825 research outputs found
Circular Separability of Polygons
Two planar sets are circularly separable if there exists a circle enclosing one of the set and whose open interior disk does not intersect the other set. This paper studies two problems related to circular separability. A linear-time algorithm is proposed to decide if two polygons are circularly separable. The algorithm outputs the smallest separating circle. The second problem asks for the largest circle included in a preprocessed, convex polygon, under some point and/or line constraints. The resulting circle must contain the query points and it must lie in the halfplanes delimited by the query lines
Computing largest circles separating two sets of segments
A circle separates two planar sets if it encloses one of the sets and its
open interior disk does not meet the other set. A separating circle is a
largest one if it cannot be locally increased while still separating the two
given sets. An Theta(n log n) optimal algorithm is proposed to find all largest
circles separating two given sets of line segments when line segments are
allowed to meet only at their endpoints. In the general case, when line
segments may intersect times, our algorithm can be adapted to
work in O(n alpha(n) log n) time and O(n \alpha(n)) space, where alpha(n)
represents the extremely slowly growing inverse of the Ackermann function.Comment: 14 pages, 3 figures, abstract presented at 8th Canadian Conference on
Computational Geometry, 199
A Multi-Temporal Object-Based Image Analysis to Detect Long-Lived Shrub Cover Changes in Drylands
Climate change and human actions condition the spatial distribution and structure of vegetation, especially in drylands. In this context, object-based image analysis (OBIA) has been used to monitor changes in vegetation, but only a few studies have related them to anthropic pressure. In this study, we assessed changes in cover, number, and shape of Ziziphus lotus shrub individuals in a coastal groundwater-dependent ecosystem in SE Spain over a period of 60 years and related them to human actions in the area. In particular, we evaluated how sand mining, groundwater extraction, and the protection of the area affect shrubs. To do this, we developed an object-based methodology that allowed us to create accurate maps (overall accuracy up to 98%) of the vegetation patches and compare the cover changes in the individuals identified in them. These changes in shrub size and shape were related to soil loss, seawater intrusion, and legal protection of the area measured by average minimum distance (AMD) and average random distance (ARD) analysis. It was found that both sand mining and seawater intrusion had a negative effect on individuals; on the contrary, the protection of the area had a positive effect on the size of the individuals’ coverage. Our findings support the use of OBIA as a successful methodology for monitoring scattered vegetation patches in drylands, key to any monitoring program aimed at vegetation preservation
Intersection of three-dimensional geometric surfaces
Calculating the line of intersection between two three-dimensional objects and using the information to generate a third object is a key element in a geometry development system. Techniques are presented for the generation of three-dimensional objects, the calculation of a line of intersection between two objects, and the construction of a resultant third object. The objects are closed surfaces consisting of adjacent bicubic parametric patches using Bezier basis functions. The intersection determination involves subdividing the patches that make up the objects until they are approximately planar and then calculating the intersection between planes. The resulting straight-line segments are connected to form the curve of intersection. The polygons in the neighborhood of the intersection are reconstructed and put back into the Bezier representation. A third object can be generated using various combinations of the original two. Several examples are presented. Special cases and problems were encountered, and the method for handling them is discussed. The special cases and problems included intersection of patch edges, gaps between adjacent patches because of unequal subdivision, holes, or islands within patches, and computer round-off error
Dispersion and polarization conversion of whispering gallery modes in arbitrary cross-section nanowires
We investigate theoretically the optical properties of Nano-Wires (NWs) with
cross sections having either discrete or cylindrical symmetry. The material
forming the wire is birefringent, showing a different dielectric response in
the plane and along the axis of the wire, which is typically the case for wires
made of wurtzite materials, such as ZnO or GaN. We look for solutions of
Maxwell`s equations having the proper symmetry. The dispersions and the
linewidths versus angle of incident light for the modes having high momentum in
the cross-section plane, so called whispering gallery modes, are calculated. We
