Formation of Surfaces Under Kinetic Displaying

The paper is devoted to the consideration of a number of issues related to the creation of an algorithmic complex designed to solve positional and metric problems with quadrics on a projection model . A feature of the complex is the active use of geometric schemes and algorithms involving imaginary geometric images. In the paper has been presented a detailed description of constructive geometric algorithms for constructing of conics, quadrics and associated geometric images in a system of constructive geometric modeling – Simplex. All the discussed algorithms are available for independent repetition by the reader. In the paper have been presented and implemented algorithms for constructing conic from a point, a polar, and three points; constructing conic from two pairs of complex conjugate points and one real point; determination of a point on a quadric’s surface; setting a quadric by nine points in three-dimensional space. A new alternative frame of the quadric has been considered, based on which have been solved problems of constructing a tangent and a normal to the quadric, finding an intersection line of an arbitrary plane with the quadric, and performing polar and inverse transformations with respect to the quadric. Have been proposed algorithms for constructing an autopolar tetrahedron with respect to the quadric, and for constructing a conic from an autopolar triangle and two points. Have been considered problems of determining a collinear transformation in three-dimensional space and control the quadric through it. The implementation of the algorithms considered in the paper allowed conclude that there is an urgent need to develop tools for modeling imaginary conics, without which the complex of solving problems with quadrics cannot be taken for the complete one.

Algorithmic Complex for Solving of Problems with Quadrics Using Imaginary Geometric Images

The paper proposes a computational method for solving differential equations of mathematical physics by approximating the desired solution using geometric objects of multidimensional space passing through predetermined points. The essence of the method is to simulate an approximating geometric object of a multidimensional affine space constructed on a regular multidimensional network of points. In this case, the response function values satisfying the solution of the original differential equation are calculated at the nodal points of the network. Modeling of approximating geometric object is carried out by means the arcs of algebraic curves passing through predetermined points. It should be noted that taking into account the boundary conditions does not require changes in the geometric algorithm or point equations. It is sufficient to use the necessary coordinates of the nodal boundary points corresponding to the boundary conditions of the solution of the differential equation. To achieve the required accuracy of the solution of differential equations, it is sufficient to compact the reference network of points. Under such conditions, it is possible to use as a single geometric object to approximate the solution of the differential equation, and composite, based on the simulation of multidimensional contours on a regular network of points of multidimensional space. A geometric classification of differential equations depending on the number of parameters determining the approximating geometric object in multidimensional space is proposed. An example of solving the inhomogeneous heat equation by means of an approximating response surface passing through 16 predetermined points is given. In this case, the required approximating compartment of the response surface passes through 3 straight lines that correspond to the boundary conditions and satisfies the solution of the original differential equation at the nodal points of the 16-point network. A comparison of the results of solving the inhomogeneous heat equation approximated by a 16-point compartment of the response surface with the reference compartment of the surface obtained by the method of separating variables is also presented.

Modeling Approximating the 16-Point Compartment the Response Surface With Respect To the Solution of the Inhomogeneous Heat Equation

Based on the published data, the essence of computational graphics has been laid down. Have been reported examples of new results obtained only through accurate computer constructions and measurements. The work content is a presentation of new ways to achieve the goal by solving non-traditional engineering problems. The author's method of projection with time-stamps, which, in fact, is a computer descriptive geometry, allows solve multi-parameter (not to be confused with multi-dimensional) problems with 9 variables [1–3; 13]. The author’s method of two-axis equal-sized evolvement [11; 12; 17] allows quantitatively measurements of solid angles. The addition of trigonometric functions (sinuses, sinusoids, etc.) can also be considered as a novelty [10; 11; 14]. At the junction of analytic (AG) and descriptive geometries have been calculated parameters of dodecahedron and has been given its mathematical description. In the traditional AG task, the required parameters have been calculated graphically, including a point’s speed of movement. Has been presented the author’s method for determining the instantaneous center in theoretical mechanics. For the first time, the equality of the angles of rotation for points and the link as a whole has been established, and a continuous centroid has been built. By decomposition of vectors a new way for summing up theirs vertical projections has been demonstrated. The developed method of projections with time-stamps allows simultaneously consider such parameters as spatial coordinates of moving objects (two or more) in time, their speeds and even sizes, including the variable ones. Has been shown the possibility for graphical programming while solving systems of equations, as well as for graphical solution of algebraic and stereometric problems. This publication aims to disseminate computer methods for engineering problems solving.

