NURBS Toolbox for Matlab

【实例简介】
NURBS Toolbox for Matlab,用户手册。包含了函数的调用和例子。
TIVISTELMA Lappeenrannan teknillinen yliopisto Teknistaloudellinen tiedekunta Tietotekniikan osasto Antti Karjalainen NURBS-ohjelmakirjasto Matlab-ymparistoin Kandidaatintyo 2008 32 siva. 8 kuvaa, 6 taulukkoa ja 2 liitetta 1豆丁 Tarkastaja: D.Sc. (Tech. Arto Kaarna Hakusanat: NURBS, Matlab, ohjelmakirjasto, Mex docIn co NURBS: t ovat laajasti kaytettyja parametrisia approksimointi kayria tai pintoja NURBS eja voidaan hyodyntaa monissa sovelluksissa, naita sovelluksia ovat esimerkiksi geometri- an esittaminen tietokoneella ja laaketieteelliset sovellukset. NURBS: ien kaytto on hyvin intuitiivista Tassa tyoissa tehtin NURBS-ohjelmakirjasto Matlab-ymparistoon Matlab on moninaisin laskentatarkoituksiin kaytetty ohjelma ja ohjelmointikieli. Tassa tyossi toteutettu ohjel makiriasto tarjoaa kayttajalle mahdollisuuden kayttaa NURBS: ien laskentaa osana kayt- tajan omia ohjelmia. Toteutettu NURBS-ohjelmakirjasto sisaltaa NurBS: ien laskentaan tarkoitetut funktiot. Ohjelmakirjasto on toteutettu siten, etta sita voidaan laajentaa ja paran taa tuleyaisuudessa PREFACE This work was made at Lappeenranta University of Technology in Laboratory of Infor- mation Technology in summer and autumn 2008. I want thank Laboratory of Information Technology for good advices. Work atmosphere in the Laboratory of Information Tech nology was especially good Lappeenranta, November 24th, 20 Antti Karjalainen docn豆丁 www.docin.com CONTENTS 1 INTRODUCTION 1. 1 Background..,............ 1.2 Objectives and restrictions 1.3 Structure of the thesis 4 2 BACKROUND 5 2.1 NURBS 2. 2 Matlab 7 2.3 Research problem and how to solve it 8 PRACTICAL PART 3.1 Structure of nUrBs toolbox 3.1.1 Function calls ,,10 3. 1.2 Algorithms ....... 12 3.2 Examples of using NURBS toolbox 13 3.2.1 NURBS curves 13 3.2.2 NURBS surfaces 15 4 PERFORMANCE TESTS 9 5 DISCUSSION 5.1 Future 2 6 SUMMARY o docin. com REFERENCES 25 APPENDICES ABBREVIATIONS AND SYMBOLS CAD Computer-Aided Design MEX Matlab executable NURBs Non-Uniform Rational B-Splines docn豆丁 www.docin.com 1 INTRODUCTION 1. 1 Background Non-uniform Rational B-Splines(NURBS) are parametric curves or surfaces. NURBS can be shaped intuitively and with NURBS many kinds of different shapes can be pre sented. First NURBS have been used in CAD applications l1]. NURBS have been devel oped from B-splines which are developed from Bezier curves and surfaces [1, 2 After the first CAD applications nurbs have been used in various applications [1]. Usu ally NURBS are used to present some shapes. Example of this is modeling of a heart of a hunan[3]. Sometimes mathematical properlies of NURBS are used in simulations. Ex- ample of this is modeling of a blood vessels using NURBS and then simulating the blood flow in the vessels [4] 1.2 Objectives and Restrictions 豆丁 The purpose of this work was to implement NURBS toolbox into Matlab environment Later on NURBS toolbox refers to the toolbox implemented in this work. It can be used to any kind of tasks where one needs to evaluate NurBs. Maintenance of nurbs toolbox should be easy. So further development of the NURBS toolbox won't be problem. The NURBS Toolbox should be efficient and also easy and intuitive lo use The nurbs toolbox should be able to evaluate nUrbs curves and surfaces in 3-dimensional space. User will give the order of NURBS, control points, weights and a knot vector to NURBS toolbox and also parametric points, where NURBs will be eval uated. The NURBS toolbox returns curve and surface values corresponding to the given parametric points. For curves the values are thought of as single points or as a continuous curve consisting of line segments. Evaluation of surfaces is divided into two cases. In the first case single scattered parametric points are evaluated and in the second, a regular net of the parametric points is evaluated. Calculations will be done using Matlab MEX-functions, which are extensions to Matlab M-functions. Only very basic tools will be implemented. The nUrBS toolbox will only evaluate nurbs at the given parametric points For example evaluations of derivatives of NURBS curves or surfaces will not be implemented, neither knot-insertion or knot removal algorithms In this work we will give examples which show how changing of values of parameters will affect to the shape of a curve or a surface. This will help users of the nurbs toolbox to use it in an efficient way. Users will also learn about NurBs via examples shown. The computation efficiency of the NUrBs toolbox will be compared to existing toolboxes 1.3 Structure of the thesis Chapter 2 contains the theory behind NURBS. Some properties of NURBS and few ap plications of NURBS are presented in Chapter 2.1. In Chapter 2.2 we explain what is Matlab. Some concepls of Matlab are introduced. In Chapter 2.3 we introduce research problem of this work In Chapter 3. 