A geometric surface model is formed taking into account given functional, structural, technological, economic, aesthetic requirements. These requirements are formulated in geometric terms and are expressed in terms of the surface parameters. The surface is modeled either Kinematics manner, or by way of a twodimensional interpolation. In accordance with in accordance with the kinematic method, the surface is formed as a continuous oneparameter many curves that form simulated in the surface. In accordance with the interpolation method, the surface consists of a set of elementary two-dimensional fragments. The article considered cinema optical method based on the use of curves of the second order of change-nests of the eccentricity as the main shaping element. To control the shape of the design surfaces are used for guide ruled surfaces (cilindroidy and conoid). Computer program is compiled, which determines the eccentricity of the forming curves of the second order depending on the boundary conditions. The program allows you to plot curve of the second order, given an arbitrary set of five coplanar points and tangents. When modeling the surface of the passing through a closed circuit, is used the mapping of this contour in four-dimensional space. Such mapping gives more possibilities for control surface shape. It is shown that the kinematics method computer simulation of the surface has technological advantages properties instead of interpolation method.

Second-order curves are used as shape-generating elements in the design of technical devices and architectural structures. In such a case, a need for reconstruction task solution may emerge. The reconstruction is called the definition of the main axes and asymptotes of the second-order curve by its incomplete image containing n points and m tangents (n + m = 5). In CAD graphical systems there is no possibility for construction of the second order curve, given by real and imaginary points and tangents. Therefore, the second-order curve reconstruction cannot be made with the standard set of computer graphics tools. In this paper are proposed geometrically accurate algorithms for reconstruction of the secondorder curve, given by a mixed set of real and imaginary elements. A specialized software package has been developed for constructive realization of these algorithms. Imaginary geometric images are pair-conjugated, so there are only seven possible combinations of given data with imaginary elements participation: five points, two of which are imaginary ones; five points, four of which are imaginary ones; three real points, two imaginary tangents; a real point, four imaginary tangents; a real point, two imaginary points, two imaginary tangents; a real point, two imaginary points, two real tangents; two real points, two imaginary points, a real tangent. For reconstruction problem solution is used the main property of polar matching: if P and p are the pole and polar relative to the conic g, the harmonic homology with center P and axis p transforms the curve g in itself. The method of solution based on projective transformation of required conic into a circle. It has been shown that in some cases for reconstruction problem solution its necessary to apply the quadratic involution conversion, resting on plane by a conic beam. The developed technique and software package expand the capabilities of the computer geometric simulation for processes occurring with the second-order curves participation.

A beam of second order surfaces which are in a real or imaginary double-tap is considered. The analysis of all possible events related to bi-quadratic curve disintegration in two plane curves of second order has been performed. The synthetic proof of generalized Dandelin theorem has been presented.

Birational (Cremona) correspondences of two planes П, П' or Cremona transformations on the combined plane П = П' represent an effective tool for design of smooth dynamic curves and two-dimensional lines. The simplest birational correspondence is a quadratic map Ω of plane fields on one another. In the projective definition of the quadratic correspondence can participate two pairs of imaginary complex conjugate F-points set as double points of elliptic involutions on the lines associated with the third pair of F-points. In this case, the imaginary projective F-bundles cannot be used for generation of points corresponding in Ω. A generic constructive algorithm for design of corresponding points in a quadratic mapping Ω(П ↔ П'), set both by real and imaginary F-points is proposed in this paper. The algorithm is based on the use of auxiliary projective correspondence Δ between the points of the planes П, П' and Hirst’s transformation Ψ with the center in the one of real F-points. A theorem on the existence of an invariant conic common to Ω and Δ mappings has been proved. Has been demonstrated a possibility for quadratic mapping’s presentation as a product of collinearity and Hirst’s transformation: Ω = ΔΨ. Has been considered a solution for auxiliary problems arising during the generic constructive algorithm’s implementation: buildup a conic section, that is incident to imaginary points, and plotting the corresponding points in collinearity set by imaginary points. It has been demonstrated that there are two or four possible versions of collinearity for plane fields П, П', set by with participation of the imaginary corresponding points, due to an uncertainty related to the order of their relative correspondence. Have been completely solved the problem of mapping a straight line in a conic section within the quadratic Cremona correspondance of fields П ≠ П', set by a pair of real fundamental points, and two pairs of imaginary ones. It has been demonstrated that in general case the problem has two solutions. The obtained results are useful for the development of the geometric theory related to imaginary elements and this theory’s application in linear and non-linear descriptive geometry, operating projective images of first and second orders.

The geometric correspondence between the points of two planes can be considered well defined only when base data for its establishing is available, and a construction method by which its possible on the basis of these data for each point in one plane to find the corresponding points in the other one. Quadratic Cremona transformation can be specified by pointing out in the combined plane seven pairs of corresponding points. Naturally there is a need to establish a method for constructing any number of corresponding points. An outstanding Russian geometer K.A. Andreev indicated the linear construction based on the consideration of two correlations by which for each eighth point in the one plane is found the corresponding point of the other one. But in his work was not set up a problem to construct excluded (fundamental) points of quadratic Cremona transformation specified by seven pairs of points. There are many constructive ways to obtain the quadratic transformation in the plane. For example, it can be obtained by using two pairs of projective pencils of straight lines with vertices at the fundamental points (F-points). K.A. Andreev noted that this method for establishing of quadratic correspondence spread only to those cases when all F-points are the real ones. This statement is true for the 19th century’s level of geometric science, but today it’s too categorical. The theory of imaginary elements in geometry allows to develop a universal algorithm for construction of corresponding points in a quadratic transformation, given both by real and imaginary F-points. Summarizing the K.A. Andreev task, we come to the problem of finding the fundamental points (F-points) for a quadratic transformation specified by seven pairs of corresponding points. Almost one and half century the K.A. Andreev generalized task remained unsolved. The formation of this task’s constructive solution algorithm and its practical implementation has become possible by means of modern computer geometric modeling. According to proposed algorithm, the construction of F-points is reduced to the construction of second order auxiliary curves, on which intersection are marked the required F-points. The result received in this paper is used for development of the Cremona transformations’ theory, and for further application of this theory in the practice of geometric modeling.