Geometrik modelleme ve sentetik eğrilerin analizi
Geometric modelling and synthetic curves
- Tez No: 66646
- Danışmanlar: PROF. DR. TEOMAN KURTAY
- Tez Türü: Yüksek Lisans
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Eğriler, Geometrik modeller, Tasarım, Curves, Geometric models, Design
- Yıl: 1997
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Makine Mühendisliği Ana Bilim Dalı
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET Geometrik model bir planlamanın,dizayn ve üretimin başlangıç noktasıdır. Bu yüzden parça dizaynının, bir üretimde 1. kısmı olması ve parçanın onsuz tanımlanamaması nedeniyle başlangıç noktasıdır. Günümüzde artık bilgisayarlar her türlü alanda kullanılmaktadırlar bunlardan bir taneside imalattan önceki safha olan,tasarım alanıdır, imalatı yapılacak parçanın şeklinin bilgisayarda tanımlanabilmesi için çeşitli eğri metodları geliştirilmiştir. Bu eğrilerin arasında bildiğimiz basit eğriler daire, çizgi gibi analitik eğrilerde kullanılmaktadır. Yanlız bu eğriler çoğu zaman imalatı yapılacak parçanın çiziminde yeterli olmamaktadır. Bu yüzden daha kompleks eğri yüzeylerini tanıtmamak için sentetik eğriler oluşturulmuştur. Sentetik eğriler hem parçaya istenilen formun verilmesinde kolaylık hemde istenilen kısmın değiştirilmesinde bir kolaylık sağlamaktadır. îleriki bölümlerde sentetik eğrilerin bu avantajlarına detaylı olarak değinilmiştir. Bahsedilen başlıca sentetik eğriler ; Hermit, bezier, B-spline ve oransal eğrilerdir. Bilgisayarlı tasarımda en çok kullanılan eğri türü günümüzde oransal eğrilerdir. İlerideki bölümlerde bu eğrilerin avantajları, dezavantajları, birbirlerine karşı üstün tarafları ve matematiksel eşitlikleri örnekler ile detaylı olarak anlatılmıştır.
Özet (Çeviri)
SUMMARY GEOMETRIC MODELING and SYNTHETIC CURVES This work discusses geometric modeling and its relevance to CAD / CAM. Early CAD /CAM systems focused on modeling enginering objects. As a results geometric models, that once were more than adequate for drafting purposes are not acceptable for engineering applications. A basic requiriment.therefore. Is that a geometric model should be an unambiguous representation of its corresponding object. That is to say, the model should be unique and complete to all engineering functions from documantation (drafting and shading ) to engineering analysis to manufacturing. i A geometric model of an object and its related database have 3 types of geometrict models, wireframes »surfaces and solids. Users usually have to decide on the type of modeling technique based on the ease of the using the technique during the construction phase and on the expected utilization of the resulting database later in the design and manufacturing processes. Regardles of the chosen technique, the usuer constructs a geometric model of an object on a CAD/CAM system. To software data into a mathematical representation which it stores in the model database for later use. The user may retrieve and modify the model during the design and manufacturing processes. To convey the importance of geometric modeling to the CAD/ CAM proces, one may refer to other engineering disciplines and make the following anology. Geometric modeling to CAD/ CAM is as important as governing equilibrium equations to classical engineering fields as machanics and thermal fluids. From an engineering point of view modeling of objects is by its self unimportant. Rather,it is ameans (tool) to enable useful engineering analysis and judgment. As a mater of fact, the amount of time and effort a designer spends in creating a geometric model cannot be justified unless the resulting database is utilized by the aplication module. The need to study the mathematical basis of geometric modeling is many fold. XIFrom a strictly modeling point of view, it provides a good understanding of a terminology encountered in the CAD/CAM field as well as CAD / CAM system documantations. From an engineering and design point of view.studying geometric modeling provides engineers and designers with new sets of tools and capabalities that they can use in their daily engineering assigments. This is an important issue because,,historically,engineers cannot think in terms of tools they have not learned to use or been exposed to. The tools are powerful if utilized innovativelyin engineering applications. It