Second Order Curves on Computer Screen

2018 ◽  
Vol 6 (2) ◽  
pp. 100-112 ◽  
Author(s):  
Виктор Короткий ◽  
Viktor Korotkiy ◽  
Е. Усманова ◽  
E. Usmanova
Keyword(s):  
Computer Graphics ◽  
Computer Program ◽  
Geometric Modeling ◽  
Main Tool ◽  
Second Order ◽  
Modern Computer ◽  
Order Curve ◽  
Junction Points ◽  
Tools And Techniques ◽  
Straight Lines

Modern computer graphics is based on methods of computational geometry. The curves and surfaces’ description is based on apparatus of spline functions, which became the main tool for geometric modeling. Methods of projective geometry are almost not applying. One of the reasons for this is impossibility to exactly construct a second-order curve passing through given points and tangent to given straight lines. To eliminate this defect a computer program for second order curves construction has been developed. The program performs the construction of second-order curve’s metric (center, vertices, asymptotes, foci) for following combinations: • The second-order curve is given by five points; • The second-order curve is given by five tangent lines; • The second-order curve is given by a point and two tangent lines with points of contact indicated on them; • The parabola is given by four tangent lines; • The parabola is given by four points. In this paper are presented algorithms for construction a metric for each combination. After construction the metric the computer program written in AutoLISP language and using geometrically exact projective algorithms which don’t require algebraic computations draws a second-order curve. For example, to construct vertices and foci of two parabolas passing through four given points, it is only necessary to draw an arbitrary circle and several straight lines. To construct a conic metric passing through five given points, it is necessary to perform only three geometrically exact operations: to construct an involution of conjugate diameters, to find the main axes and asymptotes; to note the vertices of desired second-order curve. Has been considered the architectural appearance of a new airport in Simferopol. It has been demonstrated that a terminal facade’s wavelike form can be obtained with a curve line consisting of conic sections’ areas with common tangent lines at junction points. The developed computer program allows draw second-order curves. The program application will promote the development of computer graphics’ tools and techniques.

scholarly journals APPLICATION OF CIRCLES FOR CONJUGATION OF FLAT CONTOURS OF THE FIRST ORDER OF SMOOTHNESS

2021 ◽  
pp. 162-171
Author(s):  
Galyna Koval ◽  
Margarita Lazarchuk ◽  
Liudmila Ovsienko
Keyword(s):  
Coordinate System ◽  
Geometric Modeling ◽  
Second Order ◽  
First Order ◽  
Order Curve ◽  
First Case ◽  
Affine System ◽  
Coordinate Lines ◽  
Point Of Intersection ◽  
Projective Coordinate

In geometric modeling of contours, especially for conjugation of sections of flat contours of the first order of smoothness, arcs of circles can be applied. The article proposes ways to determine the equations of a circle for two ways of its problem: the problem of a circle with a point and two tangents, none of which contains a given point, and the problem of a circle with three tangents. The equations of the circles were determined in both cases using a projective coordinate system. In the first case, when a circle is given by a point and two tangents, neither of which contains this point, the center of the conjugation circle is defined as the point of intersection of two locus of points - the bisector of the angle between the tangents and the parabola, the focus of which is a given point. given tangents. In the general case, there are 2 conjugation circles for which canonical equations are defined. Parametric equations of conjugate circles, the parameters of which are equal to 0 and ∞ on tangents and equal to one at a given point, with the help of affine and projective coordinates of points of contact are determined first in the projective coordinate system, and then translated into affine system. For the second case, when specifying a circle using three tangent lines, the equation of the second-order curve tangent to these lines is first determined in the projective coordinate system. The tangent lines are taken as the coordinate lines of the projective coordinate system. The unit point of the projective coordinate system is selected in the metacenter of the thus obtained base triangle. The equation of the tangent to the base lines of the second order contains two unknown variables, positive or negative values ​​which determine the location of four possible tangents of the second order. After writing the vector-parametric equation of the tangent curve of the second order in the affine coordinate system, the equation is written to determine the parameters of cyclic points. In order for the equation of the tangent curve of the second order obtained in the projective plane to be an equation of a circle, it must satisfy the coordinates of the cyclic points of the plane, which allows to write the second equation to determine the parameters of cyclic points. By solving a system of two equations, we obtain the required equations of circles tangent to three given lines.

