Constructing a G2-Smooth Compound Curve Based on Cubic Bezier Segments
About: This article is published in Geometry & Graphics.The article was published on 2021-11-11 and is currently open access. It has received 4 citations till now.
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08 Apr 2022
TL;DR: The theoretical foundations of point calculus for geometric modeling, invariant relating to parallel projection, have been presented in this article , where specific examples have been described, including affine ratio of three points of a straight line, intersection of two straight lines and intersection of a line with a plane, parallel translation and tangent to a curve.
Abstract: The purpose of this paper is to familiarize experts in geometric and computer modeling with specific tools for point calculus; demonstrate the possibilities of point calculus as a mathematical apparatus for modeling of multidimensional space’s geometric objects. In the paper with specific examples have been described the basic constructive tools for point calculus, having invariant properties relating to parallel projection. These tools are used to model geometric objects, including: affine ratio of three points of a straight line, intersection of two straight lines, intersection of a straight line with a plane, parallel translation and tangent to a curve. The theoretical foundations of point tools for geometric modeling, invariant relating to parallel projection, have been presented. For example, instead of traditional determination for straight lines intersection point by composing and solving a system of equations in coordinate form, zeroing of a moving triangle’s area is used. This approach allows to define geometric objects in multidimensional spaces keeping the symbolic representation of point equation, as well as to perform its coordinate-wise calculation at the last stage of modeling, which allows to significantly reduce computing resources in the process of solving the problems related to engineering geometry and computer graphics.
The local results of the research presented in this paper, which served as examples for the use of point calculation constructive tools, are: definition of the cubic Bezier curve as a curve of one relation in point and coordinate form; determination of excessive parameterization of the plane and bypass arcs based on it; determination of the tangent to the spatial curve by differentiation the original curve with respect to a current parameter, followed by parallel transfer of the obtained segment to the tangency point; the general point equation for the torso surface has been obtained on account of its definition as a geometric place of tangents to its cusp edge, and examples for the construction of torso surfaces based on the cubic Bezier curve and a transcendental space curve have been presented.
4 citations
08 Apr 2022
TL;DR: The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.
Abstract: In this paper has been considered the main content and distinctive features of the “Geometric Modeling” training course for the “Mathematics and Computer Science” training program 02.03.01 (“Mathematical and Computer Modeling” specialization).
The goal of the “Geometric Modeling” course study is the assimilation of mathematical methods for construction of geometric objects with complex curved shapes, and techniques for their computer visualization by using polygons of curves and surfaces. Methods for construction of structures’ curved shapes using spline representations, as well as techniques for construction of surfaces and volumetric geometries using motion operations and basic logical operations on geometric objects are considered. The spline representations include linear and bilinear splines, Hermite cubic splines and Hermite surfaces, natural cubic and bicubic interpolation splines, Bezier curves and surfaces, rational Bezier splines, B-splines and B-spline surfaces, NURBS-curves and NURBS-surfaces, transfinite interpolation methods, and splines of surfaces with triangular form. Logical operations for intersection of two spline curves, and intersection of two parametric surfaces are considered. The principles of scientific visualization and computer animation are considered in this course as well.
Some examples for visualization of initial data and results of curves and surfaces construction in two- and three-dimensional spaces through the software shell developed by authors and used by students while doing tests have been demonstrated. The software shell has a web interface with the WebGL library graphic support. Tasks for four practical studies in a computer classroom, as well as several variations of homework are represented.
The problems occurring in preparation materials for some course sections are discussed, as well as the practical importance of acquired knowledge for the further progress of students.
The paper may be interesting for teachers of “Geometric Modeling” and “Computer Graphics” courses aimed to students with a specialization in mathematics and information, as well as to those who independently develop software interfaces for algorithms of geometric modeling.
1 citations
10 Oct 2022
TL;DR: In 2021, in terms of the success of the seminar "Geometry and Graphics" and the conference "Problems of Engineering Geometry", the success rate increased and judging by the number of reports, scientific work on the profile of the department is carried out in a small number of departments.
