Mathematical Modeling of Some Curves in Space and their Visualization in Practical Demonstration
Helena Baraníková Slovak University of Agriculture in Nitra Faculty of Economics and Management, Department of Mathematics Tr. A. Hlinku 2 949 76 Nitra, Slovak Republic e-mail: [email protected]
Abstract Interpolation and approximation of curves in mathematics an important theoretical and practical part of the foundations of computer graphics with which the SPU students in practical demonstrations quite often encounter. Problems of mathematical modelling curves in the plane and in space is complex and difficult if you do that successfully managed, it is necessary to control a complex mathematical apparatus with its, in some cases quite difficult the understanding of those concepts such as: identifying peaks in Approximation curve, their impact for forming the shape of the resulting curve, shaping the character of the curve parameters and their impact on the change and others.
Key words: mathematics, curves in space, terminology
JEL Classification: C61, C63
"MATHEMATICS - A KEY TO THE ALL HUMAN KNOWLEDGE" L. Euler Leonhard Paul Euler (* April 15, 1707; Basel, Switzerland; - † September 18, 1783; St. Petersburg, Russia) was a Swiss mathematician and physicist who spent most of his life in Russia and Germany. He made important findings in areas as diverse as calculus and graph theory. Also introduced much of the mathematical terminology and nomenclature, particularly in mathematical analysis, such as writing a mathematical function. Famous for it also work in other areas such as in mechanics, optics and astronomy.
1. Introduction National Program of Education of the Slovak Republic has defined the direction of education in Slovakia over the next 10 to 20 years and as its first point includes: "Adapting the content and process of education and training to learning the Information Society (awakening of interest in education, strengthening creativity and ability to learn, collaborate, identify and solve problems, communicate, develop called, core competencies, supporting information and communication technologies, broadening and deepening of language skills, informal support and distance learning)", which would be useful and even necessary for the future to meet. In the past, people came to knowledge and its benefits for college. In the future, the knowledge comes to people regardless of where they are located. Not relevant to what school the student attended, but what is in it and learned what really knows or has reached the required standards and has the appropriate competence. School as a social institution providing education is part of the economic and social development of society and that is why cannot ignore the developments in information and communication technologies. 1.1 Surfaces in computer graphics and their interpretation At almost no area of which it would during its development of computer graphics and its use has not intervened. The term "Computer" means a very extensive area of creating and processing the artwork on the computer. Graphics and interpretation as such, can be divided into 2D and 3D graphics. Graphics 2D can be even further divided into vector and raster graphics. Among them is a difference in interpretation of the individual graphical objects. Raster graphics objects cover the various points and these are then saved as a bitmap. Currently, the most widely used type of graphics and for the average user and most affordable
255 way to capture the image on your computer is a photograph. Basic imaging element in this case, is the point. Raster image consists of a number of pixels, stored in fixed rows and columns. For each pixel (square) is required in addition to the position encode the color, respectively, additional parameters to the resulting object, for example transparency and others. The more points the better the image, but also more room to save them to disk. Vector graphics are composed of objects and geometric shapes. It is used to show the various geometric constructions where basic imaging element of the vector. Below a certain way digitization is then understood how the graph or picture, image on your computer encoded. In principle, there are two (main) ways of encoding methods: pixel (bitmap) graphics and vector graphics. There is a third way, and it is a metafile format that is a combination of the previous two methods. Corel Draw editor can generate all the basic types of vector objects such as curves, lines, spirals (Fig.3), stars, polygons, curves, and also calligraphic handwritten text or drawings. So created for each shape is also here you select either of its specific characteristics. Thus created objects can be further shaped, combine, split, rotate, add nodes, nodes move in etc. Working with curves is thus the same as in other editors. Drawing Tools calligraphic curves is interesting in function to Submit curve along another curve. Displayed objects can thus be easily transferred to raster output devices. Most printers and monitors work on this very principle. We argue that Corel Draw is the most widely used and very powerful ICT tools typical for vector graphics, suitable for making even the most demanding masters. It includes a number of other programs, allowing the conversion of pixel image into vector form using Corel Trace. A similar system works also used Adobe Illustrator, Zoner Draw or tool " drawing pictures " in MS Word. For animation is used, for example Corel Movie, 3D Studio MAX, True 3D Space special type of CAD softwares, Autodesk is probably at present the most widely used CAD systems representative, intended for creation of engineering drawings, but also allows for the construction and civil engineering and 2D drawings to create spatial objects either in the so called. "A wire mode", or by using the so called. Rendering the object model to actual masters. Making electrical diagrams and PCB design is the domain of Or CAD. Figure 1: Representation of curves, Bezier, Ferguson-Hermite, Coons and NURBS curves
Source: Baraníková, PhD thesis, 2007; Zaťko, 1996 Three D Graphics is characterized by its three dimensions and width, height and depth, together forming the space. The result is a three dimensional figure a model. More models can form the scene. Model is a set of points with exactly the location, in the space of polygons of connected lines (Országhová, Gregáňová, Baraníková & Tóthová, 2010). Together formed the so called, wire frame, which is applied to a raster image texture, or simulating the effect of a
256 surface or material. Pattern thus formed is placed in the scene must be generated and the resulting image. Objects in the scene can be enriched by the movement, which may be the result animation. Today's software allows you to create these graphics indistinguishable from reality. For vector graphics objects are described using various properties of geometric figures. Of these, great importance are the different types of curves that allow representation and creating more complex objects, such as curve (Fig.1-3) under the teaching of mathematics, informatics and the very foundations of computer graphics, as well as other objects and their use. Multidimensional generalization of quadratic curves and quadratic surfaces in Euclidean spaces of any dimensions using the spectral theory of symmetric linear operators, where the main tool for this analysis is the application accessible to the situation of our interest in the topic of the geometric figure, in the plane of the conic and quadratic surfaces in three dimensional space (Fig.2-3) are illustrated by the PC and E Beam Triumph 90 boards of the Department of Mathematics. "Vector editing" a majority is an interactive program that has had a long development. From the first imperfect programs over time developed comprehensive and complex graphical tools. Producers of programs increasingly include in their products, so a large number of tools, filters, functions and enhancements to the original image vector editors spread to almost unimaginable proportions. While the theoretical basis and mathematical curves of all known and freely available long enough to make him their software authors to examine and take into their solutions. Freeware editor Inkscape is an open source vector graphics editor, with capabilities similar to programs like Illustrator, Free Hand, Corel Draw, or Xara X using the W3C's standard Scalable Vector Graphics SVG. Inkscape is an open source vector graphics editor, with capabilities similar to programs like Illustrator, Free Hand, Corel Draw, or Xara X using the W3C's standard Scalable Vector Graphics SVG. Figure 2: A sphere, hyperboloid, one piece, two piece conical, surface and a paraboloid, rotating, hyperbolic, rotating cylindrical surface
