On the G-Similarities of two open B-spline curves in R^3

OpenAlex · İngilizce

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.