1 we explain structure of the NURBs toolbox. Some examples about using NURBS curves and surfaces are shown in Chapter 3.2 In Chapter 4.1 we compare our NURBS toolbox to other toolboxes. In Chapter 4.2 we will tell which parts of the nurbs toolbox could be developed further www.docin.com 2 BACKROUND 2.1 NURBS NURBS curves and surfaces are generalizations of B-splines curves and surfaces. Sim ilarly, B-splines curves and surfaces are generalizations of Bezier curves and surfaces B-spline basis function, used in this work, has published 1972 15, 6]. Earlier formula for basis function also exists but the one used in this work is better for computer implementa tions. Earlier formula is published at late 1940s [1]. First applications of B-splines have been a tool for CAd 1. CAd techniques using B-splines was introduced first at 1973 [1 Earliest study using NURBS has been published in 1975 [11 NURBS curves are parametric. Definition of NURBS curve C(u)is Nip(uop C a<,u,, b ∑=0Mn(4) where Ni k(u are basis functions. Definition of Ni k(u)is 豆丁 1if≤l<lil1 0 otherwise L Nip( p-u.p-1(u)+ N (2) 4i+1 where ui are elements of non-uniform and non-periodic knot vector U={ L p+1 m-p-1,b…,b 3) P are the control points, @i are the weights and p is the a degree of the polynomial curve n+ 1 is an amount of control points Pi. m+ l is a length of the knot vector U.Between d p holds n+pt (4) Order of the curve is p+l. Because values of the irsl p +l elements of the knot veclor U are all a nurbs curve begins from the first control point. Because values of the last p+l elements of the knot vector U are all b NURBS curve ends to the last control point For these definitions see [1, 2, 5, 6,7,8.9, 10, 111 NURBS Surface S(u, v) is defined similarly as a curve using a tensor product ∑20∑=0Mnp()N( 5 i=02j=01V,n)V 1) N(u and N(v)are defined in Eq. 2 [1, 2, 7, 8, 9, 101 Reasons why Bezier curves and surfaces has been developed to B-spline curves surfaces and finally to NURBS curves and surfaces is that NURBS have many nice properties [1 2]. First important property is that NUrBS offer a mathematical form of different kinds of shapes. NURBS can present analytic shapes as well as free-form shapes. An important property for computer applications is that the evalualion of NURBS is compuTationally stable. By manipulating the knot vector, control points, weights and degree, NURBS can get very many different shapes In addition LO CAD applications NURBS has been applied to various purposes[1, 2]. First example is from the field of simulation [12]. Tn this example a geometrical structure of antennas is modeled, These models can be used in simulations of radiation patterns Next example is about presenting real world's objects with NURBS [13, 14, 15]. The presentation is done by fitting a NURBS surface to a 3-dimensional data set of points from a surface of a real object. In one study these 3-dimensional surfaces are measured from a 2-dimensional image [13]. NURBS can be used in modeling macromolecular structures [16]. In this case makes NURBS presenting shape of a molecule easier. In another study nurbs are used to represent sets of spectral signatures [17] NURBS have some applications in the field of medical science [3, 4, 18,19]. Here are presented few examples. NURBS are used to develop torso phantom of the measured data of a patient 33, 18]. From developed model can doctors examine human body as 3- dimensional model. In another similar study nurbs are used to present skeletal muscular model of a human body [19]. Last example is about modeling living tissue 14. NUrBs are used to model blood vessels. Then these nurbs models can be used in modeling of blood flows in the vessels [4]. An example of a blood vessel model is shown in Figure 1 【实例截图】
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