Computer-aided geometric design (CAGD) deals with the mathematical description, and computational aspects of geometric objects, such as parametric curves and surfaces. It is a field of mathematical nature with relevant use in computer science and engineering. Approximation theory studies the process of approximating general functions by simple functions such as polynomials or splines. Therefore, it plays a central role in the analysis of numerical methods, particularly in approximation of partial differential equations. During the lecture, I will show how my research interests connect the fields of Computer-aided geometric design and Approximation theory. The following two research topics, in which I have been most active in the last decade, will be briefly presented. Parameterization of curves and surfaces are basic concepts in CAGD. In several applications, it is desirable for these curves and surfaces to have some particular properties, such as polynomial arc-length and rational offsets. A special class of curves that satisfies these requirements are Pythagorean-hodograph curves, of which the first part of my talk will be devoted. In the second part, we will get acquainted with a recent method for numerical solution of partial differential equations, i.e., Isogeometric Analysis. The main advantage of this method is in the use of the same standard CAGD spline functions both for describing the geometry and for the numerical simulation of partial differential equations. The increased smoothness of the spline functions compared to traditional finite elements allows for improved stability and convergence properties.

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