28 research outputs found
From rubber bands to rational maps: A research report
This research report outlines work, partially joint with Jeremy Kahn and
Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal
surfaces with boundary. One one hand, this lets us tell when one rubber band
network is looser than another, and on the other hand tell when one conformal
surface embeds in another.
We apply this to give a new characterization of hyperbolic critically finite
rational maps among branched self-coverings of the sphere, by a positive
criterion: a branched covering is equivalent to a hyperbolic rational map if
and only if there is an elastic graph with a particular "self-embedding"
property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example
Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds
Orbifolds are a modern mathematical concept that arises in the research of
hyperbolic geometry with applications in computer graphics and visualization.
In this paper, we make use of rooms with mirrors as the visual metaphor for
orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we
provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon
to match the orbifold. This polygon is then used to create a room for which the
polygon serves as the floor and the ceiling. With our system that implements
M\"obius transformations, the user can interactively edit the scene and see the
reflections of the edited objects. To correctly visualize non-Euclidean
orbifolds, we adapt the rendering algorithms to account for the geodesics in
these spaces, which light rays follow. Our interactive orbifold design system
allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In
addition, our mirror-based orbifold visualization approach has the potential of
helping our users gain insight on the orbifold, including its orbifold notation
as well as its universal cover, which can also be the spherical space and the
hyperbolic space.Comment: IEEE VIS 202
Tutte Embeddings of Tetrahedral Meshes
Tutte's embedding theorem states that every 3-connected graph without a
or minor (i.e. a planar graph) is embedded in the plane if the outer
face is in convex position and the interior vertices are convex combinations of
their neighbors. We show that this result extends to simply connected
tetrahedral meshes in a natural way: for the tetrahedral mesh to be embedded if
the outer polyhedron is in convex position and the interior vertices are convex
combination of their neighbors it is sufficient (but not necessary) that the
graph of the tetrahedral mesh contains no and no , and all
triangles incident on three boundary vertices are boundary triangles
Mathematical Imaging and Surface Processing
Within the last decade image and geometry processing have become increasingly rigorous with solid foundations in mathematics. Both areas are research fields at the intersection of different mathematical disciplines, ranging from geometry and calculus of variations to PDE analysis and numerical analysis. The workshop brought together scientists from all these areas and a fruitful interplay took place. There was a lively exchange of ideas between geometry and image processing applications areas, characterized in a number of ways in this workshop. For example, optimal transport, first applied in computer vision is now used to define a distance measure between 3d shapes, spectral analysis as a tool in image processing can be applied in surface classification and matching, and so on. We have also seen the use of Riemannian geometry as a powerful tool to improve the analysis of multivalued images.
This volume collects the abstracts for all the presentations covering this wide spectrum of tools and application domains