36,887 research outputs found
Focal surfaces of discrete geometry
The differential geometry of smooth three-dimensional surfaces can be interpreted from one of two perspectives: in terms of oriented frames located on the surface, or in terms of a pair of associated focal surfaces. These focal surfaces are swept by the loci of the principal curvatures' radii. In this article, we develop a focal-surface-based differential geometry interpretation for discrete mesh surfaces. Focal surfaces have many useful properties. For instance, the normal of each focal surface indicates a principal direction of the corresponding point on the original surface. We provide algorithms to robustly approximate the focal surfaces of a triangle mesh with known or estimated normals. Our approach locally parameterizes the surface normals about a point by their intersections with a pair of parallel planes. We show neighboring normal triplets are constrained to pass simultaneously through two slits, which are parallel to the specified parametrization planes and rule the focal surfaces. We develop both CPU and GPU-based algorithms to efficiently approximate these two slits and, hence, the focal meshes. Our focal mesh estimation also provides a novel discrete shape operator that simultaneously estimates the principal curvatures and principal directions.Engineering and Applied Science
Discrete asymptotic nets and W-congruences in Plucker line geometry
The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added
reference
On organizing principles of Discrete Differential Geometry. Geometry of spheres
Discrete differential geometry aims to develop discrete equivalents of the
geometric notions and methods of classical differential geometry. In this
survey we discuss the following two fundamental Discretization Principles: the
transformation group principle (smooth geometric objects and their
discretizations are invariant with respect to the same transformation group)
and the consistency principle (discretizations of smooth parametrized
geometries can be extended to multidimensional consistent nets). The main
concrete geometric problem discussed in this survey is a discretization of
curvature line parametrized surfaces in Lie geometry. We find a discretization
of curvature line parametrization which unifies the circular and conical nets
by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is
slightly changed and umbilic points are discusse
Geometric discretization of the Bianchi system
We introduce the dual Koenigs lattices, which are the integrable discrete
analogues of conjugate nets with equal tangential invariants, and we find the
corresponding reduction of the fundamental transformation. We also introduce
the notion of discrete normal congruences. Finally, considering quadrilateral
lattices "with equal tangential invariants" which allow for harmonic normal
congruences we obtain, in complete analogy with the continuous case, the
integrable discrete analogue of the Bianchi system together with its geometric
meaning. To obtain this geometric meaning we also make use of the novel
characterization of the circular lattice as a quadrilateral lattice whose
coordinate lines intersect orthogonally in the mean.Comment: 26 pages, 7 postscript figure
Focal rigidity of hyperbolic surfaces
In this note, we consider the rigidity of the focal decomposition of closed
hyperbolic surfaces. We show that, generically, the focal decomposition of a
closed hyperbolic surface does not allow for non-trivial topological
deformations, without changing the hyperbolic structure of the surface. By
classical rigidity theory this is also true in dimension . Our
current result extends a previous result that flat tori in dimension
that are focally equivalent are isometric modulo rescaling.Comment: Published in Annales Academiae Scientiarum Fennicae Mathematica,
Volumen 37, 2012, 149-159, 11 page
Geometric discretization of the Koenigs nets
We introduce the Koenigs lattice, which is a new integrable reduction of the
quadrilateral lattice (discrete conjugate net) and provides natural integrable
discrete analogue of the Koenigs net. We construct the Darboux-type
transformations of the Koenigs lattice and we show permutability of
superpositions of such transformations, thus proving integrability of the
Koenigs lattice. We also investigate the geometry of the discrete Koenigs
transformation. In particular we characterize the Koenigs transformation in
terms of an involution determined by a congruence conjugate to the lattice.Comment: 17 pages, 2 figures; some spelling and typing errors correcte
Integrable Systems and Discrete Geometry
This is an expository article for Elsevier's Encyclopedia of Mathematical
Physics on the subject in the title. Comments/corrections welcome.Comment: 22 pages, 7 figure
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