35,195 research outputs found
Feature Lines for Illustrating Medical Surface Models: Mathematical Background and Survey
This paper provides a tutorial and survey for a specific kind of illustrative
visualization technique: feature lines. We examine different feature line
methods. For this, we provide the differential geometry behind these concepts
and adapt this mathematical field to the discrete differential geometry. All
discrete differential geometry terms are explained for triangulated surface
meshes. These utilities serve as basis for the feature line methods. We provide
the reader with all knowledge to re-implement every feature line method.
Furthermore, we summarize the methods and suggest a guideline for which kind of
surface which feature line algorithm is best suited. Our work is motivated by,
but not restricted to, medical and biological surface models.Comment: 33 page
A finite element approach for vector- and tensor-valued surface PDEs
We derive a Cartesian componentwise description of the covariant derivative
of tangential tensor fields of any degree on general manifolds. This allows to
reformulate any vector- and tensor-valued surface PDE in a form suitable to be
solved by established tools for scalar-valued surface PDEs. We consider
piecewise linear Lagrange surface finite elements on triangulated surfaces and
validate the approach by a vector- and a tensor-valued surface Helmholtz
problem on an ellipsoid. We experimentally show optimal (linear) order of
convergence for these problems. The full functionality is demonstrated by
solving a surface Landau-de Gennes problem on the Stanford bunny. All tools
required to apply this approach to other vector- and tensor-valued surface PDEs
are provided
Nonlinear Holomorphic Supersymmetry on Riemann Surfaces
We investigate the nonlinear holomorphic supersymmetry for quantum-mechanical
systems on Riemann surfaces subjected to an external magnetic field. The
realization is shown to be possible only for Riemann surfaces with constant
curvature metrics. The cases of the sphere and Lobachevski plane are elaborated
in detail. The partial algebraization of the spectrum of the corresponding
Hamiltonians is proved by the reduction to one-dimensional quasi-exactly
solvable sl(2,R) families. It is found that these families possess the
"duality" transformations, which form a discrete group of symmetries of the
corresponding 1D potentials and partially relate the spectra of different 2D
systems. The algebraic structure of the systems on the sphere and hyperbolic
plane is explored in the context of the Onsager algebra associated with the
nonlinear holomorphic supersymmetry. Inspired by this analysis, a general
algebraic method for obtaining the covariant form of integrals of motion of the
quantum systems in external fields is proposed.Comment: 24 pages, new section and refs added; to appear in Nucl. Phys.
Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds
We develop exterior calculus approaches for partial differential equations on
radial manifolds. We introduce numerical methods that approximate with spectral
accuracy the exterior derivative , Hodge star , and their
compositions. To achieve discretizations with high precision and symmetry, we
develop hyperinterpolation methods based on spherical harmonics and Lebedev
quadrature. We perform convergence studies of our numerical exterior derivative
operator and Hodge star operator
showing each converge spectrally to and . We show how the
numerical operators can be naturally composed to formulate general numerical
approximations for solving differential equations on manifolds. We present
results for the Laplace-Beltrami equations demonstrating our approach.Comment: 22 pages, 13 figure
- …