Abstract: The article is devoted to the annual All-Russian scientific and methodological conference "Problems of Engineering Geometry" and the annual All-Russian scientific and methodological seminar "Geometry and Graphics" in 2021. Statistical information about the conference and seminar is provided: the number of participants, universities, the number of cities and countries in which universities are located -participants. Using the expression proposed earlier, the activity of participation of the departments of graphic disciplines in the conference "Problems of Engineering Geometry" and the seminar "Geometry and Graphics", held in 2021, was determined. The comparison of the number of participants and reports of the conference and seminar in 2021 with the number of participants and reports is given and analyzed International Internet conferences "Quality of graphic training" at the Perm National Research Polytechnic University. The results of the All-Russian Seminars "Geometry and Graphics" and the All-Russian Conferences "Problems of Engineering Geometry" of the last two years are compared with each other. In order to compare conferences and seminars quantitatively, not qualitatively, a relationship has been proposed. The content of the reports of the participants of the conference and the seminar is briefly considered. Conclusions are drawn: 1) in 2021, in terms of the success of the seminar "Geometry and Graphics" and the conference "Problems of Engineering Geometry", we managed to move forward - the success rate increased; 2) judging by the number of reports, scientific work on the profile of the department is carried out in a small number of departments. This is due to shortcomings in the staffing of departments of graphic disciplines by teachers. One of them is a lack of understanding that the winners or participants of All-Russian and regional Olympiads who have undergone appropriate training should work as teachers.
1 citations
TL;DR: The article describes the model of group pursuit of a single target by the chase method, where each object has its own tasks, implemented by the chased method, and can be in demand in the design of unmanned aerial vehicles with elements of autonomous control and artificial intelligence.
Abstract: The article describes the model of group pursuit of a single target by the chase method. All objects participating in the pursuit model move with a constant modulo speed. One of the participants in the process moves along a certain trajectory and releases objects at specified intervals, the task of which is to achieve the goal by the chase method. All objects have restrictions on the curvature of the motion path. A single target, in turn, is tasked with achieving the target that releases objects using the parallel approach method. For each pursuing object, a detection area is formed in the form of two beams. The object's velocity vector is directed along the bisector of the angle formed by such rays. If the target enters the detection area, then the object starts pursuit and the velocity vector is directed to the target. If the target leaves the detection area, then the object makes a uniform and rectilinear movement. The task is to implement a dynamic model of multiple group pursuit, where each object has its own tasks, implemented by the chase method. As an example, where the model developed in the article could be in demand, the following example can be given. Consider the movement of a low-maneuverable object that is overtaking a faster target. As a means of protection, instead of releasing passive heat traps, it is proposed to drop a variety of autonomously controlled weapons. An analysis of existing studies has shown that such means of protecting aircraft do not exist. The results of the research can be in demand in the design of unmanned aerial vehicles with elements of autonomous control and artificial intelligence.
References
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01 Jan 2014
TL;DR: Allgemeine anwendungen mathematik de de kunstobjekten verschlungene, de kundenrezensionen geometrie und ihre, überall geometries wissenschaft, €n sachunterricht 3 4 klasse natur und leben.
Abstract: Geometrie und ihre anwendungen institut für kunst und. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen bibliothek. canon 7d manual pdf download pdf download. geometrie und ihre anwendungen in kunst natur und technik. geometrie wikipedie. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. allgemeine anwendungen mathematik de. geometrisches modellieren von kunstobjekten verschlungene. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. anwendungen für das smartphone gt pc und internet. sachunterricht 3 4 klasse natur und leben. de kundenrezensionen geometrie und ihre. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. lwb kurs 8r 4 halbjahr sommersemester 2020. ge glaeser. buchtipp geometrie und ihre anwendungen in kunst natur. geometrie und ihre anwendungen in kunst natur und technik. darstellende geometrie. geometrie und ihre anwendungen in kunst natur. geometrie und ihre anwendungen in kunst natur und technik. der mathematische werkzeugkasten anwendungen in natur und. buchkritik zu geometrie und ihre anwendungen in kunst. geometrie und ihre anwendungen in kunst natur und technik. leinwandbild nahtlose muster in modernen skandinavischen. geometrie und ihre anwendungen in kunst natur und technik. geometrie und ihre anwendungen in kunst natur und technik. überall geometrie wissenschaft
38 citations
TL;DR: In technical inventions related to innovative developments, the role of one of the main components belongs to geometry, and in technical inventions the geometrical component is the determining one, this statement is proved by examples developed based on geometry of following inventions.
