Yes / Bezier curves and surfaces are two very useful tools in Geometric Modeling, with many applications. In this paper, we will offer a new method to provide approximations of regular curves and surfaces by Bezier ones, with the corresponding examples.

In this paper, we study Bézier surfaces in View the MathML source three-dimensional Minkowski space. In particular, we focus on timelike and spacelike cases for Bézier surfaces. We also deal with the Plateau¿Bézier problem in View the MathML source, obtaining conditions over the control net to be extremal of the Dirichlet function for both timelike and spacelike Bézier surfaces. Moreover, we provide interesting examples showing the behavior of the Plateau¿Bézier problem in View the MathML source and illustrating the relationship between it and the corresponding Plateau¿Bézier problem in the Euclidean space R3.

<p>In this report, we show how to implement direct ray tracing of B´ezier surfaces on graphics processing units (GPUs), in particular bicubic rectangular Bézier surfaces and nonparametric cubic Bézier triangles. We use Newton’s method for the rectangular case and show how to use this method to find the ray-surface intersection. For Newton’s method to work we must build a spatial partitioning hierarchy around each surface patch, and in general, hierarchies are essential to speed up the process of ray tracing. We have chosen to use bounding box hierarchies and show how to implement stackless traversal of such a structure on a GPU. For the nonparametric triangular case, we show how to find the wanted intersection by simply solving a cubic polynomial. Because of the limited precision of current GPUs, we also propose a numerical approach to solve the problem, using a one-dimensional Newton search.</p>

In this report, we show how to implement direct ray tracing of B´ezier surfaces on graphics processing units (GPUs), in particular bicubic rectangular Bézier surfaces and nonparametric cubic Bézier triangles. We use Newton’s method for the rectangular case and show how to use this method to find the ray-surface intersection. For Newton’s method to work we must build a spatial partitioning hierarchy around each surface patch, and in general, hierarchies are essential to speed up the process of ray tracing. We have chosen to use bounding box hierarchies and show how to implement stackless traversal of such a structure on a GPU. For the nonparametric triangular case, we show how to find the wanted intersection by simply solving a cubic polynomial. Because of the limited precision of current GPUs, we also propose a numerical approach to solve the problem, using a one-dimensional Newton search.

This thesis introduces new algorithms for solving curve/curve and ray/surface intersections. These algorithms introduce the concept of a geometric interval to extend the technique of Bézier clipping. A geometric interval is used to tightly bound a curve or surface or to contain a point on a curve or surface. Our algorithms retain the desirable characteristics of the Bézier clipping technique such as ease of implementation and the guarantee that all intersections over a given interval will be found. However, these new algorithms generally exhibit cubic convergence, improving on the observed quadratic convergence rate of Bézier clipping. This is achieved without significantly increasing computational complexity at each iteration. Timing tests show that the geometric interval algorithm is generally about 40-60% faster than Bézier clipping for curve/curve intersections. Ray tracing tests suggest that the geometric interval method is faster than the Bézier clipping technique by at least 25% when finding ray/surface intersections.

This diploma evaluates methods for verification of key characteristics of a product using digital image processing techniques. At first, reasons why this work has been done are described followed by a list of all methods that were used in this diploma such as Hough Circle Transform and Flood Fill (Seed Fill) algorithm. Also, a new approach how to compensate non regularly illuminated scene, which is based on surface modeling with Bézier Surfaces, was developed. Moreover, the algorithm was implemented in the C++ programming language and some of the parts were also simulated using the MATLAB environment. The algorithm was evaluated based on the percentage level of recognition of the required parameters. Efficiency of the implementation is also important for the author.

The master's thesis deals with Navier-Stokes equations in curvilinear coordinates and their solution for quasi-potential flow. The emphasis is on detailed description of curvilinear space and its expression using Bézier curves, Bézier surfaces and Bézier bodies. Further, fundamental concepts of hydromechanics are defined, including potential and quasi-potential flow. Cauchy equations are derived as a result of the law of momentum conservation and continuity equation is derived as a result of principle of mass conservation. Navier-Stokes equations are then derived as a special case of Cauchy equations using Cauchy stress tensor of Newtonian compressible fluid. Further transformation into curvilinear coordinates is accomplished through differential operators in curvilinear coordinates and by using curvature vector of space curve. In the last section we use results from previous chapters to solve boundary value problem of quasi-potential flow, which was solved by finite difference method using Matlab environment.