is usually left to the individual imagination to apply these tools usefully to applications in a new contex. Having established the need for geometric modeling, what is the most useful geometric model to engineering applications? Unfortunately, ther is no direct answer to this question. Newrtheless, the follpwing answer has two levels. At one level »engineers may agree that some sort of geometry is required to carry enginering analysis. the degree of geometric details depend on the analysis procedure that utilizes the geometry. Engineers may also agree that there is no model that is sufficent to study all behavioural aspects of an engineering component or asystem.Aa machine part, for example,can be modeled as a lumped mass rigid body on one occasion or a distributed mass continuum on other occasion. At the second level, the adequace of geometry or a geometric model to an analysis procudure is decided by its related useful atributes to that procedure. Atributes of geometry is never an issue for manual procedures because the engineer's mind coordinates all the related facts and information. This work covers the avaliable types and most useful mathematical represantations of curves. There are two categories of curves ; analytic and synthetic. Analytic curves are defined as those that can be described by analytic equations such as lines,circles,and conies. synthetic curves are the ones that are described by a set of data points (control points) such as splines and bezier curves. Parametric polynomials usually fit the control points. while analytic curves provide very compact forms to represent shapes and simplify the computation of related properties such as areas and volumes,they are not attractive to deal with interactively.Alternatively, synthetic curves provide designer with greater flexibility and control of a curve shape by changing the positions of the control points. Analytic curves are usually not sufficent to meet geometric design requirements of mechanical parts. Products such as car bodies. ship hulls, airplane fuselage and wings, propeller blades and bottles are a few examples that require free-form,or synthetic »curves and surfaces. The xnneed for synthetic curves in design arises on two occasions : When a curve is represented by a collection of measured data points and when an existing curve must change to meet new design requirements. In the later occasion, the designer would need a curve represantation that is directly related to the data points and flexible enough to bend,twist,or change the curve shape by changing one more data points. Data points are usually called control points and the curve itself called an interpoland if it passes through all the data points. Mathematically, synthetic curves represent a curve-fitting problem to constuct a smoot curve that passes through given data points. Therefore, polynomials are the typical form of these curves. Various continuity requirements can be specified at the data points to impose various degrees of smoothness of the resulting curve. The order of contiunity becomes important when a complex curve is modeled by several curve segments pieced together end to end. Zero order continuity yiels a position continuous curve. First and second order continuityies imply slope and curvature continuous curves respectively curve is the minumum acceptable curve for engineering design. Figure 1 shows a geometrical interpretion of these order of continuity. - J anccnts Center of curvature \ + Cimtinl point (<i) Zero-order continuity if" curve). Tangent -. - Tangent lb) FirM-imJcr continuity (C1 curve) Center of curvature li) Scnmil-mctcr continuity (C2 curve) Figure 1 Various orders of continuity of curves In addition,the cubic polynomial is the lowest- degree polynomial that permits inflection within acurve segment and that allows represantation nonplanar three - dimensional curves in space. Higher -order polynomials are not common used in CAD / CAM because they tend to oscillate about control points, are computationally inconvenient, and are uneconomical of stroring curve and surface representation in the computer. xniThe type of input data and its influence on the control of the resulting synthetic curve determine the use and effectiveness of the curve in design. For example,curve segments that require positions of