1,180 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Riemannian geometries on spaces of plane curves
We study some Riemannian metrics on the space of regular smooth curves in the
plane, viewed as the orbit space of maps from to the plane modulo the
group of diffeomorphisms of , acting as reparameterizations. In particular
we investigate the metric for a constant : G^A_c(h,k) :=
\int_{S^1}(1+A\ka_c(\th)^2)
|c'(\th)| d\th where \ka_c is the curvature of the curve and
are normal vector fields to . The term A\ka^2 is a sort of geometric
Tikhonov regularization because, for A=0, the geodesic distance between any 2
distinct curves is 0, while for the distance is always positive. We give
some lower bounds for the distance function, derive the geodesic equation and
the sectional curvature, solve the geodesic equation with simple endpoints
numerically, and pose some open questions. The space has an interesting split
personality: among large smooth curves, all its sectional curvatures are , while for curves with high curvature or perturbations of high frequency,
the curvatures are .Comment: amslatex, 45 pagex, 8 figures, typos correcte
Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms
The -metric or Fubini-Study metric on the non-linear Grassmannian of all
submanifolds of type in a Riemannian manifold induces geodesic
distance 0. We discuss another metric which involves the mean curvature and
shows that its geodesic distance is a good topological metric. The vanishing
phenomenon for the geodesic distance holds also for all diffeomorphism groups
for the -metric.Comment: 26 pages, LATEX, final versio
Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks
This paper deals with the computation of sectional curvature for the
manifolds of landmarks (or feature points) in D dimensions, endowed with
the Riemannian metric induced by the group action of diffeomorphisms. The
inverse of the metric tensor for these manifolds (i.e. the cometric), when
written in coordinates, is such that each of its elements depends on at most 2D
of the ND coordinates. This makes the matrices of partial derivatives of the
cometric very sparse in nature, thus suggesting solving the highly non-trivial
problem of developing a formula that expresses sectional curvature in terms of
the cometric and its first and second partial derivatives (we call this Mario's
formula). We apply such formula to the manifolds of landmarks and in particular
we fully explore the case of geodesics on which only two points have non-zero
momenta and compute the sectional curvatures of 2-planes spanned by the
tangents to such geodesics. The latter example gives insight to the geometry of
the full manifolds of landmarks.Comment: 30 pages, revised version, typos correcte
On the nonlinear statistics of range image patches
In [A. B. Lee, K. S. Pedersen, and D. Mumford, Int. J. Comput. Vis., 54 (2003), pp. 83–103], the authors study the distributions of 3 × 3 patches from optical images and from range images. In [G. Carlsson, T. Ishkanov, V. de Silva, and A. Zomorodian, Int. J. Comput. Vis., 76 (2008), pp.
1–12], the authors apply computational topological tools to the data set of optical patches studied by Lee, Pedersen, and Mumford and find geometric structures for high density subsets. One high density subset is called the primary circle and essentially consists of patches with a line separating a light and a dark region. In this paper, we apply the techniques of Carlsson et al. to range patches.
By enlarging to 5×5 and 7×7 patches, we find core subsets that have the topology of the primary circle, suggesting a stronger connection between optical patches and range patches than was found by Lee, Pedersen, and Mumford
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