Computational Graphics in Solving of Non-Traditional Engineering Problems

A brief journey into the history of industrial revolutions has been presented. It is noted that our society has entered the third industrial revolution’s era. In this regard, the main consequences of the third industrial revolution have been noted. The stages of development for design methods and the basic science providing the design process have been considered. The historical necessity and significance of Gaspar Monge’s descriptive geometry appearance has been considered as well. Modern products design approaches using CAD systems are described. It is stated that design has again returned to three-dimensional space, in fact prior to the Monge’s era, but at a new stage of development. The conclusion is drawn that, taking into account the realities and needs of modern production, it is necessary to modernize the descriptive geometry course for technical high educational institutions. The author's suggestions on course content changing are presented related to extension of one sections and reducing of another ones, taking into account the fact that in real design practice the designer solves geometric problems in three-dimensional space, rather than in a complex drawing. It is noted that in connection with the extensive use of CAD systems, the design stages and the composition of design documentation developed at each stage are changed. Such concepts as \"electronic model\" and \"electronic document\" have appeared and are widely used, that is confirmed by adoption of new USDD standards. In such a case the role and significance of some types of drawings may change in the near future, since modern CAD systems allow transfer to production not 2D drawings, but electronic models and product drawings.

The Content of Descriptive Geometry Course For High Educational Institution’s In the Third Industrial Revolution’s Era

Computer Simulation of Kinematic Surfaces

Graphic Reconstruction Algorithms of the Second-Order Curve, Given by the Imaginary Elements

In this paper are considered historically the first (the 60’s of the 20th century) computational methods for algebraic cubic curves constructing. The analysis of a general cubic curve equation r(t)=a3t3+a2t2+a1t+a0 \nhas been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt. \nBased on the general cubic curve equation have been obtained equations of a cubic curve passing through two predetermined points and having predetermined tangents at these points. \nThe equations have been presented both in Ferguson and Bézier forms. It has been shown that the cubic curve vector equation (for example, the standard equation of a Bezier curve) can be represented in a point form. Have been considered examples for constructing segments of cubic curves meeting the given boundary conditions. \nThe generalized cubic curve equation, containing weight coefficients, has been obtained by the method of exit into four-dimensional space. Has been considered a vector parametric equation of a conical section, passing through two given points and touching predetermined straight lines at these points. The conical section is considered as a special case of a cubic curve. \nCurvature can be specified as an additional boundary condition. Has been considered the possibility for constructing a cubic curve with fixed positions of contacting planes at end points and given radii of curvature. Has been proposed an algorithm for constructing a plane cubic curve with a given curvature at the end points. \nHave been considered algorithms for constructing smooth compound Ferguson-Bezier curves. Smoothness conditions are imposed on a compound curve: 1) at any of its points, the curve must have a tangent (no fractures are allowed), 2) the curvature vector must be changed continuously from point to point (no discontinuous jump in the curvature vector is allowed neither in modulus no in direction). Have been proposed examples for constructing compound Ferguson-Bézier curves. \nHas been performed comparison of polynomial cubic spline with compound parametrically defined curves. Have been given examples for constructing cubic splines with fastened and free ends. \nThe paper is for educational purposes, and intended for in-depth study of computer graphics basics.

Cubic Curves in Engineering Geometry

The fundamental issue of constructing a nine-point quadric was frequently discussed by mathematicians in the 19th century. They failed to find a simple linear geometric dependence that would join ten points of a quadric (similar to Pascal's theorem, which joins six points of a conic section). Nevertheless, they found different algorithms for a geometrically accurate construction (using straightedge and compass or even using straightedge alone) of any number of points of a quadric that passes through nine given points. While the algorithms are quite complex, they can be implemented only with the help of computer graphics. The paper proposes a simplified computer-based realization of J.H. Engel’s well-known algorithm, which makes it possible to define the ninepoint quadric metric. The proposed graphics algorithm can be considered an alternative to the algebraic solution of the stated problem. The article discusses two well-known graphical algorithms for constructing a quadric (the Rohn — Papperitz algorithm and the J.H. Engel algorithm) and proposes a simplified version of the J.H. algorithm. For its constructive implementation using computer graphics. All algorithms allow you to determine the set of points and the set of flat sections of the surface of the second order, given by nine points. The Rohn — Papperitz algorithm, based on the spatial configuration of Desargues, is best suited for its implementation on an axonometric drawing using 3D computer graphics. Algorithm J.H. Engel allows you to solve a problem on the plane. The proposed simplified constructive version of the algorithm J.H. Engel is supplemented with an algorithm for constructing the principal axes and symmetry planes of a quadric, given by nine points. The construction cannot be performed with a compass and a ruler, since this task reduces to finding the intersection points of two second-order curves with one known general point (third degree task). For its constructive solution, a computer program is used that performs the drawing of a second order curve defined by an arbitrarily specified set of five points and tangents (both real and imaginary). The proposed graphic algorithm can be considered as an alternative to the algebraic solution of the problem.

Viktor Korotkiy

Papers

Computer Simulation of Kinematic Surfaces

Graphic Reconstruction Algorithms of the Second-Order Curve, Given by the Imaginary Elements

Cubic Curves in Engineering Geometry

Graphic Algorithms for Constructing a Quadric, Given Nine Points

Surface as an Object of Computer Geometric Modelling