Abstract: In technical inventions related to innovative developments, the role of one of the main components belongs to geometry. A follow hypothesis has been adopted: in technical inventions the geometrical component is the determining one. This hypothesis applied to technical inventions can be confirmed by any copyright certificate, any patent both in Russia and abroad. In proposed paper this statement is proved by examples developed based on geometry of following inventions. 1. Screen feeder for sticky masses. Screen feeder’s grates are made in pairs, and between grate pairs there are gaps for screening of material’s size-defined fractions. In the screen-feeder has been proposed such geometry of grates that grates of each pair could clean each other, thereby preventing sticking on the surface and destroying the gaps between the pairs, which transforms the usual screen-feeder with cylindrical grates at an ordinary feeder. 2. Double-screw mixer for paste-like masses. The mixer consists of two contiguous worms. Their surfaces are the helical ones, in cross-section consisting of two quarters of circles stacked at the ends. Such cross-section allows homogenize the mixed paste-like material in the best way, and also deliver it under higher pressure in an extrusion head, that improves a final product. 3. Machine for processing of multi-faceted surfaces. This invention serves for manufacture of worms with a cross-section composed of two, three, etc. pieces of circles of the same radius and angle. Worms, made with this machine, are designed for the above mentioned two-screw mixer. 4. Method of mechanical processing. This method is also intended to manufacture of worms for two-screw mixer.
26 citations
TL;DR: The problem related to improvement of quality in engineering training for students of technical high educational institutions, that today is considered as one of the main tasks for the system of higher professional education, is solved by proposed method.
Abstract: Using the computer graphics tools in the design allows improve the design quality and speed, as well as provide the qualitative front end engineering design. In this paper the problem related to improvement of quality in engineering training for students of technical high educational institutions, that today is considered as one of the main tasks for the system of higher professional education. A method proposed by authors allows solve this problem in the frame of students training in disciplines of graphic cycle, and directed to introducing the computer technologies in the teaching process of students. This method provides the development of students’ professional skills in the area of products and electronic devices design, and in the front end engineering design. In such a case, design is regarded from the standpoint of project-process approach. Project-process approach is a combination of interrelated projects implemented in the frame of process. A process is considered as a group of projects aimed at achievement of a planned result – a design object model. Design objects models in the area of mechanical engineering, electronics and electric power engineering, are presented as drawings, schemes and 3D models. That is why the emphasis is upon models and drawings building by means of CAD software (Autodesk AutoCAD, Autodesk Inventor, Kompas-Grafik, Kompas 3D). A high level of students’ competences can be achieved by modernization the educational content so that from their first teaching year the students could see the relation of learned graphical disciplines with their future professional occupation and prospects of production development and project activities. State-oftheart software for devices design provides for students an opportunity to extent their possibilities in the learning of graphical disciplines’ courses. The project-process approach is necessary for identifying and studying the relationship between the design as a process and the reformative human activity in general.
20 citations
13 citations
TL;DR: 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 and examples for constructing segments of cubic curves meeting the given boundary conditions are considered.
Abstract: 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
has been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt.
Based 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.
The 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.
The 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.
Curvature 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.
Have 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.
Has 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.
The paper is for educational purposes, and intended for in-depth study of computer graphics basics.
12 citations