The proposed method of specifying the surface of a particular discrete frame with a flat 3-tissue. A one-parameter non-parametrized family of curves is given, i.e. a discrete frame of lines. There are two cases of specifying the lines of a discrete surface frame: point series or equations. In the article the questions of the task spatial 3-a fabric flat. In this case, the basic concepts of a proper tetrahedron and a complete quadrilateral are used. If the fabric M has two families of diagonal surfaces, then it also has a third similar family.

Keywords: descriptive geometry, graphics, octahedral 3-tissue, spatial 3-tissue, full quadrilateral, topology, affine geometry, regular 3-tissue, projective geometry, involution

The algorithm of forming of kinematic surfaces with a constant cross-sectional area based on a complex equiaffine transformation of the plane (elliptic rotation) is considered. Functional dependences of elliptic rotation parameters for the formation of a one-parameter family of equiaffine lines on a plane are determined. At the same time the received families of lines can not include the line prototype and the line image at the set constant parameters of rotation. Conditions of receiving the line prototype and the line image at the set parameter of family and the parametrical equations of shift of the geometrical center of a curve are defined. The conditions for the formation of central surfaces are determined. It is established that the surfaces obtained can be periodic. The domains of admissible values of the parameters and functions included in the parametric equations of one-parameter families of curves are determined. It is shown that as a closed loop one can use not only an analytically determined curve, but also a polyline (for example, a polygon). Examples of kinematic surfaces are given.

Keywords: kinematic surface, equiaffine transformations, algorithm, elliptic rotation, affine-like curves, parametric equations, line-image, prototype line

In the article the algorithm of construction of an array of points, put in correspondence to points of a compartment of a surface is resulted. Calculation formulas for the coordinates of points are shown. The possible cases of the location of the corner points of the array are considered. And the conclusion is made that this algorithm will help in modeling acoustic, optical and other processes.

Keywords: array of points, surface area, algorithm for matching array points, geometric modeling, radix-vector

We will call the Geometrical Determinant (GD) set of geometrical elements which define a fabric task 3 for the planes. GO curvilinear 3 fabrics can be only in various way organized 3 families of curves. Rectilinear 3 fabric can be set by various GO.

Keywords: Flat 3 fabric, geometrical determinants, curvilinear 3 fabric, rectilinear 3 fabric

The way of a task of a surface certain by a discrete framework by means of flat 3 fabrics is offered. The one-parametrical not parametrized family of curves, i.e. a discrete framework of lines is set. Two cases of a task of lines of a discrete framework of a surface are possible: dot ranks or equations

Keywords: Discrete, framework, surface, flat, 3 fabric, spline, equations, 3 fabrics, surfaces

The process of forming the surfaces is the first step in designing of a physical model of the product. In the article one of ways of formation of surfaces, which lines are lines of congruence, is considered. In the article the model of the first order’s congruence of the equi-affine images of the circles, which are received by elliptic rotation of the plane, is developed. This congruence does not have focal lines. Equi-affine images of the circle are ellipses, which are equal on the area. Structural elements of the received congruence are specified, types of coordinate lines of curvilinear coordinates are considered. The parametrical equations of u-congruence and its surfaces, formed by immersion of any line in congruence, are synthesized. Constraints for the inputting parameters of the equations are specified. Examples of surfaces at immersion of a straight line, ellipse and the spiral line with the image of the immersed line on the received surface are given.

Keywords: Congruence, equi-affine transformation, elliptic rotation, parametrical equations, generatrix, circle

On the base of geometrical model of interior revolving one axoid by another for the pairs of contacted cylinders and cones analytical representations, and computer visualization of the new constructed kinematical surfaces is developed in this research. The described geometrical model corresponds to the case when one axoid is located in the interior of another axoid. As this takes place, there are two possible variants of mutual arrangement of moving and fixed axoids. In the first variant, the moving axoid is located in the interior of the fixed axoid. In the second variant, the fixed axoid is located in the interior of the moving axoid. In the first variant, the outside surface of the moving axoid revolves around the interior surface of the fixed axoid. In the second variant, the interior surface of the moving axoid revolves around the outside surface of the fixed axoid. For pairs of axoids “cylinder – cylinder” or “cone – cone” both variants of geometrical model for constructing kinematical ruled surfaces are considered on the base of rolling one axoid along another one. As a result of the moving one axoid along another the kinematical ruled surface is generated by one of the generating lines of moving axoid. Computer graphics of the new constructed kinematical ruled surfaces based on the proposed geometrical models has been accomplished by the previously developed software application “ArtMathGraph”.

Keywords: Mathematical Modeling, Analytical Geometry; Kinematical Ruled Surfaces, Computer Graphics

The geometrical model of interior revolving one axoid by another for the pairs of contacted one-sheet hyperboloids of revolution, as the methodical base for constructing of the new kinematical ruled surfaces is proposed in this article. Analytical representation of the constructed kinematical ruled surfaces is based on the previously developed model of complex moving one axoid along another. For the case of pair of contacted one-sheet hyperboloids of revolution the geometrical model of the complex moving can be represented as a superposition of some interrelated movements: rotational movement of the moving axoid around its axis; rotational movement of the moving axoid’s axis around the fixed axoid’s axis; translational movement of the moving axoid along the common generating line of both axoids. The proposed geometrical model of interior revolving one axoid by another corresponds to the case when one axoid is located in the interior of another axoid. As this takes place, there are two possible variants of mutual arrangement of moving and fixed axoids. In the first variant, the moving axoid is located in the interior of the fixed axoid. In the second variant, the fixed axoid is located in the interior of the moving axoid. In the first variant, the outside surface of the moving axoid revolves around the interior surface of the fixed axoid. In the second variant, the interior surface of the moving axoid revolves around the outside surface of the fixed axoid. As a result of the complex moving one axoid along another the kinematical surface is generated by one of the generating lines of moving axoid. Computer graphics of the new constructed kinematical ruled surfaces has been accomplished by the previously developed software application “ArtMathGraph”.

