this fact, a number of properties of cubic curves with a node (acnode) in the Euclidean plane $E_2$ are substantiated. Let us present some of the proved properties, denoting the cubic curve by the symbol $\sigma$, and its node or acnode by the symbol $I$. If the tangents of $\sigma$ at an acnode $I$ pass through circle points of the plane $E_2$, then the angle between any two lines, each of which connects the point $I$ with the inflection point of this curve, is equal to $\pi / 3$. The pseudotangent at the point $I$ divides the strip between the parallel tangents of $\sigma$ passing through $I$, in the ratio of three to one, counting from the tangent of $\sigma$ at the conjugate point with $I$, if and only if, when the line of the inflection points of $\sigma$ coincides with the absolute line of the plane $E_2$. The tangent of $\sigma$ at the conjugate point with $I$ divides the strip between the mutually parallel pseudotangents at the point $I$ and the line of the inflection points of $\sigma$ in the ratio of three to one, counting from the pseudotangent, if and only if, when the tangent line of $\sigma$ at the point $I$ coincides with the absolute line of the plane $E_2$. The line of the inflection points of the curve $\sigma$ divides the strip between the parallel tangents to $\sigma$ passing through $I$ in the ratio of three to one, counting from the tangent line of $\sigma$ at the point $I$, if and only if when the pseudotangent of the curve $\sigma$ at the point $I$ coincides with the absolute line of the plane $E_2$. The tangent of the curve $\sigma$ at the point $I$ divides the strip between the line of inflection points and the parallel to it pseudotangent of $\sigma$ at $I$ in the ratio of three to one, counting from the line inflection points, if and only if the tangent of the curve $\sigma$ at the conjugate point with $I$ coincides with the absolute line of the plane $E_2$.

Détail

Keywords: cubic curve, inflection point, node of a cubic curve, acnode of a cubic curve.

Bibliographie

[1] Atanasyan L. S., Bazylev V. T., Geometry, Textbook for students of physical and mathematical faculties of pedagogical institutes. In 2 p., v. 2, Education, M., 1987, 352 pp.

[2] Klein F., Non-Euclidean geometry, ONTI NKTP USSR, M.–L., 1936, 356 pp.

[3] Rosenfeld B. A., Non-Euclidean spaces, Science, M., 1969, 548 pp.

[4] Romakina L. N., Geometry of a hyperbolic plane of positive curvature, in 4 party, v. 1, Trigonometry, Publishing House Sarat. university, Saratov, 2013, 244 pp.

[5] Romakina L. N., Geometry of co-Euclidean and copseudo-Euclidean planes, Publishing house «Scientific book», Saratov, 2008, 279 pp.

[6] Romakina L. N., “The study of remarkable curves of the pseudo-Euclidean plane in the system of training teachers of mathematics”, Collection of scientific papers, Teacher – student : problems, searches, finds, 8, IC «Science», Saratov, 2010, 61–64

[7] Romakina L. N., “Rations for the inflection points of a cubic curve with a node or acnode”, Algebra, number theory and discrete geometry : modern problems, applications and problems of history, Proceedings of the XVII Intern. conf., dedicated centenary of the birth of professors B. M. Bredikhin, V. I. Nechaev and S. B. Stechkin, Tula state. ped. univ. of L. N. Tolstoy, 2020, 427–429

[8] Savelov A. A., Plane curves. Systematics, properties, applications (reference guide), State publishing house of physical and mathematical literature, M., 1960, 294 pp.

[9] Smogorgevsky A. S., Stolova E. S., Handbook of third order plane curve theory, State publishing house of physical and mathematical literature, M., 1961, 263 pp.

[10] Walker R., Algebraic curves, Foreign Languages Publishing House, M., 1952, 236 pp.

[11] Shafarevich I.R., “Foundations of algebraic geometry”, Uspekhi Mat. Nauk, 24:6(150) (1969), 3–184 | DOI | MR | Zbl

[12] Lawrence J. D., A Catalog of Special Plane Curves, Dover, New York, 2014

[13] Romakina L. N., Bessonov L.V., Chernyshkova A.A., “Modeling of curls of Agnesi in non-Euclidean planes”, AIP Conference Proceedings, 2037, American Institute of Physics, 2018, 020022-1–020022-6 | DOI

[14] Romakina L. N., “Construction of cubic curves with a node”, Beitr Algebra Geom., 60:4 (2019), 761–781 | DOI | MR | Zbl

[15] Weisstein E. W., Maclaurin Trisectrix. MathWorld — a wolfram web resource, http://mathworld.wolfram.com/MaclaurinTrisectrix.html