26,411 research outputs found
B\'ezier curves that are close to elastica
We study the problem of identifying those cubic B\'ezier curves that are
close in the L2 norm to planar elastic curves. The problem arises in design
situations where the manufacturing process produces elastic curves; these are
difficult to work with in a digital environment. We seek a sub-class of special
B\'ezier curves as a proxy. We identify an easily computable quantity, which we
call the lambda-residual, that accurately predicts a small L2 distance. We then
identify geometric criteria on the control polygon that guarantee that a
B\'ezier curve has lambda-residual below 0.4, which effectively implies that
the curve is within 1 percent of its arc-length to an elastic curve in the L2
norm. Finally we give two projection algorithms that take an input B\'ezier
curve and adjust its length and shape, whilst keeping the end-points and
end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure
Generalized thermodynamics and Fokker-Planck equations. Applications to stellar dynamics, two-dimensional turbulence and Jupiter's great red spot
We introduce a new set of generalized Fokker-Planck equations that conserve
energy and mass and increase a generalized entropy until a maximum entropy
state is reached. The concept of generalized entropies is rigorously justified
for continuous Hamiltonian systems undergoing violent relaxation. Tsallis
entropies are just a special case of this generalized thermodynamics.
Application of these results to stellar dynamics, vortex dynamics and Jupiter's
great red spot are proposed. Our prime result is a novel relaxation equation
that should offer an easily implementable parametrization of geophysical
turbulence. This relaxation equation depends on a single key parameter related
to the skewness of the fine-grained vorticity distribution. Usual
parametrizations (including a single turbulent viscosity) correspond to the
infinite temperature limit of our model. They forget a fundamental systematic
drift that acts against diffusion as in Brownian theory. Our generalized
Fokker-Planck equations may have applications in other fields of physics such
as chemotaxis for bacterial populations. We propose the idea of a
classification of generalized entropies in classes of equivalence and provide
an aesthetic connexion between topics (vortices, stars, bacteries,...) which
were previously disconnected.Comment: Submitted to Phys. Rev.
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