Study of the problem on constructing quadrics at the assigned tangent cones - Аnpilogova, Vira; Botvinovska, Svitlana; Zolotova, Alla; Sulimenko, Hanna (Eastern-European Journal of Enterprise Technologies, 2019)
The research focuses on solving the problems, related to modeling the second-order surfaces (quadrics), the determinant of which includes tangent cones. All research was performed by the paradigm of using constructive methods of creation of algorithms. This is caused by the fact that there is a possibility to rely on a significant number of basic geometric tasks implemented in the DSS. The problem of modeling of quadrics by tangent cones is relevant because there are at least its two important applications. The first is the construction of the surfaces by its contour line on perspective images. In this case, the point of view and the perspective contour line assign the enveloping cone, which in the case of quadrics coincides with the tangent cone. Such problems are solved in the context of problems of technical aesthetics and architectural design. The second application is found in the problems of constructing the quadrics that are conjugated by the assigned curves or in the problems of conjugation of two quadrics by the third one. The problem of conjugation of surfaces is of wide practical significance, which is proved by the interest of users and developers of computer modeling systems. Within the framework of this research, the existing theoretical geometric properties for modeling the quadrics, the determinant of which includes tangent cones, were accumulated, and a series of new geometric properties were found. We developed the method, by which through assigning the contact line on one cone, there is a possibility to find a contact line on the second cone, as well as to find the center of the quadric inscribed in these two cones. An alternative method for modeling the described surfaces was also proposed. In this way, the cross-sections of all quadrics, tangent to two cones, are inscribed in quadrangulars, the vertices of which belong to the lines of the intersection of the assigned cones. Based of structural geometric research, the algorithms for computer realization of problems of modeling objects by their contour lines on their perspective images were developed. The research results in the form of the theoretical calculations and examples of their application show the effectiveness of the proposed algorithms. The described approach to the solution of the stated tasks makes it possible to extend the possibilities of existing computer systems in their being applied in the work of designers and greatly simplify the process of the creation of actual objects