control points and tanjant vektors at these points are easier to deal with and gather data for than those that might require curvature information. Also, the designer may prefer to control the shape of the curve locally instead of globally by changing the control points. If changing a control points result in changing the curve locally in the vicinity of that point, local control of the curve is achieved ;otherwise global control results. Major CAD/CAM systems provide three types of synthetic curves: Hermit cubik spline,Bezier and B-spline curves.The cubic spline curves passes through the data points and therefore is an interpolant. Bezier and B- spline curves in general approximate the data points, that is, they dont pass throgh them. Both the cubic spline and bezier curves have the first order continuity and the B-spline curve has a second -order continuity. The formulation of each curve is discussed below. Hermit cubic spline : Parametric spline curves are defined as piecewise polynomial curves with a certain order of continuity. Paramrtric cubic splines are used to interpolate to given data, not to design free-form curves as bezier and b-spline curves. The parametric cubic spline curve connects two data points and utilizes a cubic equation The parametric equation of a cubic spline segment is given by: P(u) = (2U3 -3U2+ 1) P0 + (-2U3 + 3U2) Pj + (U3-2U2+U) P0' + (U3 - U2 ) Pi' O < U<1 ( 1 ) In This equation U is the parameter. And the tanjant vektör of this curve is; P'(u) = (6U2 -6U) Po + (-6U2 + 6U) P, + (3U2-4U+1 ) Po' + (3U2 -2U) P^ O < U^l (2) The function of U in Eqs. 1 and 2 are called blending functions. The first two functions blend P0 and Pi and the second two blend P0' and P^ to procedure the left -handside in each equation. Equation 1 describes the cubic spline curve in terms of its two endpoints and their tanjant vektors.The equation shows that the curve passes through the endpoints (U=0 and 1). It also shows that curve shape can be controlled by chaging its endpoints or its tanjant vektors. xivThe use of the cubic splines in design applications is not very popular compared to Bezier or B-spline curves. The control of the curve is not very obvious from the input data due to its global control characteristics. Bezier curves : Another alternative to create curves is to use approximation techniques which produce curves that dont pass through the given data points. Instead, these points are used to control the shape of the resulting curves. most often, approximation techniques are prefered over interpolation techniques in curve design due to the adde flexibility and the additional intuitive feel provided by the former. The shape of bezier curve is controlled by its defining points only. The order or the degree of bezier curve is variable and is related to the number of points defining it; n+1 points define an nth degree curve which permits higher order continuiy.The bezier curve is smooter than the cubic spline because it has higher -order derivatives. Pk Control points (vertices) Characteristic polygon Figure 2. Cubic Bezier curve Mathematically, for n+1 control points, the bezier curve is defined by the following polynomial of degree n; />(«) = £ ^4», o<u<i 1=0 (3) Where P(u) is any point on the curve and Pi is a control points. Bi>n(u) are the Berstein polynomials. Thus the bezier curve has a berstein basis. The Berstein polynomial serves as the blending or basis funtion for the bezier curve and is given by ; xvBu(u) = C(n,i) U^ı-u)01 (4) C(n,i); binomial coefficient c<w> = TîÖ^Ö! (5) While bezier cuve seems superior to a cubic spline curve, it steel has some disadvantages.First the curve does not pass through the control points which may be inconvenient to some designers. Second, the curve lacks local control. It only has the global control nature. If one control point is changed.the whole curve changes. Therefore,the designer cannot selectively change parts of the curve. B-Spline Curves : B spline curves provide another effective method,besides that of bezier, of generating curves defined by polygons. In fact, B-spline curves are the proper and powerful generalization of bezier curves. In additin to sharing most of the characteristics of bezier curves they enjoy some other unique advanteges. They provide local control of the curve shape as opposed to