Keywords: Mathematical Modeling, Analytical Geometry; Kinematical Ruled Surfaces, Computer Graphics

Article is devoted to designing of surfaces on the basis of flat 3-fabrics.Such families 3 of lines which block some area of the plane so are called as flat 3 fabric that through each point of this plane passes 3 lines of different families. Functional determinants of this of 3 fabrics anywhere in area doesn't address in zero, two curves of various families have no more than one general point. 3-fabric used in our case is the hexagonal, i.e. consisting of families parallel straight lines. Each line of 3 families bears on itself information on parameters of lines of the modelled surface. On the basis of information which is born on itself by each straight line of three families, some surface is modelled.

Keywords: designing of modeling, computer graphics, modeling of surfaces,hexagonal 3-fabrics

Article is devoted to a subject: in space some closed curve is set; among all possible surfaces passing through this curve to find such for which the part it concluded in a curve would have the smallest area. Curvature of any curve on a surface is equal in the set its point to curvature of flat section of a surface the adjoining curve plane.

Keywords: minimum surfaces, modeling, 3 fabric, curvature of a surface, equation of average curvature

Linear surfaces of congruence of parabolic rotation which sections are Equiaffine figure (the secant plane is parallel to the plane xOy) are described. The coefficients of quadratic forms of surfaces are determined. A special case of the surface is considered which is obtained by immersion of a circle in a parabolic congruence. The relationships of equi-affine transformation of a circle and the location of the secant plane are determined.

Keywords: linear surface, a congruence of parabolic rotation, Equi-affine transformation,quadratic form, invariant, parametric equations

Problem research of display the orthogonal projection of four dimensional hypersurface defined by the equation in an implicit form one and two hyperplane. To communicate and bypass hypersurface Discriminants assumes that the hypersurface of the two-parameter family of two-dimensional display received surfaces in five-dimensional space. The analysis of kriminant four dimensional hypersurface orthogonal projection onto the coordinate hyperplane in two coordinate axes. The intersection of three dimensional hypersurfaces axiom (kriminant) defines a two-dimensional surface that is the envelope of the two-parameter family of two-dimensional surfaces. Are the necessary and sufficient conditions of existence of this envelope.
The results obtained are used to study the Discriminants four-dimensional hypersurface, the display of two-parameter family of spheres in a five-dimensional space.  

Keywords: a family of surfaces, hypersurface, envelope, the display features, а discriminant.

  This paper presents the mathematical model for the construction of visual images of the intricate geometrical objects by means of computer graphics of transformed initial well-known surfaces. The transformed surfaces are a result of applying the mathematical transformations to the analytical representations of initial well-known classical surfaces such as plane, cone, cylinder, sphere, ellipsoid, etc. Graphical means of computer visualization of the geometrical objects is included  in the previously developed software application “ArtMathGraph” (AMG). The AMG application provides the possibility to interactive combine the individual images into computer compositions of derivable intricate geometrical objects as computer models of technical, natural or architectural visual images.    

Keywords: Mathematical Modeling, Analytical Surfaces, Computer Visualization

  New mathematical model of complex moving one axoid along another for the case of one-sheet hyperboloid of revolution as fixed and moving axoids has been proposed. On the base of this model analytical development and computer graphics of the new kinematic ruled surfaces are realized.   

Keywords: Mathematical Modeling, Computer Graphics, Kinematic Surfaces

  In many applications of profiling cutting tool, define the envelope surface collection. Along with the classic approach to the definition of the envelope of the recently used and new. So, if your family schedule projected two-dimensional surfaces in space R4, then get some three dimensional hypersurface Σ. Kriminanta this is the envelope of family of surfaces. The authors study the surface Σ when setting its parametric equations and equation in implicit form held in the works.  Found some new properties of such surfaces. Due to the fact that when profiling cutting tool set family of surfaces, coordinate transformation equations, an important task is the study received thus hypersurface. On the basis of article studies obtained in a general manner the necessary payment according to calculate kriminanty and diskriminanty hypersurface.  They show examples of how to define envelopes two families of spheres. These are family formulas transform coordinates defining the translational and helical movement. The results are illustrated by computer polygonal models.  

Keywords: a family of surfaces, hypersurface, profiling, feature display, cutting tool

  Questions contact ruled developable surfaces along their common generator. The properties of such surfaces and their striction for the initial order of contact. The obtained results can be used as a basis for engineering design of complex engineering ruled surfaces, consisting of segments of line, docked on the conditions of contact.

Keywords: ruled surface, the order of contact, the dual vector differences, ruled strips

  The article contains geometric and computer simulation of helical surface corner cutter. When the geometric modeling shows how to get the analytical dependences for determination of envelope surface collection features both display the hyper surface in the hyper plane. Shows the polygonal model envelope and one of the sections of hyper surface. Computer solid modeling process forming opportunity of obtaining models srezaemyh layers and their quality characteristics.

Keywords: Computer modeling, shaping, angular milling, helical surface