Equations of negative curves of blow-ups of Ehrhart rings of rational convex polygons

Despite the extent of subsistence farmland in Africa, little is known about endangered species that persist within them. The Cape Vulture (Gyps coprotheres) is regionally endangered in southern Africa and at least 20% of the population breeds in the subsistence farmland area previously known as the Transkei in the Eastern Cape province of South Africa. To understand their movement ecology, adult Cape Vultures (n = 9) were captured and fitted with global positioning system/global system for mobile transmitters. Minimum convex polygons (MCPs),and 99% and 50% kernel density estimates (KDEs) were calculated for the breeding and non breeding seasons of the Cape Vulture. Land use maps were constructed for each 99% KDE and vulture locations were overlaid. During the non-breeding season, ranges were slightly larger(mean [± SE] MCP = 16 887 km2 ± 366 km2) than the breeding season (MCP = 14 707 km2 ± 2155 km2). Breeding and non-breeding season MCPs overlapped by a total of 92%. Kernel density estimates showed seasonal variability. During the breeding season, Cape Vultures used subsistence farmland, natural woodland and protected areas more than expected. In the non-breeding season, vultures used natural woodland and subsistence farmland more than expected, and protected areas less than expected. In both seasons, human-altered landscapes were used less, except for subsistence farmland.Conservation implications: These results highlight the importance of subsistence farm land to the survival of the Cape Vulture. Efforts should be made to minimise potential threats to vultures in the core areas outlined, through outreach programmes and mitigation measures.The conservation buffer of 40 km around Cape Vulture breeding colonies should be increased to 50 km.

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Small distances in convex polygons

Discrete Mathematics ◽

10.1016/j.disc.2013.05.007 ◽

2013 ◽

Vol 313 (18) ◽

pp. 1767-1782 ◽

Cited By ~ 1

Author(s):

Filip Morić

Keyword(s):

Convex Polygons

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Densest translational lattice packing of non-convex polygons

Computational Geometry ◽

10.1016/s0925-7721(01)00051-7 ◽

2002 ◽

Vol 22 (1-3) ◽

pp. 205-222 ◽

Cited By ~ 6

Author(s):

Victor J. Milenkovic

Keyword(s):

Lattice Packing ◽

Convex Polygons

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AN APPROXIMATION ALGORITHM FOR LOCATING MAXIMAL DISKS WITHIN CONVEX POLYGONS

International Journal of Computational Geometry & Applications ◽

10.1142/s0218195911003858 ◽

2011 ◽

Vol 21 (06) ◽

pp. 661-684

Author(s):

HIROFUMI AOTA ◽

TAKURO FUKUNAGA ◽

HIROSHI NAGAMOCHI

Keyword(s):

Approximation Algorithm ◽

Convex Polygon ◽

Computation Time ◽

Approximation Ratio ◽

Convex Polygons ◽

The Given

This paper considers a problem of locating the given number of disks into a container so that the area covered by the disks is maximized. In the problem, the radii of the disks can be changed arbitrarily unless they overlap outside of the container, and the disks are allowed to overlap with each other. We present an approximation algorithm for this problem assuming that the container is a convex polygon. Our algorithm achieves approximation ratio (0.78 - ϵ) for any small ϵ > 0. Since the computation time of our algorithm depends on the number of corners of the convex polygon exponentially, we also give a heuristic to reduce the number of corners.

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An Algorithm for a Near Optimal NC Path Generation in Staircase (Lase) Traversal of Convex Polygonal Surfaces

Journal of Manufacturing Science and Engineering ◽

10.1115/1.538895 ◽

1999 ◽

Vol 122 (1) ◽

pp. 182-190 ◽

Cited By ~ 2

Author(s):

S. V. Kamarthi ◽

S. T. S. Bukkapatnam ◽

S. Hsieh

Keyword(s):

Rapid Prototyping ◽

Path Length ◽

Tool Path ◽

Path Generation ◽

Convex Polygons ◽

Sweep Angle ◽

Pocket Milling ◽

Different Shapes ◽

Path Lengths ◽

Better Than

This paper presents an analytical model of the tool path for staircase traversal of convex polygonal surfaces, and an algorithm—referred to as OPTPATH—developed based on the model to find the sweep angle that gives a near optimal tool path length. The OPTPATH algorithm can be used for staircase traversal with or without (i) overlaps between successive sweep passes, and (ii) rapid traversal along edge passes. This flexibility of OPTPATH renders it applicable not only to conventional operations such as face and pocket milling, but also to other processes such as robotic deburring, rapid prototyping, and robotic spray painting. The effective tool path lengths provided by OPTPATH are compared with those given by the following two algorithms: (i) a common industrial heuristic—referred to as the IH algorithm—and (ii) an algorithm proposed by Prabhu et al. (Prabhu, P. V., Gramopadhye, A. K., and Wang, H. P., 1990, Int. J. Prod. Res., 28, No. 1, pp. 101–130) referred to as PGW algorithm. This comparison is conducted using 100 randomly generated convex polygons of different shapes and a set of seven different tool diameters. It is found that OPTPATH performs better than both the IH as well as PGW algorithms. The superiority of OPTPATH over the two algorithms becomes more pronounced for large tool diameters. [S1087-1357(00)71501-2]

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