Simplicial map

Abstract: We introduce an efficient algorithm for producing provably injective mappings of tetrahedral meshes with strict bounds on their tetrahedra aspect-ratio distortion.The algorithm takes as input a simplicial map (e.g., produced by some common deformation or volumetric parameterization technique) and projects it on the space of injective and bounded-distortion simplicial maps. Namely, finds a similar map that is both bijective and bounded-distortion. As far as we are aware, this is the first algorithm to produce injective or bounded-distortion simplicial maps of tetrahedral meshes. The construction of the algorithm was made possible due to a novel closed-form solution to the problem of finding the closest orientation-preserving bounded-distortion matrix to an arbitrary matrix in three (and higher) dimensions.The algorithm is shown to have quadratic convergence, usually not requiring more than a handful of iterations to converge. Furthermore, it is readily generalized to simplicial maps of any dimension, including mixed dimensions. Finally, it can deal with different distortion spaces, such as bounded isometric distortion. During experiments we found the algorithm useful for producing bijective and bounded-distortion volume parameterizations and deformations of tetrahedral meshes, and improving tetrahedral meshes, increasing the tetrahedra quality produced by state-of-the-art techniques.

Abstract: We present a new method for (1) automatically generating multiple Levels Of Detail (LODs) of a polygonal surface, (2) progressively loading, or transmitting, and displaying a surface, and for (3) changing interactively the LOD when displaying. We build the LODs using any algorithm that performs edge collapses and certain vertex removals to simplify surfaces, and provides an ordered list of ordered vertex pairs (edge collapse specifications). We propose a Surface Partition for encoding surface LODs: we define vertex and triangle levels during simplification; vertices and triangles are partitioned and sorted according to their level, and are passed to the display algorithm in decreasing level order, one level at a time, together with a vertex representatives array. Each level of vertices and triangles, together with higher levels and the vertex representatives, form a valid surface. The vertex representatives array encodes a succession of simplicial maps between the highest resolution LOD and other LODs. We propose a data structure using a Directed Acyclic Graph (DAG) for recording a partial ordering among edge collapses, and varying the LODs across the surface. We describe an implementation of our method in VRML. Key-words : Simplicial Map, Edge Collapse, Vertex and Triangle Levels, Surface Levels of Detail, Surface Partition, Progressive Transmission and Display, Dynamic Simplification.

Abstract: Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0. We prove that a simplicial map lambda from C(R) to C(R) is superinjective if and only if it is induced by a homeomorphism of R. As a corollary, we prove that if K is a finite index subgroup of Mod_R^* and f is an injective homomorphism from K to Mod_R^*, then f is induced by a homeomorphism of R and f has a unique extension to an automorphism of Mod_R^*. This extends the author's previous results about closed connected orientable surfaces of genus at least 3, to the surface R.

Abstract: In this paper, we prove that each injective simplicial map of the arc complex of a compact, connected, orientable surface with nonempty boundary is induced by a homeomorphism of the surface. We deduce, from this result, that the group of automorphisms of the arc complex is naturally isomorphic to the extended mapping class group of the surface, provided the surface is not a disc, an annulus, a pair of pants, or a torus with one hole. We also show, for each of these special exceptions, that the group of automorphisms of the arc complex is naturally isomorphic to the quotient of the extended mapping class group of the surface by its center.