##### Abstract

The ﬁrst part of this thesis will build up an overview of a small section of the ﬁeld of subdivision, namely binary univariate schemes. Since most of the theory on subdivision is based on a uniform spacing of the initial control points, and dyadic reﬁnement at the

subsequent levels, we will try to uncover some of the results which are based on irregular spacing of the original points, and non-dyadic insertion of the new points. The four-point scheme is a special member of a family of interpolatory schemes ﬁrst introduced by

Dubuc and Deslauriers. The scheme was independently discovered and analysed by Dyn, Levin and Gregory, who also introduced the scheme with a tension parameter ω. Here the scheme will be a reappearing character throughout. Secondly, we will introduce a general subdivision scheme for interpolating both the

given function values and one or more derivative values at the end point. The formulation of the subdivision scheme will be given for most the general case, while smoothness-, and

with it Hölder regularity, results will be given for the cubic case, where the ﬁrst order end point derivative is assumed to be provided as well as function values. This cubic case can be view as a modiﬁcation of the usual four-point scheme, in the sense that the

subdivision rule is only changed at the ﬁrst odd point. In deﬁnition of this scheme we will use a cubic osculatory interpolant to calculate the ﬁrst new odd point, while we for

the rest of the new odd points still will work with rules generated by ordinary Lagrangian

cubic interpolation. The need for the two extra function values at the ends is replaced

by one derivative value. In addition, we will investigate whether the given derivative value is interpolated by the limit curve. As in the paper by Floater, the main idea is to view the limit function as the limit of piecewise polynomials of odd degree and

deduce smoothness results from this. An application of our new univariate scheme can be how to join two curves in a smooth fashion.

The cubic case is also generalized to a (bicubic) tensor product scheme. We do expect a similar smooth join of surfaces based on our bivariate scheme. Numerical examples will be given, and for completeness we introduce the interpolation

theory used in the derivations and analysis where appropriate.