Rational Points on the Unit Sphere: Approximation Complexity and Practical Constructions

Rational Points on the Unit Sphere

The unit sphere, centered at the origin in R, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point v on the unit sphere in R, and every > 0, there is a point r = (r1, r2, . . . , rn) such that: • ||r− v||∞ < . • r is also a point on the unit sphere; P r i = 1. • r has rat...

Points at Rational Distance from the Vertices of a Unit Polygon

A free group acting without fixed points on the rational unit sphere

We prove the existence of a free group of rotations of rank 2 which acts on the rational unit sphere without non-trivial fixed points. Introduction. The purpose of this paper is to prove that the group SO3(Q) of all proper orthogonal 3 × 3 matrices with rational entries has a free subgroup F2 of rank 2 such that for all w ∈ F2 different from the identity and for all ~r ∈ S2 ∩Q3 we have w(~r ) 6...

Weighted quadrature formulas and approximation by zonal function networks on the sphere

Let q ≥ 1 be an integer, Sq be the unit sphere embedded in the Euclidean space Rq+1. A Zonal Function (ZF) network with an activation function φ : [−1, 1] → R and n neurons is a function on Sq of the form x 7→ nk=1 akφ(x · ξk), where ak’s are real numbers, ξk’s are points on Sq. We consider the activation functions φ for which the coefficients {φ̂(`)} in the appropriate ultraspherical polynomial...

Representation of a digitized surface by triangular faces

This paper proposes a triangulation method for a digitized surface whose points are located on a regular lattice. The method relies on an iterative and adaptive splitting of triangular faces of an initial polyhedral surface. Assuming a bijection between the digitized surface and its approximation, a partition of the data base is performed. The method allows the measurement of the local quality ...