put a special emphasis on the case of hexagonal cross sections. The energy
positions of the modes for a set of azimuthal quantum numbers are shown. We
demonstrate the dependence of the energy splitting between TE and TM modes
versus birefringence. The polarization conversion from TE to TM with increase
of the axial wave vectoris discussed for both cylindrical and discrete
symmetry.Comment: 9 pages, 10 figure
On a discretization of confocal quadrics. I. An integrable systems approach
Confocal quadrics lie at the heart of the system of confocal coordinates
(also called elliptic coordinates, after Jacobi). We suggest a discretization
which respects two crucial properties of confocal coordinates: separability and
all two-dimensional coordinate subnets being isothermic surfaces (that is,
allowing a conformal parametrization along curvature lines, or, equivalently,
supporting orthogonal Koenigs nets). Our construction is based on an integrable
discretization of the Euler-Poisson-Darboux equation and leads to discrete nets
with the separability property, with all two-dimensional subnets being Koenigs
nets, and with an additional novel discrete analog of the orthogonality
property. The coordinate functions of our discrete nets are given explicitly in
terms of gamma functions.Comment: 37 pp., 9 figures. V2 is a completely reworked and extended version,
with a lot of new materia
On reconfiguration of disks in the plane and related problems
We revisit two natural reconfiguration models for systems of disjoint objects in the plane: translation and sliding. Consider a set of n pairwise interior-disjoint objects in the plane that need to be brought from a given start (initial) configuration S into a desired goal (target) configuration T, without causing collisions. In the translation model, in one move an object is translated along a fixed direction to another position in the plane. In the sliding model, one move is sliding an object to another location in the plane by means of an arbitrarily complex continuous motion (that could involve rotations). We obtain various combinatorial and computational results for these two models: (I) For systems of n congruent disks in the translation model, Abellanas et al. showed that 2n − 1 moves always suffice and ⌊8n/5 ⌋ moves are sometimes necessary for transforming the start configuration into the target configuration. Here we further improve the lower bound to ⌊5n/3 ⌋ − 1, and thereby give a partial answer to one of their open problems. (II) We show that the reconfiguration problem with congruent disks in the translation model is NPhard, in both the labeled and unlabeled variants. This answers another open problem of Abellanas et al. (III) We also show that the reconfiguration problem with congruent disks in the sliding model is NP-hard, in both the labeled and unlabeled variants. (IV) For the reconfiguration with translations of n arbitrary convex bodies in the plane, 2n moves are always sufficient and sometimes necessary
Separating bichromatic point sets in the plane by restricted orientation convex hulls
The version of record is available online at: http://dx.doi.org/10.1007/s10898-022-01238-9We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let R and B be two disjoint sets of red and blue points in the plane, and O be a set of k=2 lines passing through the origin. We study the problem of computing the set of orientations of the lines of O for which the O-convex hull of R contains no points of B. For k=2 orthogonal lines we have the rectilinear convex hull. In optimal O(nlogn) time and O(n) space, n=|R|+|B|, we compute the set of rotation angles such that, after simultaneously rotating the lines of O around the origin in the same direction, the rectilinear convex hull of R contains no points of B. We generalize this result to the case where O is formed by k=2 lines with arbitrary orientations. In the counter-clockwise circular order of the lines of O, let ai be the angle required to clockwise rotate the ith line so it coincides with its successor. We solve the problem in this case in O(1/T·NlogN) time and O(1/T·N) space, where T=min{a1,…,ak} and N=max{k,|R|+|B|}. We finally consider the case in which O is formed by k=2 lines, one of the lines is fixed, and the second line rotates by an angle that goes from 0 to p. We show that this last case can also be solved in optimal O(nlogn) time and O(n) space, where n=|R|+|B|.Carlos Alegría: Research supported by MIUR Proj. “AHeAD” no 20174LF3T8. David Orden:
Research supported by Project PID2019-104129GB-I00 / AEI / 10.13039/501100011033 of the Spanish
Ministry of Science and Innovation. Carlos Seara: Research supported by Project PID2019-104129GB-I00 /
AEI / 10.13039/501100011033 of the Spanish Ministry of Science and Innovation. Jorge Urrutia: Research
supported in part by SEP-CONACYThis project has received funding from the European Union’s Horizon 2020 research and innovation
programme under the Marie Skłodowska–Curie Grant Agreement No 734922.Peer ReviewedPostprint (published version
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