User:ChapoBlanchard

Circular crunodal cubics with an infinite real inflexion point in a Minkowski plane. Crunodal cubics of both Euclidean and non-Euclidean planes can be used in modeling of various geometric objects and physical processes. In this regard, we set up a problem of classifying and studying the crunodal cubics in non-Euclidean planes. In this direction, we start with the study of circular cubics in a Minkowski plane R^2_1. Given the variety of types of such curves, at the first stage, we limit ourselves to considering cubics with an inflection point on the absolute of the plane R^2_1. The asymptotes of such cubics are isotropic lines. To simplify calculations when classifying cubics and derive canonical equations, we use both the orthogonal coordinate system of the Minkowski plane, which is an analogue of the Cartesian system of the Euclidean plane, and projective frames (see, for instance) A projective plane P2 with an infinitely removed line l∞ and a fixed pair of real points K1, K2 on it is the Cayley – Klein projective model of the Minkowski plane R^2_1. The set of objects l∞, K1, and K2 is called the absolute of the plane R21.The fundamental group of the plane R^2_1 consists of all projective automorphisms of the absolute and is the group of similarity transformations. By choosing the projective frame R = {A1, A2, A3, E} of the plane R2 1 due course, we define the elements of the absolute so that l∞(0 : 0 : 1) = A1A2, K1(1:1:0) = A1 + A2, K2(1 : −1:0) = A1 − A2. In this case, the quadratic form (x1/x3)^2 −(x2/x3)^2 that corresponds to the absolute setting in the frame R, induces a pseudo-Euclidean metric in the plane R^2_1 and determines the isometries group of this plane. We prove that each circular crunodal cubic with the real infinite inflection point belongs to one of six base types. These types depend on the location of the cubic with respect to the absolute. We derive the canonical equations of each type of the cubics and find out the geometric meaning of the coefficients of the equations. In each case, we find a generating set and a set of independent invariants of the cubic.Continuing the classification process, we distinguish different classes in four of the six types of the considered cubics. Let us consider, for example, the following canonical equations of the studied cubics

k^2(x^2−y^2)(x−y)+2k(x−y)^2−(x+y) = 0, k no equal to 0, (x^2−y^2)(x−y)+axy = 0, a no equal to 0.

A circular crunodal cubic whose node and real inflection point coincide with the circle-points K1,K2 of the plane R^21can be given by the equation. Each such cubic is one of the possible analogues of a strophoid of the Euclidean plane and the cubic of stationary curvature in terms of the Burmester theory, that is, the locus of points of the mobile plane whose four corresponding positions lie on circles of the fixed plane. Also, the cubic of stationary curvature can be defined as the locus of points, whose trajectories have a vanishing differential of the curvature with respect to the natural parameter at zero position. A circular crunodal cubic whose loop vertice and real inflection point coincide with the circle-points can be given by the equation .Any cubic of each of presented here types has a single independent invariant, which determines the distance from the node of the cubic to a point on the asymptote along an axis. In the first case, the axes consist the loop vertice of the cubic and harmonicaly devide isotropic lines in this point. In the second case, the axes are orthogonal tangents of the cubic in its node.

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