695,911 research outputs found
Generalizing the Convex Hull of a Sample: The R Package alphahull
This paper presents the R package alphahull which implements the ñ-convex hull and the ñ-shape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the ñ-convex hull and the ñ-shape are able to reconstruct non-convex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations. The usefulness of the package is illustrated with two small simulation studies on boundary length estimation.
A Multi Centerpoint Theorem via Fourier analysis on the Torus
The Centerpoint Theorem states that for any set of points in , there exists a point such that any hyperplane goes through that point divides the set. For any half-space containing the point , the amount of points in that half-space is no bigger than of the whole set. This can be related to how close can any hyperplane containing the point comes to equipartitioning for a given shape . For a function from unit circle to real number, it has a Fourier interpretation. Using Fourier analysis on the Torus, I will try to find a multi centerpoint theorem for many points in the plane such that any hyperplanes go through those points are close to equipartitioning a given shape
Distributed boundary tracking using alpha and Delaunay-Cech shapes
For a given point set in a plane, we develop a distributed algorithm to
compute the shape of . shapes are well known geometric
objects which generalize the idea of a convex hull, and provide a good
definition for the shape of . We assume that the distances between pairs of
points which are closer than a certain distance are provided, and we show
constructively that this information is sufficient to compute the alpha shapes
for a range of parameters, where the range depends on .
Such distributed algorithms are very useful in domains such as sensor
networks, where each point represents a sensing node, the location of which is
not necessarily known.
We also introduce a new geometric object called the Delaunay-\v{C}ech shape,
which is geometrically more appropriate than an shape for some cases,
and show that it is topologically equivalent to shapes
Generalizing the convex hull of a sample: The R package alphahull
This vignette presents the R package alphahull which implements the α-convex hull and the α-shape of a finite set of points in the plane. These geometric structures provide an informative overview of the shape and properties of the point set. Unlike the convex hull, the α-convex hull and the α-shape are able to reconstruct non-convex sets. This flexibility make them specially useful in set estimation. Since the implementation is based on the intimate relation of theses constructs with Delaunay triangulations, the R package alphahull also includes functions to compute Voronoi and Delaunay tesselations
3D printing dimensional calibration shape: Clebsch Cubic
3D printing and other layer manufacturing processes are challenged by
dimensional accuracy. Several techniques are used to validate and calibrate
dimensional accuracy through the complete building envelope. The validation
process involves the growing and measuring of a shape with known parameters.
The measured result is compared with the intended digital model. Processes with
the risk of deformation after time or post processing may find this technique
beneficial. We propose to use objects from algebraic geometry as test shapes. A
cubic surface is given as the zero set of a 3rd degree polynomial with 3
variables. A class of cubics in real 3D space contains exactly 27 real lines.
We provide a library for the computer algebra system Singular which, from 6
given points in the plane, constructs a cubic and the lines on it. A surface
shape derived from a cubic offers simplicity to the dimensional comparison
process, in that the straight lines and many other features can be analytically
determined and easily measured using non-digital equipment. For example, the
surface contains so-called Eckardt points, in each of which three of the lines
intersect, and also other intersection points of pairs of lines. Distances
between these intersection points can easily be measured, since the points are
connected by straight lines. At all intersection points of lines, angles can be
verified. Hence, many features distributed over the build volume are known
analytically, and can be used for the validation process. Due to the thin shape
geometry the material required to produce an algebraic surface is minimal. This
paper is the first in a series that proposes the process chain to first define
a cubic with a configuration of lines in a given print volume and then to
develop the point cloud for the final manufacturing. Simple measuring
techniques are recommended.Comment: 8 pages, 1 figure, 1 tabl
How to Cover a Point Set with a V-Shape of Minimum Width
A balanced V-shape is a polygonal region in the plane contained in the union
of two crossing equal-width strips. It is delimited by two pairs of parallel
rays that emanate from two points x, y, are contained in the strip boundaries,
and are mirror-symmetric with respect to the line xy. The width of a balanced
V-shape is the width of the strips. We first present an O(n^2 log n) time
algorithm to compute, given a set of n points P, a minimum-width balanced
V-shape covering P. We then describe a PTAS for computing a
(1+epsilon)-approximation of this V-shape in time O((n/epsilon)log
n+(n/epsilon^(3/2))log^2(1/epsilon)). A much simpler constant-factor
approximation algorithm is also described.Comment: In Proceedings of the 12th International Symposium on Algorithms and
Data Structures (WADS), p.61-72, August 2011, New York, NY, US
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