*Visnyk LNAU: Architecture and Farm Building 2018 №19: 28-32*

## GEOMETRIC MODELING IN ARCHITECTURE AND TECHNICAL OF CONJUGATE SURFACES OF THE SECOND ORDER

**I. Kernytskyy.**, Doctor of Technical Sciences

**O. Nikitenko.**, Candidate of Technical Sciences

WULS, Faculty of Civil and Environmental Engineering

**I. Stukalec.**, Candidate of Technical Sciences

Lviv National Agrarian University

https://doi.org/10.31734/architecture2018.19.028
#### Annotation

In modern construction and architecture, various complex curvilinear surfaces are widely used. Computer technologies allow to supplement the mathematical methods of determining surfaces and their calculations by three-dimensional computer models. Of particular importance are modern design technologies and methods for calculating curved surfaces in architecture, in particular, in the design of architectural constructions, for example, spatial coatings (domes) or spacious objects of complex shape (cooling towers). Modeling of surface curves also applies in solutions to technical problems.

Conjugated surfaces are constructed for revolution surfaces of the second order ones in this paper. The line of contact of such surfaces is a geodesic line the surfaces. The task of constructing such surfaces is finding geodetic lines and cal-culation of the required parameters. All of the surfaces are drawn in the graphics program AutoCAD.

In the article, the concept of “conjugate surfaces” is associated with the concept of “tangent surfaces”, which are used in differential geometry to describe geodetic lines. The difference between the conjugate and the tangent surfaces is determined taking into account the location of the axes: in the tangent surfaces of the axis occupy a respectable position, and in the conjugated – only the passage-way; that is, conjugate surface-no regarded as a partial case of tangent surfaces. In differential geometry, it has been proved that such tangent surfaces are conjugated by geodesic lines.

The proposed method for constructing conjugate surfaces of rotation of the second order is simple enough. The examples discussed are the on-line demonstration of the method itself and the connection of graphic modeling with a mathematical one. In the future, it is advisable to use the developed method for the consistence of the rotation surfaces formed by the rotation of the transcendental curves.

#### Key words

conjugate surfaces, an ellipsoid of revolution, paraboloid of revolution, elliptical torus, hyperboloid of revolution, elliptical globoid

#### Link

- Androsov A., Grebenyuk G. The gear transmission with an elliptical profile of the tooth as an element of scientific and technological progress in the machine-building. CAD and a graph of 8'2005. URL: http://www.sapr.ru/Article.aspx?id=7812.
- Nikitenko O.A. Geometric modeling of curvilinear conjugate on-surface with the use of a diagram of a kinematic screw. Computer-integrated technologies: education, science, production. Lutsk: LNTU, 2015. Is. 19. P. 129-132.
- Nikitenko O.A. Construction and analytical description of conjugate surfaces of ob-rats. Bulletin of Kherson National Technical University. 2015. Is. 3 (54). P. 595-598.
- Pogorelov A.V. Lectures on differential geometry. Kharkiv: Publishing house of Kharkiv State University, 1956. 180 p.