Towards Uniform Online Spherical Tessellations

by   Paul C. Bell, et al.

The problem of uniformly placing N points onto a sphere finds applications in many areas. An online version of this problem was recently studied with respect to the gap ratio as a measure of uniformity. The proposed online algorithm of Chen et al. is upper-bounded by 5.99, which is achieved by considering a circumscribed dodecahedron followed by a recursive decomposition of each face. We analyse a simple tessellation technique based on the regular icosahedron, which decreases the upper-bound for the online version of this problem to around 2.84. Moreover, we show that the lower bound for the gap ratio of placing up to three points is approximately 1.618. The uniform distribution of points on a sphere also corresponds to uniform distribution of unit quaternions which represent rotations in 3D space and has numerous applications in many areas.



page 1

page 2

page 3

page 4


Advice Complexity of Online Non-Crossing Matching

We study online matching in the Euclidean 2-dimesional plane with non-cr...

A Tight Lower Bound for Uniformly Stable Algorithms

Leveraging algorithmic stability to derive sharp generalization bounds i...

The Randomized Competitive Ratio of Weighted k-server is at least Exponential

The weighted k-server problem is a natural generalization of the k-serve...

Attenuate Locally, Win Globally: An Attenuation-based Framework for Online Stochastic Matching with Timeouts

Online matching problems have garnered significant attention in recent y...

A practical algorithm to calculate Cap Discrepancy

Uniform distribution of the points has been of interest to researchers f...

An enumerative formula for the spherical cap discrepancy

The spherical cap discrepancy is a widely used measure for how uniformly...

On the Number of Faces and Radii of Cells Induced by Gaussian Spherical Tessellations

We study a geometric property related to spherical hyperplane tessellati...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.