Source: Baraníková, Didmattech, 2013 This allows the creation of vector graphics on par with commercial editors, where even in this software is an interesting feature insertion curves along the other curves, with modifications allowing them to deform. The program can handle not only all the basic operations with vector graphics, but also includes a few extra features such as sympathetic. Microsoft Expression 3.3, which is indeed very wide application. Conic, Hermit curves and Bezier curves of low grades and their representation. Parametric coherence and its use in the construction of spline curves 2 and 3 degree. Cardinal splines. Geometric continuity. Beta spline curve in matrix form. Quadratic area, rotary and ruler's surface, Hermit and Bezier patches 3°. Uniformed biquadratic and bicubic B spline surfaces, Nurbs, Simple redistribution scheme (Sudivision) for determining smooth surfaces. Representation of smooth surfaces polygonal mesh screen. Hermit and Bezier polynomial representation of segments. Can
257 import Adobe Illustrator up to version 8, also supports all the standard vector operation, one of the few programs of its application can draw in addition to Bezier and B spline curves, Hermitian, Picard approximation and other planar and spatial curves ( Fig.1). Since knowledge of some calculations in engineering practice for agricultural engineers indispensable familiarize yourself with some of its approximate numerical values based just on the approximation of function graphs splines curve. Polynomial and the partial polynomial curve. Geometric continuity of composite spline curves and its applications. Bilinear interpolant four points, And Gordon Coons patches, Bezier triangles, Nurb curves and surfaces, Parametric and geometric continuity of composite surfaces and structures, Coons and B spline elements. Graphical input and output devices. Basic techniques of computer graphics: Halftoning, font generation, tessellation surfaces, trimming and penetrations, rasterization, filling areas. Special data structure, representation of objects. Methods and modeling techniques of computer graphics: Winged edges, DCEL, polygonal representation (meshes), B rep and sweeping, CSG, implicit representations and F rep. Spatial redistribution techniques, wavelets, procedural, and deformable multi techniques. 1.2 Interpolation function graphs intermediate curves
Hermitian curve is the interval 〈0,1〉 given by
H (t) = V0 H 0 (t) + v0 H1 (t) + +v1H 2 (t) +V1H 3 (t) , t ∈ 〈0,1〉 , (1) where V0 , V1 are the start and end points of the curve are vectors v0 ; v1 in touch them and known Hermitian 3° third degree polynomials
2 3 2 3 2 3 2 2 H 0 (t) = 1− 3t + 2t ; H1 (t) = t − 2t + t ; H 2 (t) = −t + t ; H 3 (t) = 3t − 2t (2) and we also know the value of all the nodal points. For this purpose, we will first graph of f on the interval 〈a, b〉 interpolate a sub Hermitian curve. Let us divide the interval 〈a, b〉 i = 0, 1, ..., n − 1. into n equal parts Segment si (x) the graph of the function f in the interval 〈xi , xi+1〉 , i =
0, 1, ..., n: h = xi+1 − xi = (b – a) / n..
Segment si (x) the graph of the function f in the interval 〈xi , xi+1〉 , i = 0, 1, ..., n
2 3 2 3 x − x x − x x − x x − x x − x i i ′ i i i Gi (x) = f (xi )1− 3 + 2 + f (xi ) − 2 + + h h h h h − 2 − 3 − 2 − 3 x xi x xi x xi x xi ; + f ′(xi+1 )− + + f (xi+1 )3 − 2 h h h h
if h = xi+1 − xi = (b – a) / n; (3) i separating points: x = a + (b − a) , i = 0,1,,n . Let: Us make a sub intervals, 〈x , x 〉 ; i n i i+1 i=0,1,...,n−1can be approximated Hermitian curve following.
x n xi +1 i +1 x − x 1 (x − x )n+1 1 hn+1 h Then: i dx = i = ⋅ = ; n = 0,1, 2, 3 ... (4) ∫ h hn n +1 hn n +1 n +1 xi xi 1 n−1 1 ′ ′ (5) = h ( f (x0 ) + f (xn ) + ∑ f (xi )) + ( f (x0 ) − f (xn )) 2 i=1 12 b b − a 1 1 n−1 i Consequently ≈ + + ′ − ′ + x = a + (b − a) . (6) ∫ f (x) dx ( f (a) f (b)) ( f (a) f (b)) ∑ f (xi ) i a n 2 12 i=1 n
2. Results and discussion
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2.1 Example: Find a local extremes in space f (x, y) = x3 + y3 − 6xy .