global control by using a special set of blending functions that provide local influence. They also provide the ability to add control points without increasing the degree of the curve. In contrast to bezier curves,the theory of B-spline curves separates the degree of the resulting curve from the number of the given control points. Similar to bezier curves, the B-spline curve defined by n+1 kontrol points Pi is given by; P(u)=S W,*(«) 0<U<U (6) <=0 Nu(.) = N)^+(^-»)|f (7) f 1 Mi<u< UM Nil = 1° otherwise (8) Ni)k are the B-spline functions. Thus B-spline curves have a B-spline basis. The control points form the vertices of the control polygon. The parameter k kontrols the degree (k-1) of the resulting B- spline curve and is usually independent of the number of control points except as restricted. XVIThe maximum limit of the parameter u is no longer unity as it was so chosen arbitrarily for bezier curves. The Ui are called parametric knots or knot values. The values of the Ui depend on whether the B-spline curve is an open or closed curve For an open curve,they are given by ; Uj = Where f0 j<k j-k + 1, k<j<n n-k + 2 j>n 0 < j < n+k (9) (10) and the range of U is ; 0 < U < n-k+2 (11) While the degree of the resulting b-spline curve is controlled by k, the range of the parameter u as given by Eq 11 implies that there is alimit on k that is determined by the number of the given control points. The local control of the curve can be achieved by changing the position of a control point 5using multiple control points by placing several points at the same location,or by choosing a different degree (k-1).As mentioned earlier changing one control point affects only k segments.Figure 3 shows the local control for a cubic B-spline curve by moving P3 to P3* and P3* *. The four curve segments surrounding P3 change only. Figure 3 Local control of B-spline curves. The closed B-spline curve of degree (k-1) or order k defined by (n+1) kontrol points is given by Eq.6 as the open curve.However,for closed curves : xvi iNi>k (u) = N0>k [(u-i+n+1) mod(n+l)] Uj=j, 0<J£n+l (12) 0<J £ n+1 and the range of U ; 0 < U < n+1 Closed B-splines share the same characteristics of the open curves excpt that they dont pass through the first and last control points and therefore are not tanjant to the first and last segments of the control polygon Rational curves : A rational curve is defined by the algebraic ratio of two polynomials A rational b-spline curve defined by n+1 kontrol points ; ?(u) =fdPiRit(u) 0<U^Umax (13) /so R;,k (u) are the rational B-spline basis functions an are given by Ri,k (u) = ' * ( 1 4) EiloVW) The above equation shows that Ri>k (u) are a generalization of the nonrational basis functions. The affective use of these analytic and synthetic curves in a design and manufacturing enviroment depends mainly on their manupulations to achieve goals in hand. These manipulations are ; displaying, Blending, segmentation, trimming, itersection and transformation. xvm
Benzer Tezler
- 3D computer modelling and sliding failure analysis of jointed rock slopes
Eklemli kaya şevlerinin 3 boyutlu bilgisayar modellemesi ve kayma analizi
ADEM ÖCAL
Doktora
İngilizce
2000
Maden Mühendisliği ve MadencilikOrta Doğu Teknik ÜniversitesiMaden Mühendisliği Ana Bilim Dalı
PROF. DR. ABDURRAHİM ÖZGENOĞLU
- Cam yapısı ve sulu piridin-2,6-dikarboksilat komplekslerinde terbium luminesansının deneysel ve teorik incelenmesi
Theoretical and experimental studies on terbium luminescence in glass matrix and in aqueous pyridine-2,6- dicarboxylate complexes
TUĞBA TÜĞSÜZ
- Texture mapping on geometrical models
Başlık çevirisi yok
OKTAY AYDIN AÇIKGÖZ
Yüksek Lisans
İngilizce
1989
Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve Kontrolİhsan Doğramacı Bilkent ÜniversitesiPROF. DR. BÜLENT ÖZGÜÇ
- Bilgisayar destekli grafik modelleme ve yüzey modelleme esasları
Başlık çevirisi yok
ALPER YILDIZ
Yüksek Lisans
Türkçe
1994
Bilgisayar Mühendisliği Bilimleri-Bilgisayar ve KontrolYıldız Teknik ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
YRD. DOÇ. DR. MESUT ÖZGÜRLER
- Surface modelling based on bezier and B-spline techniques
Bezier ve B-spline teknikleri ile yüzey modelleme
MURAT YABAN
Yüksek Lisans
İngilizce
1993
Makine MühendisliğiOrta Doğu Teknik ÜniversitesiMakine Mühendisliği Ana Bilim Dalı
DOÇ. DR. MUSTAFA İLHAN GÖKLER