By Jin Akiyama, Hiro Ito, Toshinori Sakai
This ebook constitutes the completely refereed post-conference lawsuits of the sixteenth eastern convention on Discrete and computational Geometry and Graphs, JDCDGG 2013, held in Tokyo, Japan, in September 2013.
The overall of sixteen papers incorporated during this quantity was once rigorously reviewed and chosen from fifty eight submissions. The papers characteristic advances made within the box of computational geometry and concentrate on rising applied sciences, new method and purposes, graph concept and dynamics.
Read or Download Discrete and Computational Geometry and Graphs: 16th Japanese Conference, JCDCGG 2013, Tokyo, Japan, September 17-19, 2013, Revised Selected Papers PDF
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Additional resources for Discrete and Computational Geometry and Graphs: 16th Japanese Conference, JCDCGG 2013, Tokyo, Japan, September 17-19, 2013, Revised Selected Papers
Any point on a sphere of ﬁxed radius can be described by the two angles of spherical coordinates: θ (polar angle) and φ (azimuthal angle). The underlying strategy is to spiral with constant slope much in the way the 4 sides of the cube were wrapped. Figure 6 shows how the strip wraps a sphere by 0 φ π 2 π θ Fig. 6. Strip wrapping a sphere. Bold lines are mapped without any contraction. maintaining constant dθ/dφ. We focus on only the top hemisphere as the bottom follows by symmetry. 40 A. Cole et al.
The image of g is just a subset of F , but we can extend the domain of f to all of F by mapping the unused region to the boundary of F . The map f sends the line going through the centers of four faces of the cube to an equator of the sphere without any contraction. If S is the sidelength of the cube, then the resulting sphere will have radius R = 2S/π. This also shows f is optimal: no contractive mapping can produce larger spheres from a cube. 42 A. Cole et al. √ Theorem 3. S-tetrahedra contractively map to S/(2 3 arccos √13 )-spheres.
Previous lower bounds (Lower Bounds 1–4) are plotted as black dots. Using Theorem 1, discrete constructions for cube lower are trans√ √ bounds 2 × 2 wrapping of formed into a continuum. One surprise here is that the 1/ √ the 1/(2 2)-cube is less eﬃcient than a rescaling of a construction from Lower Bound 2. Other results in Sect. 4 provided signiﬁcant improvements over previous known bounds across a variety of aspect ratios. The two new spherical upper bounds from Sect. 3 greatly improve upon previous bounds, especially for intermediate values of x.
Discrete and Computational Geometry and Graphs: 16th Japanese Conference, JCDCGG 2013, Tokyo, Japan, September 17-19, 2013, Revised Selected Papers by Jin Akiyama, Hiro Ito, Toshinori Sakai