2 2 Solution : First order partial derivatives are fx′ = 3x − 6y, f y′ = 3y − 6x . Solving simultaneous equations fx′ = 0, f y′ = 0 . Second order partial derivatives are. Thus, we solve the system 3x2 − 6y = 0 and 3y2 − 6x = 0 . Find stationary point
M 0 = [0,0], M 1 = [2,2]. The stationary point bode M 0 = [0,0] is A = f xx′′ (0,0) = 0 , A B 0 − 6 B = f ′′ (0,0) = −6 ; f ′′ (0,0) = 0 . Determinant D = = < 0. Function xy yy B C − 6 0
f (x, y) in a stationary point M 0 = [0,0] local extremum allowed. Figure 3: Representation of Example 2.1 by of illustration of using the Microsoft graphic calculator
Source: Országhová, 2010 ′′ The stationary point bode M 1 = [2,2] is A= fxx′′ (2,2) = 12 , B = fxy′′ (2,2) = −6 and f yy (2,2) = 12. 12 − 6 Determinant D = = 108 > 0 . Because D > 0 while A= 12 > 0 the function f (x, y) at − 6 12 point M1 local minimum fmin = f (2,2) = 8 + 8 − 24 = −8.
2.2 Example: Write the equation of the tangent plane to the spherical surface of x 2 + y 2 + z 2 = 9 the touch point A = [1, 2,2].
Solution: We get the function f (x, y) = ± 9 − x 2 − y 2 (resp. z = ± 9 − x 2 − y 2 ). The square root is defined for nonnegative numbers, so the expression under the square root must satisfy the condition 9 − x 2 − y 2 ≥ 0. We can assure you that truly meet the coordinates of point A, if we substitute the coordinates of the original equation. ∂z − x ∂z − y ∂(1,2) −1 ∂(1,2) If = , = ; then = , = −1. 2 2 ∂x 9 − x 2 − y 2 ∂y 9 − x − y ∂x 2 ∂y −1 Equation and find the tangent plane is λ : (z − 2) = (x −1)− (y − 2) and the after the 2 adjustment λ : x + 2y + 2z − 9 = 0 (see Figure 4).
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The research was conducted on a sample of 150 students and 4 teachers providing instruction, training and the testing of these students at SUA, where his sub-objective was to investigate the use of multimedia in their education. Figure 4: Representation of Example 2.2 by of illustration of using the Microsoft graphic calculator
Source: Baraníková, Didmattech, 2012 The research showed that multimedia programs used in teaching 34% of respondents surveyed, the majority of respondents do not use dial multimedia programs. Similar results were also recorded in the use of educational programs (see Figure 5, 6).
Figure5: Left:The use of multimedia programs in education, teaching and preparing it teaching; Figure 6: Right:The use of software programs in education, teaching and preparing it teaching
Source: own
3. Conclusion Working with presentations is simple and can be done individually for each student. Between individual Slide moves forward and backward arrows, or may return to the beginning and start learning again. Each concept is explained by illustrative examples. At the end of each presentation is repetition in the form of questions with or with the addition of the correct answer. Identical to the e-learning. On the question of the possibilities student answers by clicking on the answer, the correct answer will take this further. When incorrect answer is
260 alerted and returned to the wrong Questions Answered that can be repaired. For questions you cant correct the word by over time. This training is appropriate for inclusion in each phase of the teaching process and the appropriate individual learning for all students .Multimedia math programs not only allow detailed disclosure of the issue , consulting and repetition of the substance, but also the preparation and printing of various tests. Offer educational software that handle such matters disciplines, is the world market satisfactory, it can be used effectively as meet the required criteria and didactic aspects. In this paper, we point out several options for implementation of computer technology not only in numerical calculations, graphical interpretation of the basic concepts of higher mathematics in the plane and in space, but we have outlined their approximation aspects (Országhová, Gregáňová, Baraníková & Tóthová, 2010, Harvey Mudd College, 2014). The Faculty of Economics and Management Slovak University of Agriculture in Nitra is subject Mathematics compulsory subject for students from different types of study of bachelor, master, doctorate never intended for the various faculties of the university. In this paper we want to point out, as a university teacher can make use of modern information and communication technologies (ICT) in teaching the subject of higher mathematics, the aspect of a reference to the illustrative interpret. In this paper we want to point out, as a university teacher can make use of modern information and communication technologies in teaching the subject of higher mathematics, the aspect of a reference to interpret the visual.
Acknowledgements Contribution originated within the project KEGA 021SPU-4/2011: Teaching mathematics with applications - content changes in university mathematical education (Solution in 2011- 2013).
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