The geometry of Bezier curves and surfaces in three-dimensional Euclidean space

This thesis consists of five main chapters. In the first part of the thesis, a brief history of Computer Aided Geometric Design (CAGD) is presented and information about the stages of creating a proper design model within the scope of CAGD is given. The second part of the thesis is devoted to the studies carried out in the field and an attempt is made to present a wide collection to researchers who plan to conduct studies on CAGD. In the third section, general information is included. That is the basic geometric definitions, polynomial function spaces, Bernstein polynomials and the definitions of classical Bezier curves and surfaces are given. In addition, Bezier-like curves and surfaces with a series of shape parameters are mentioned, in which new blending functions are used, unlike Bernstein polynomials. The fourth chapter is the original part of the thesis. In this section, new blending functions were obtained based on three different methods, and for each blending function the novel Bezier curves and surfaces were created. Additionally, a series of cross-sectional surface designs were created using the Bezier like curves obtained from the generalized α−Bernstein operators. Some of these surfaces have evolved into physical models with the help of a 3D printer. In addition, parametric continuity conditions were created for all of the new Bezier curves defined in the thesis. In the fifth and last chapter of the thesis, a brief summary of the thesis is outlined and the future studies planned to be carried out are mentioned.