APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values ​​which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

The Gorenstein Property for Projective Coordinate Rings of the Moduli of Parabolic $\mathrm{SL}_2$-Principal Bundles on a Smooth Curve

Using combinatorial methods, we determine that a projective coordinate ring of the moduli of parabolic principal $\mathrm{SL}_2-$bundles on a marked projective curve is not Gorenstein when the genus and number of marked points are greater than $1$.

A class of projective coordinate spaces over modules

In this paper, we study a class of local rings that are not isomorphic to the class of two local rings we have been working on before. This class is also an isomorphism to a special matrix algebra. We construct a projective coordinate space over a module defined on this special algebra class. Specifically, in a 3-dimensional projective coordinate space, the incidence matrix for a line that combines the certain two points and also all points of a line given with the incidence matrix are found by the help of Maple programme.

On projective coordinate spaces

In the present study, an (n+1)-dimensional module over the local ringK = Mmm(R) is constructed. Further, an n-dimensional projective coordinate space over this module is constructed with the help of equivalence classes. The points and lines of this space are determined and the points are classified. Finally, for a 3-dimensional projective coordinate space, the incidence matrix for a line that goes through the given points and also all points of a line given with the incidence matrix are found by the use of Maple commands.