On the G-Similarities of two open B-spline curves in R3

Abstract

Let G be a transformation group in R3. Any two vectors x and y in R3 are called G-equivalence vectors if there exist a transformation g G such that y = gx satisfies. In this paper the transformation group G will be considered as similarity transformations group or its any subgroup. So if given two vectors x and y in R3 are G-equivalence vectors then these vectors x and y are called G-similar. i.e. rotational, reflectional, translational or scaling similarity. B-spline curves are used basically in Computer Aided Design (CAD), Computer Aided Geometric Design (CAGD), Computer Aided Modeling (CAM). In determining the invariants of spline curves and surfaces at any point, it is necessary to find the analytical equation of each curve and surface and calculate its invariants such as curvature, torsion, principal curvatures, mean and Gaussian curvatures at the desired point. However, it can be very difficult to find the curve or surface to be designed analytically. For example, when a car is designed, the aerodynamic curves in the car will be different from the known surface equation of the car. It is very difficult to write this equation exactly. For these curves and surfaces we designed, the way to overcome this difficulty is to design them with spline curves and surfaces. In this paper the G- equivalence conditions of given two open B-spline curves are studied in case G is similarity transformations group or its any subgroup.

Keywords

• open B-spline curves
• similarity groups
• G-similar splines

R3 de Açık B-Spline Eğrilerinin G-Benzerlikler

Abstract

G, R3 de bir dönüşüm grubu olsun. R3 te herhangi iki x ve y vektörleri verildiğinde eğer bir g G dönüşümü y = gx şartını sağlayacak şekilde bulunabilirse bu iki vektöre G- denk vektörler denir. Bu çalışmada G dönüşüm grubu olarak benzerlik dönüşümleri grubu ve bu grubun tüm altgrupları dikkate alınacaktır. Böylece R3 te herhangi iki x ve y vektörleri G- denk vektörler ise bu vektörlere G-benzer denir. Döndürülme, yansıtılma, ötelenme, ya da germe benzerliği gibi. B-spline eğrileri temelde Bilgisayar Destekli Tasarım (BDT), Bilgisayar Destekli Geometrik Tasarım (BDGT) ya da Bilgisayar Destekli Modelleme (BDM) alanlarında kullanılır. Herhangi bir noktada spline eğri ve yüzeylerinin invaryantlarını belirlemede eğri ve yüzeyin analitik denklemini bulmak ve istenilen noktada eğrilik torsiyon, asal eğrilikler, ortalama ve Gauss eğriliklerini hesaplamak gerekmektedir Oysa ki tasarlanan eğri ve yüzeyde bunu analitik olarak bulmak oldukça zordur. Örneğin, bir araç tasarlandığında, onun aerodinamik yapısından dolayı yüzeyin ve onun üzerindeki eğrilerin analitik denklemini tam olarak bulmak oldukça zordur. Tasarlanan bu eğri ve yüzeyler için bu zorluğun üstesinden gelmenin yolu bunları spline eğri ve yüzeyleri ile tasarlamaktır. Bu çalışmada G, benzerlik dönüşümleri grubu ve onun altgrupları olması durumunda verilen iki B-spline eğrilerinin G- denklik koşulları verilmiştir.

Keywords

• açık B-spline curves
• benzerlik grupları
• G-benzer splinelar

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