By Jin Akiyama, Hiro Ito, Toshinori Sakai

ISBN-10: 3319132865

ISBN-13: 9783319132860

ISBN-10: 3319132873

ISBN-13: 9783319132877

This booklet constitutes the completely refereed post-conference complaints of the sixteenth jap 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 used to be conscientiously reviewed and chosen from fifty eight submissions. The papers function advances made within the box of computational geometry and concentrate on rising applied sciences, new technique and purposes, graph idea and dynamics.

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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

Sample text

Three diameter-d disks can wrap an R-sphere only if d ≥ πR (see s Table 2 of [11]). Composing with the contrapositive of Proposition 3 yields R ≤ x2 + (3x)−2 /π. 1 Lower Bounds Rescaling Lower Bounds on Cubes Most lower bounds on wrapping take the form of a construction for a specific x. Here we present a method to rescale particular foldings to produce a continuous set of lower bounds. Theorem 1. If x × 1/x paper wraps an S-cube, then there exists a folding of an x × 1/x rectangle into an f (x )-cube where f (x ) = S min{x /x, x/x }.

3 Strip Folding Strip folding is a technique introduced in [3] that weaves a narrow strip of paper back and forth to cover a surface. This section sketches new strategies for strip folding that produce superior bounds on the sphere and the cube. Cubes. Refer to Fig. 5. Here we present a new technique for strip folding on the cube that is more efficient than that presented in [3]. The general strategy consists of 3 parts resembling an algorithm more than a function: Fig. 5. Strip wrapping a cube. 3D diagram (left) and edge unfolding (right).

On Wrapping Spheres and Cubes with Rectangular Paper 35 Proposition 2. An x × y stadium of flat paper mapped onto an R-sphere may occupy no more surface area than A(x, y) = 2R πR − πR cos y y + x sin . 2R 2R Proof. To bound A(x, y) we will first establish A(0, y) and then bound the derivative dA/dx. This will allow us to bound the areas of all mapped stadiums. First, consider an 0 × y stadium: a radius-y/2 disk. By definition, the disk must fall within y/2 of its center on the sphere. A radius-y/2 spherical cap has y area 2πR2 (1 − cos 2R ), proving the claim for A(0, y).

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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

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