An Algorithm for Triangulating 3D Polygons

In this thesis, we present an algorithm for obtaining a triangulation of multiple, non-planar 3D polygons. The output minimizes additive weights, such as the total triangle areas or the total dihedral angles between adjacent triangles. Our algorithm generalizes a classical method for optimally triangulating a single polygon. The key novelty is a mechanism for avoiding non-manifold outputs for two and more input polygons without compromising opti- mality. For better performance on real-world data, we also propose an approximate solution by feeding the algorithm with a reduced set of triangles. In particular, we demonstrate experimentally that the triangles in the Delaunay tetrahedralization of the polygon vertices offer a reasonable trade off between performance and optimality

Delaunay-restricted Optimal Triangulation of 3D Polygons

Triangulation of 3D polygons is a well studied topic of research. Existing methods for finding triangulations that minimize given metrics (e.g., sum of triangle areas or dihedral angles) run in a costly O(n4) time [BS95,BDE96], while the triangulations are not guaranteed to be free of intersections. To address these limitations, we restrict our search to the space of triangles in the Delaunay tetrahedralization of the polygon. The restriction allows us to reduce the running time down to O(n2) in practice (O(n3) worst case) while guaranteeing that the solutions are intersection free. We demonstrate experimentally that the reduced search space is not overly restricted. In particular, triangulations restricted to this space usually exist for practical inputs, and the optimal triangulation in this space approximates well the optimal triangulation of the polygon. This makes our algorithms a practical solution when working with real world data

Connectivity Compression for Irregular Quadrilateral Meshes

Applications that require Internet access to remote 3D datasets are often limited by the storage costs of 3D models. Several compression methods are available to address these limits for objects represented by triangle meshes. Many CAD and VRML models, however, are represented as quadrilateral meshes or mixed triangle/quadrilateral meshes, and these models may also require compression. We present an algorithm for encoding the connectivity of such quadrilateral meshes, and we demonstrate that by preserving and exploiting the original quad structure, our approach achieves encodings 30 - 80% smaller than an approach based on randomly splitting quads into triangles. We present both a code with a proven worst-case cost of 3 bits per vertex (or 2.75 bits per vertex for meshes without valence-two vertices) and entropy-coding results for typical meshes ranging from 0.3 to 0.9 bits per vertex, depending on the regularity of the mesh. Our method may be implemented by a rule for a particular splitting of quads into triangles and by using the compression and decompression algorithms introduced in [Rossignac99] and [Rossignac&Szymczak99]. We also present extensions to the algorithm to compress meshes with holes and handles and meshes containing triangles and other polygons as well as quads

Generating Surface Geometry in Higher Dimensions using Local Cell Tilers

In two dimensions contour elements surround two dimensional objects, in three dimensions surfaces surround three dimensional objects and in four dimensions hypersurfaces surround hyperobjects. These surfaces can be represented by a collection of connected simplices, hence, continuous n dimensional surfaces can be represented by a lattice of connected n-1 dimensional simplices. The lattice of connected simplices can be calculated over a set of adjacent n-dimensional cubes, via for example the Marching Cubes Algorithm. These algorithms are often named local cell tilers. We propose that the local-cell tiling method can be usefully-applied to four dimensions and potentially to N-dimensions. We present an algorithm for the generation of major cases (cases that are topologically invariant under standard geometrical transformations) and introduce the notion of a sub-case which simplifies their representations. Each sub-case can be easily subdivided into simplices for rendering and we describe a backtracking tetrahedronization algorithm for the four dimensional case. An implementation for surfaces from the fourth dimension is presented and we describe and discuss ambiguities inherent within this and related algorithms

ConTesse: Accurate Occluding Contours for Subdivision Surfaces

This paper proposes a method for computing the visible occluding contours of subdivision surfaces. The paper first introduces new theory for contour visibility of smooth surfaces. Necessary and sufficient conditions are introduced for when a sampled occluding contour is valid, that is, when it may be assigned consistent visibility. Previous methods do not guarantee these conditions, which helps explain why smooth contour visibility has been such a challenging problem in the past. The paper then proposes an algorithm that, given a subdivision surface, finds sampled contours satisfying these conditions, and then generates a new triangle mesh matching the given occluding contours. The contours of the output triangle mesh may then be rendered with standard non-photorealistic rendering algorithms, using the mesh for visibility computation. The method can be applied to any triangle mesh, by treating it as the base mesh of a subdivision surface.Comment: Accepted to ACM Transactions on Graphics (TOG