### Abstract:

The purpose of this thesis is to answer the following question: "Given the control polygons of two B'ezier curves, what restrictions should be imposed on the control points of the second curve to ensure that the two curves combine with a specified degree of continuity?" We are particularly interested in the geometric continuity of composite B�zier curves. Geometric continuity is a relatively new research area, and the concept causes a lot of confusion for those not familiar with it. We devote Chapter 1 to geometric continuity, and give a thorough explanation of "Gn" continuity and Frenet frame continuity. Most of the work is based on an article by J.A. Gregory, written in 1989, in which he compared these two types of continuity. We prove in Chapter 1 that these are the only significant continuities in geometry, since they depend only on the shape of the relevant curve. It is also proved that "Gn" continuity implies Frenet frame continuity in "n" dimensions, and that, in particular, "G2" continuity is equivalent to Frenet frame continuity in two dimensions. In Chapter 2, we study Bd'ezier curves, and prove results that we need in Chapter 3, where we answer the question stated above. Three different approaches for constructing curves are defined, namely the de Casteljan algorithm, Bd'ezier's method and Bernstein polygon approximations. We prove that these three construction methods are equivalent, and that they can be used to construct so-called B�zier curves. To investigate the continuity of B'ezier curves, we also need their derivatives, which are calculated using Bernstein polynomials. We also discuss algorithms for subdivision and for reducing the degree of a B�zier curve. The results of Chapters 1 and 2 are used in Chapter 3 to continuity of composite B�zier curves. We show that certain control points of the curves will ensure that they combine . type and degree of continuity. We then use this restriction "C1" quadratic [beta]-splines, "C2" cubic [beta]-splines and Gamma-splines is frequently used in computer aided design.