22,760 research outputs found
Stacking the Equiangular Spiral
We present an algorithm that adapts the mature Stack and Draw (SaD) methodology for fabricating the exotic Equiangular Spiral Photonic Crystal Fiber. (ES-PCF) The principle of Steiner chains and circle packing is exploited to obtain a non-hexagonal design using a stacking procedure based on Hexagonal Close Packing. The optical properties of the proposed structure are promising for SuperContinuum Generation. This approach could make accessible not only the equiangular spiral but also other quasi-crystal PCF through SaD
Combinatorial Calabi flows on surfaces
For triangulated surfaces, we introduce the combinatorial Calabi flow which
is an analogue of smooth Calabi flow. We prove that the solution of
combinatorial Calabi flow exists for all time. Moreover, the solution converges
if and only if Thurston's circle packing exists. As a consequence,
combinatorial Calabi flow provides a new algorithm to find circle packings with
prescribed curvatures. The proofs rely on careful analysis of combinatorial
Calabi energy, combinatorial Ricci potential and discrete dual-Laplacians.Comment: 17 pages, 5 figure
Online Circle and Sphere Packing
In this paper we consider the Online Bin Packing Problem in three variants:
Circles in Squares, Circles in Isosceles Right Triangles, and Spheres in Cubes.
The two first ones receive an online sequence of circles (items) of different
radii while the third one receive an online sequence of spheres (items) of
different radii, and they want to pack the items into the minimum number of
unit squares, isosceles right triangles of leg length one, and unit cubes,
respectively. For Online Circle Packing in Squares, we improve the previous
best-known competitive ratio for the bounded space version, when at most a
constant number of bins can be open at any given time, from 2.439 to 2.3536.
For Online Circle Packing in Isosceles Right Triangles and Online Sphere
Packing in Cubes we show bounded space algorithms of asymptotic competitive
ratios 2.5490 and 3.5316, respectively, as well as lower bounds of 2.1193 and
2.7707 on the competitive ratio of any online bounded space algorithm for these
two problems. We also considered the online unbounded space variant of these
three problems which admits a small reorganization of the items inside the bin
after their packing, and we present algorithms of competitive ratios 2.3105,
2.5094, and 3.5146 for Circles in Squares, Circles in Isosceles Right
Triangles, and Spheres in Cubes, respectively
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