On Piecewise Algebraic Curves And A Discrete Scheme For The Willmore Problem

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. Piecewise algebraic curve not only has its special properties in the aspect of al-gebraic geometry, but also plays an important role in the interpolation by bivariate spline functions and the geometric modeling in computer aided design. Moreover, it relates to the coloring problems in the graph theory and problems in traditional algebraic geometry. In this thesis, we mainly discuss the different properties between the traditional algebraic curves and the piecewise algebraic curves, the application of piecewise algebraic curves to the problems of interpolations by bivariate spline functions, and propose a discrete scheme for the Willmore problem. We investigate the following topics:the Bezout type theorem for linear piecewise algebraic curves over arbitrary triangulations; the Cayley-Bacharach theorem for continuous piecewise algebraic curves over cross-cut triangulations; the prop-erty of the interpolation set along linear piecewise algebraic curves and its application to constructing the interpolation set of the space of continuous bivariate spline functions; discrete schemes for mean curvature and the Willmore problem over triangular meshes.1. The Bezout number for piecewise algebraic curves of SmT (Δ) and Snt(Δ) is de-fined to be the maximum number of intersections of two piecewise algebraic curves whose intersections are finite, denoted by BN(m,r;n,t;Δ). In this thesis, by considering the maximum number of intersections between two linear piecewise algebraic curves over an odd cycle of cells, we give an upper bound of the Bezout number BN(1,0; 1,0;Δ) for the triangulationΔwith one odd interior vertex v:the difference between the number of cells ofΔand the distance from the odd interior vertex v to the boundary ofΔ. When the triangulation with an odd interior vertex satisfies a vertex coloring condition, we compute the exact value of the Bezout number BN(1,0; 1,0;Δ), which equals to the upper bound given above. Moreover, we introduce the odd-cycle covering number, compute the expres-sion of the odd-cycle covering number of triangulations, and hence give an upper bound of the Bezout number BN(1,0; 1,0;Δ) for arbitrary triangulations. The result indicates that, the Bezout number for linear piecewise algebraic curves over triangulations with odd interior vertices depends on the distances between the odd interior vertices and the distances from each odd interior vertex to the boundary of the triangulation. 2. The Cayley-Bacharach theorem considers the number of independent conditions imposed on polynomials of given degree by certain set of points in the plane. One common version says that if two algebraic curves of degrees m and n meet at mn intersections, then any algebraic curve of degree m+n-3 passing through all but one point of those intersections also passes through the last one. Our work shows that, if two continuous piecewise algebraic curves of degree m and n meet at mnT intersections over a cross-cut triangulation, then any continuous piecewise algebraic curve of degree m+n-2 passing through all but one point of those intersections also passes through the last one; if the last point of those intersections lies in a cell with two or three interior edges of the triangulation, then the degree of piecewise algebraic curve which satisfies the preceding property reaches m+n-1 or m+n. Moreover, we discuss the property of the interpolation set along piecewise algebraic curves and apply it to interpolation problems of spaces of bivariate spline functions. The interpolation set along piecewise algebraic curves, is a set of points chosen in a given piecewise algebraic curve, such that it becomes an interpolation set of the space of higher-degree spline functions, together with any interpolation set of the space of lower-degree spline functions, provided that no vertices of the latter lies in the given piecewise algebraic curve. Our work shows that, the number of the interpolation set of degree k and smoothness 0 along linear piecewise algebraic curves over a star region within each cell satisfies an alternating property. Based on this property, we propose a novel method to construct the interpolation set of the space of continuous spline functions over arbitrary triangulations. Compared with the traditional B-net method, our method depends not on the geometric locations of the points, but on the number of points-located in the piecewise algebraic curve within each cell of the triangulation.3. The classic Willmore problem is to find a surface in an admissible class of surfaces embedded in R3, which minimizes the Willmore energy (the integral of the mean curvature square). We give the convergence conditions of two traditional formulae for computing mean curvature over triangular mesh:the cotangent sum formula and quadratic fitting formula, and propose the so-called weighted cotangent sum formula based on the error function of the cotangent sum formula. The analysis and numerical results show that the new formula has good convergence property compared with the two classic formulae. We also give the discrete Willmore energy using the numerical integration of discrete mean curvature given by quadratic fitting formula. The asymptotic analysis shows that, when the mesh is quadratic fittable, the discrete Willmore energy converges to the Willmore energy of the surface. Moreover, if the discrete normal vector with cubic convergence rate is obtained, the minimizer of the discrete Willmore energy converges to the minimizer of the Willmore energy. Hence one could establish numerical schemes for simulating surface deformation of the Willmore problem based on this energetic functional.