Reconstruction of Quadratic Cremona Transformation

2017 ◽  
Vol 5 (2) ◽  
pp. 59-68
Author(s):  
Короткий ◽  
Viktor Korotkiy
Keyword(s):  
Geometric Modeling ◽  
The Other ◽  
Practical Implementation ◽  
Cremona Transformation ◽  
Quadratic Transformation ◽  
Modern Computer ◽  
Set Up ◽  
The One ◽  
Straight Lines ◽  
Base Data

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.

Some aspects of the pedagogical model of constructive geometric modeling

2020 ◽  
Author(s):  
Denis Voloshinov ◽  
K. Solomonov ◽  
Lyudmila Mokretsova ◽  
Lyudmila Tishchuk
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Educational Process ◽  
Educational Institutions ◽  
Software Systems ◽  
Teaching Methodology ◽  
Models Of Teaching ◽  
Software Product ◽  
The University

The application of constructive geometric modeling to pedagogical models of teaching graphic disciplines today is a promising direction for using computer technology in the educational process of educational institutions. The essence of the method of constructive geometric modeling is to represent any operation performed on geometric objects in the form of a transformation, as a result of which some constructive connection is established, and the transformation itself can be considered as a result of the action of an abstract cybernetic device. Constructive geometric modeling is a popular information tool for information processing in various applied areas, however, this tool cannot be appreciated without the presence of appropriate software systems and developed design techniques. Traditionally, constructive geometric modeling is used in the design of mechanical engineering, energy, aircraft and shipbuilding facilities, in architectural and design engineering. The need to study descriptive geometry at the university in recent years has something in common with the issues of mastering graphic packages of computer programs in the framework of the new discipline "Engineering and Computer Graphics". The well-known KOMPAS software product is considered the simplest and most attractive for training. It should be noted the important role of graphic packages in the teaching of geometric disciplines that require a figurative perception of the material by students. Against the background of a reduction in classroom hours, computer graphics packages are practically the only productive teaching methodology, successfully replacing traditional tools - chalk and blackboard.

scholarly journals Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Mehwish Bari ◽  
Ghulam Mustafa ◽  
Abdul Ghaffar ◽  
Kottakkaran Sooppy Nisar ◽  
Dumitru Baleanu
Keyword(s):  
Computer Graphics ◽  
Geometric Modeling ◽  
Tension Control ◽  
Free Form ◽  
Subdivision Surfaces ◽  
Smooth Curves ◽  
Practical Applications ◽  
Tension Parameter ◽  
Exact Reproduction ◽  
Free Form Surfaces

AbstractSubdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having $C^{1}$ C 1 smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

TABLES, a program to display space-group symmetry information in three dimensions

1987 ◽  
Vol 20 (6) ◽  
pp. 532-535 ◽  
Author(s):  
C. Abad-Zapatero ◽  
T. J. O'Donnell
Keyword(s):  
Computer Graphics ◽  
Computer Program ◽  
Three Dimensional ◽  
Spatial Location ◽  
Three Dimensions ◽  
Group Symmetry ◽  
Space Group ◽  
Interactive Display ◽  
Symmetry Operators ◽  
Group Information

TABLES is a computer program developed to display the crystal symmetry and the spatial location of the different symmetry operators for a given space group using interactive computer graphics. It allows the three-dimensional interactive display of the space-group information contained in International Tables for Crystallography [(1983), Vol. A. Dordrecht: Reidel]. Such a program is useful as a teaching aid in crystallography and is valuable for exploring molecular packing arrangements.

A Torus Patch Approximation Approach for Point Projection on Implicit Surfaces

2011 ◽  
Vol 346 ◽  
pp. 259-265
Author(s):  
Xiao Ming Liu ◽  
Lei Yang ◽  
Qiang Hu ◽  
Jun Hai Yong
Keyword(s):  
Iterative Algorithm ◽  
Geometric Modeling ◽  
Second Order ◽  
Implicit Surfaces ◽  
Implicit Surface ◽  
Approximation Approach ◽  
Initial Values ◽  
Point Projection

Point projection on an implicit surface is essential for the geometric modeling and graphics applications of it. This paper presents a method for computing the principle curvatures and principle directions of an implicit surface. Using the principle curvatures and principle directions, we construct a torus patch to approximate the implicit surface locally. The torus patch is second order osculating to the implicit surface. By taking advantage of the approximation torus patch, this paper develops a second order geometric iterative algorithm for point projection on the implicit surface. Experiments illustrate the efficiency and less dependency on initial values of our algorithm.

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