774,724 research outputs found
Normal forms and invariants for 2-dimensional almost-Riemannian structures
Two-dimensional almost-Riemannian structures are generalized Riemannian
structures on surfaces for which a local orthonormal frame is given by a Lie
bracket generating pair of vector fields that can become collinear.
Generically, there are three types of points: Riemannian points where the two
vector fields are linearly independent, Grushin points where the two vector
fields are collinear but their Lie bracket is not, and tangency points where
the two vector fields and their Lie bracket are collinear and the missing
direction is obtained with one more bracket. In this paper we consider the
problem of finding normal forms and functional invariants at each type of
point. We also require that functional invariants are "complete" in the sense
that they permit to recognize locally isometric structures. The problem happens
to be equivalent to the one of finding a smooth canonical parameterized curve
passing through the point and being transversal to the distribution. For
Riemannian points such that the gradient of the Gaussian curvature is
different from zero, we use the level set of as support of the
parameterized curve. For Riemannian points such that the gradient of the
curvature vanishes (and under additional generic conditions), we use a curve
which is found by looking for crests and valleys of the curvature. For Grushin
points we use the set where the vector fields are parallel. Tangency points are
the most complicated to deal with. The cut locus from the tangency point is not
a good candidate as canonical parameterized curve since it is known to be
non-smooth. Thus, we analyse the cut locus from the singular set and we prove
that it is not smooth either. A good candidate appears to be a curve which is
found by looking for crests and valleys of the Gaussian curvature. We prove
that the support of such a curve is uniquely determined and has a canonical
parametrization
Fast Fencing
We consider very natural "fence enclosure" problems studied by Capoyleas,
Rote, and Woeginger and Arkin, Khuller, and Mitchell in the early 90s. Given a
set of points in the plane, we aim at finding a set of closed curves
such that (1) each point is enclosed by a curve and (2) the total length of the
curves is minimized. We consider two main variants. In the first variant, we
pay a unit cost per curve in addition to the total length of the curves. An
equivalent formulation of this version is that we have to enclose unit
disks, paying only the total length of the enclosing curves. In the other
variant, we are allowed to use at most closed curves and pay no cost per
curve.
For the variant with at most closed curves, we present an algorithm that
is polynomial in both and . For the variant with unit cost per curve, or
unit disks, we present a near-linear time algorithm.
Capoyleas, Rote, and Woeginger solved the problem with at most curves in
time. Arkin, Khuller, and Mitchell used this to solve the unit cost
per curve version in exponential time. At the time, they conjectured that the
problem with curves is NP-hard for general . Our polynomial time
algorithm refutes this unless P equals NP
Fast B-Spline 2D Curve Fitting for unorganized Noisy Datasets
In the context of coordinate metrology and reverse engineering, freeform curve reconstruction from unorganized data points still offers ways for improvement. Geometric convection is the process of fitting a closed shape, generally represented in the form of a periodic B-Spline model, to data points [WPL06]. This process should be robust to freeform shapes and convergence should be assured even in the presence of noise. The convection's starting point is a periodic B-Spline polygon defined by a finite number of control points that are distributed around the data points. The minimization of the sum of the squared distances separating the B-Spline curve and the points is done and translates into an adaptation of the shape of the curve, meaning that the control points are either inserted, removed or delocalized automatically depending on the accuracy of the fit. Computing distances is a computationally expensive step in which finding the projection of each of the data points requires the determination of location parameters along the curve. Zheng et al [ZBLW12] propose a minimization process in which location parameters and control points are calculated simultaneously. We propose a method in which we do not need to estimate location parameters, but rather compute topological distances that can be assimilated to the Hausdorff distances using a two-step association procedure. Instead of using the continuous representation of the B-Spline curve and having to solve for footpoints, we set the problem in discrete form by applying subdivision of the control polygon. This generates a discretization of the curve and establishes the link between the discrete point-to-curve distances and the position of the control points. The first step of the association process associates BSpline discrete points to data points and a segmentation of the cloud of points is done. The second step uses this segmentation to associate to each data point the nearest discrete BSpline segment. Results are presented for the fitting of turbine blades profiles and a thorough comparison between our approach and the existing methods is given [ZBLW12, WPL06, SKH98]
Recommended from our members
Recursive Percentage based Hybrid Pattern Training for Supervised Learning
Supervised learning algorithms, often used to find the I/O relationship in data, have the tendency to be trapped in local optima as opposed to the desirable global optima. In this paper, we discuss the RPHP learning algorithm. The algorithm uses Real Coded Genetic Algorithm based global and local searches to find a set of pseudo global optimal solutions. Each pseudo global optimum is a local optimal solution from the point of view of all the patterns but globally optimal from the point of view of a subset of patterns. Together with RPHP, a Kth nearest neighbor algorithm is used as a second level pattern distributor to solve a test pattern. We also show theoretically the condition under which finding several pseudo global optimal solutions requires a shorter training time than finding a single global optimal solution. As the difficulty of curve fitting problems is easily estimated, we verify the capability of the RPHP algorithm against them and compare the RPHP algorithm with three counterparts to show the benefits of hybrid learning and active recursive subset selection. The RPHP shows a clear superiority in performance. We conclude our paper by identifying possible loopholes in the RPHP algorithm and proposing possible solutions
On fixed figure problems in fuzzy metric spaces
summary:Fixed circle problems belong to a realm of problems in metric fixed point theory. Specifically, it is a problem of finding self mappings which remain invariant at each point of the circle in the space. Recently this problem is well studied in various metric spaces. Our present work is in the domain of the extension of this line of research in the context of fuzzy metric spaces. For our purpose, we first define the notions of a fixed circle and of a fixed Cassini curve then determine suitable conditions which ensure the existence and uniqueness of a fixed circle (resp. a Cassini curve) for the self operators. Moreover, we present a result which prescribed that the fixed point set of fuzzy quasi-nonexpansive mapping is always closed. Our results are supported by examples
A Sequential Descent Method for Global Optimization
In this paper, a sequential search method for finding the global minimum of
an objective function is presented, The descent gradient search is repeated
until the global minimum is obtained. The global minimum is located by a
process of finding progressively better local minima. We determine the set of
points of intersection between the curve of the function and the horizontal
plane which contains the local minima previously found. Then, a point in this
set with the greatest descent slope is chosen to be a initial point for a new
descent gradient search. The method has the descent property and the
convergence is monotonic. To demonstrate the effectiveness of the proposed
sequential descent method, several non-convex multidimensional optimization
problems are solved. Numerical examples show that the global minimum can be
sought by the proposed method of sequential descent
- âŠ