8,078 research outputs found
Polynomial Meshes: Computation and Approximation
We present the software package WAM, written in Matlab, that generates Weakly
Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d
polynomial least squares and interpolation on compact sets with various geometries.
Possible applications range from data fitting to high-order methods for PDEs
Geometric fitting by two coaxial cylinders
Fitting two coaxial cylinders to data is a standard problem incomputational metrology and reverse engineering processes, which also arisesin medical imaging. There are many fitting criteria that can beused. One that is widely used in metrology, for example, isthat of the sum of squared minimal distance. A similarnumerical method is developed to fit two coaxial cylinders inthe general position to 3D data, and numerical examples are given
The Isbell monad
In 1966, John Isbell introduced a construction on categories which he termed
the "couple category" but which has since come to be known as the Isbell
envelope. The Isbell envelope, which combines the ideas of contravariant and
covariant presheaves, has found applications in category theory, logic, and
differential geometry. We clarify its meaning by exhibiting the assignation
sending a locally small category to its Isbell envelope as the action on
objects of a pseudomonad on the 2-category of locally small categories; this is
the Isbell monad of the title. We characterise the pseudoalgebras of the Isbell
monad as categories equipped with a cylinder factorisation system; this notion,
which appears to be new, is an extension of Freyd and Kelly's notion of
factorisation system from orthogonal classes of arrows to orthogonal classes of
cocones and cones.Comment: 21 page
Radii minimal projections of polytopes and constrained optimization of symmetric polynomials
We provide a characterization of the radii minimal projections of polytopes
onto -dimensional subspaces in Euclidean space \E^n. Applied on simplices
this characterization allows to reduce the computation of an outer radius to a
computation in the circumscribing case or to the computation of an outer radius
of a lower-dimensional simplex. In the second part of the paper, we use this
characterization to determine the sequence of outer -radii of regular
simplices (which are the radii of smallest enclosing cylinders). This settles a
question which arose from the incidence that a paper by Wei{\ss}bach (1983) on
this determination was erroneous. In the proof, we first reduce the problem to
a constrained optimization problem of symmetric polynomials and then to an
optimization problem in a fixed number of variables with additional integer
constraints.Comment: Minor revisions. To appear in Advances in Geometr
Anderson Localization in Disordered Vibrating Rods
We study, both experimentally and numerically, the Anderson localization
phenomenon in torsional waves of a disordered elastic rod, which consists of a
cylinder with randomly spaced notches. We find that the normal-mode wave
amplitudes are exponentially localized as occurs in disordered solids. The
localization length is measured using these wave amplitudes and it is shown to
decrease as a function of frequency. The normal-mode spectrum is also measured
as well as computed, so its level statistics can be analyzed. Fitting the
nearest-neighbor spacing distribution a level repulsion parameter is defined
that also varies with frequency. The localization length can then be expressed
as a function of the repulsion parameter. There exists a range in which the
localization length is a linear function of the repulsion parameter, which is
consistent with Random Matrix Theory. However, at low values of the repulsion
parameter the linear dependence does not hold.Comment: 10 pages, 6 figure
New Confocal Hyperbola-based Ellipse Fitting with Applications to Estimating Parameters of Mechanical Pipes from Point Clouds
This manuscript presents a new method for fitting ellipses to two-dimensional
data using the confocal hyperbola approximation to the geometric distance of
points to ellipses. The proposed method was evaluated and compared to
established methods on simulated and real-world datasets. First, it was
revealed that the confocal hyperbola distance considerably outperforms other
distance approximations such as algebraic and Sampson. Next, the proposed
ellipse fitting method was compared with five reliable and established methods
proposed by Halir, Taubin, Kanatani, Ahn and Szpak. The performance of each
method as a function of rotation, aspect ratio, noise, and arc-length were
examined. It was observed that the proposed ellipse fitting method achieved
almost identical results (and in some cases better) than the gold standard
geometric method of Ahn and outperformed the remaining methods in all
simulation experiments. Finally, the proposed method outperformed the
considered ellipse fitting methods in estimating the geometric parameters of
cylindrical mechanical pipes from point clouds. The results of the experiments
show that the confocal hyperbola is an excellent approximation to the true
geometric distance and produces reliable and accurate ellipse fitting in
practical settings
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