Minimal discrete curves and surfaces

In this thesis, we apply the ideas from combinatorial optimization to find globally optimal solutions arising in discrete and continuous minimization problems. Though we limit ourselves with the 1- and 2-dimensional surfaces, our methods can be easily generalized to higher dimensions. We start with a continuous N-dimensional variational problem and show that under very general conditions it can be approximated with a finite discrete spatial complex. It is possible to construct and refine the complex such that the globally minimal discrete solution on the complex converges to the continuous solution of the initial problem. We develop polynomial algorithm to find the minimal discrete solution on the complex and show that embeddability is a crucial property for such a solution to exist. Later on, we demonstrate that the spatial complex arising in the minimum weight triangulation problem is not embeddable. Linear programming becomes the main tool to investigate this problem and to develop a practical algorithm for constructing the minimum weight triangulation of a random point set. Finally, we consider two curve minimization problems arising in computer graphics. Both of them are described by the curves on the piece-wise linear triangulated surface. We revisit the geodesic problem, simplify and enhance the well-known exact algorithm and construct the fast approximate algorithm. We also show that the silhouette simplification can be done efficiently using the minimization techniques that we develop in the previous chapters. In many cases, the algorithms described in this thesis can be considered as a generalization of the classical Dijkstra's shortest path algorithm.