The contents of the work may be divided in two parts. In the first part we deal with the computation of the arc-lengths of curves and the areas of surfaces. The second part considers the problem of reparametrizing a parametric surface, in such a manner that the new parametrization is close to conformal.
The first part.
In computational geometry, one often needs to calculate length, angle, area and other intrinsic quantities. In themselves, they are interesting because they give information about the geometric object we are studying. They are also essential in almost every geometric computation or algorithm, from curve interpolation to texture mapping.
Three different ways to compute length and area are investigated, all of which share the property that they only require point evaluations and not derivatives of the given curve or surface.
The first method is based on interpolating the curve or surface by a polynomial and using numerical integration to approximate the length or area of the polynomial. By this process we are able to obtain rules of arbitrary order. Compared to traditional methods, we require one less degree of smoothness for the same order of accuracy.
The second method is based on using Richardson extrapolation to build high-order rules from simpler rules, starting with the chord-length method for curves, and the so-called 'diagonal' method for surfaces. A central issue is proving that these rules have proper error expansions in powers of h. Rules of arbitrary order may be constructed by schemes such as Romberg's.
The third method is similar to Clenshaw-Curtis quadrature in that it uses Chebyshev polynomials via the FFT for the interpolation. We also employ the FFT in order to compute the necessary derivatives of the obtained polynomial. Again, we obtain rules of arbitrary order.
The second part.
The first paper investigates the Laplace-Beltrami equation on a parametric surface, and shows several properties of its solution using PDE theory. It is shown that the so-called 'discrete harmonic' or 'cotangent' weights that have been suggested for parametrization purposes may be viewed as arising from a FEM analysis. This is obtained by using linear elements and a particular point-based quadrature to approximate the bilinear form. It is shown that in the L2-norm, this method is of second order. Also, quadratic elements are investigated and shown to have substantial accuracy advantages in examples.
The second paper applies the knowledge and FEM method investigated above to the issue of reparametrization of parametric surfaces. A reparametrization is sought such that the boundaries are parametrized by scaled arc-length, while in the interiour it solves the Laplace-Beltrami equation in both components, i.e. it is a harmonic map. This is shown to give parametrizations that are well suited for purposes such as interpolation, intersection, closest point computation and gridding.
Paper 1 M.S. Floater and A.F.Rasmussen. Point-based methods for estimating the length of a para-metric curve. J.Comp. Appl. Math.(196),pp.512-522, 2006.
Paper 2 A.F.Rasmussen and M.S.Floater. Apoint-based method for estimating surface area. To appear in Proceedings of the SIAM conference on Geometric Design and Computing, Phoenix 2005.
Paper 3 M.S. Floater, A.F.Rasmussen and U.Reif. Extrapolation methods for approximating arc-length and surface area. Numer Algor (2007) 44:235–248
Paper 4 M.S.Floater, A.F.Rasmussen and N.H.Risebro. Finite elements for the Laplace-Beltrami equation on parametric surfaces. Preprint 2006.
Paper 5 M.S.Floater, A.F.Rasmussen and N.H.Risebro. Reparametrization of surfaces using PDE methods. Preprint 2007.