16,726 research outputs found
Changing Views on Curves and Surfaces
Visual events in computer vision are studied from the perspective of
algebraic geometry. Given a sufficiently general curve or surface in 3-space,
we consider the image or contour curve that arises by projecting from a
viewpoint. Qualitative changes in that curve occur when the viewpoint crosses
the visual event surface. We examine the components of this ruled surface, and
observe that these coincide with the iterated singular loci of the coisotropic
hypersurfaces associated with the original curve or surface. We derive
formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and
show how to compute exact representations for all visual event surfaces using
algebraic methods.Comment: 31 page
A characterisation of generically rigid frameworks on surfaces of revolution
A foundational theorem of Laman provides a counting characterisation of the
finite simple graphs whose generic bar-joint frameworks in two dimensions are
infinitesimally rigid. Recently a Laman-type characterisation was obtained for
frameworks in three dimensions whose vertices are constrained to concentric
spheres or to concentric cylinders. Noting that the plane and the sphere have 3
independent locally tangential infinitesimal motions while the cylinder has 2,
we obtain here a Laman-Henneberg theorem for frameworks on algebraic surfaces
with a 1-dimensional space of tangential motions. Such surfaces include the
torus, helicoids and surfaces of revolution. The relevant class of graphs are
the (2,1)-tight graphs, in contrast to (2,3)-tightness for the plane/sphere and
(2,2)-tightness for the cylinder. The proof uses a new characterisation of
simple (2,1)-tight graphs and an inductive construction requiring generic
rigidity preservation for 5 graph moves, including the two Henneberg moves, an
edge joining move and various vertex surgery moves.Comment: 23 pages, 5 figures. Minor revisions - most importantly, the new
version has a different titl
Surface embedding, topology and dualization for spin networks
Spin networks are graphs derived from 3nj symbols of angular momentum. The
surface embedding, the topology and dualization of these networks are
considered. Embeddings into compact surfaces include the orientable sphere S^2
and the torus T, and the not orientable projective space P^2 and Klein's bottle
K. Two families of 3nj graphs admit embeddings of minimal genus into S^2 and
P^2. Their dual 2-skeletons are shown to be triangulations of these surfaces.Comment: LaTeX 17 pages, 6 eps figures (late submission to arxiv.org
Reidemeister/Roseman-type Moves to Embedded Foams in 4-dimensional Space
The dual to a tetrahedron consists of a single vertex at which four edges and
six faces are incident. Along each edge, three faces converge. A 2-foam is a
compact topological space such that each point has a neighborhood homeomorphic
to a neighborhood of that complex. Knotted foams in 4-dimensional space are to
knotted surfaces, as knotted trivalent graphs are to classical knots. The
diagram of a knotted foam consists of a generic projection into 4-space with
crossing information indicated via a broken surface. In this paper, a finite
set of moves to foams are presented that are analogous to the Reidemeister-type
moves for knotted graphs. These moves include the Roseman moves for knotted
surfaces. Given a pair of diagrams of isotopic knotted foams there is a finite
sequence of moves taken from this set that, when applied to one diagram
sequentially, produces the other diagram.Comment: 18 pages, 29 figures, Be aware: the figure on page 3 takes some time
to load. A higher resolution version is found at
http://www.southalabama.edu/mathstat/personal_pages/carter/Moves2Foams.pdf .
If you want to use to any drawings, please contact m
Edge contraction on dual ribbon graphs and 2D TQFT
We present a new set of axioms for 2D TQFT formulated on the category of cell
graphs with edge-contraction operations as morphisms. We construct a functor
from this category to the endofunctor category consisting of Frobenius
algebras. Edge-contraction operations correspond to natural transformations of
endofunctors, which are compatible with the Frobenius algebra structure. Given
a Frobenius algebra A, every cell graph determines an element of the symmetric
tensor algebra defined over the dual space A*. We show that the
edge-contraction axioms make this assignment depending only on the topological
type of the cell graph, but not on the graph itself. Thus the functor generates
the TQFT corresponding to A.Comment: accepted in Journal of Algebra (22 pages, 13 figures
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