Suboptimal covering of multidimensional unit sphere
Each of the text files given below provides a covering of the unit sphere,
which is the surface of the d-dimensional unit ball.
The covering was constructed by using the Stepwise Covering Replenishment (SCR) method
that applies adaptive approximating of the unit ball by a polytope.
In the SCR method, vertices of the approximating polytope are used as the covering base,
i.e. a system of points of the sphere, which epsilon-neighborhood covers the sphere.
In a text file, any list of points, which begins with the first line of the file, provides a covering base.
In addition to a covering base, information of a different type is provided in a text file.
It can be used for calculating the value of epsilon, which is guaranteed by the covering base.
The value of epsilon can be found by using the formula
epsilon = square root of (2 h),
where the value h describes the deviation of the unit ball from the approximating polytope given
by its vertices that are the points listed above the information concerning h.
Such information is given after those points of the text files that start with multiple dashes.
The achieved value of h is given only after those points that change the value of h.
The text files given below contain the covering bases with a certain number of points
(the number depends upon the dimension d).
To obtain a covering with a desired number of points N
(which is not greater than the total number of points of the covering base in the file),
one has to take the first N points.
Alternatively, one can start with a desired value of epsilon,
then calculate a related value of h,
and select the points that are above the calculated value of h.
Downloadable covering bases
- Covering base (85984 points) for d=3 (9222K text file)
- Covering base (49118 points) for d=4 (6372K text file)
- Covering base (23484 points) for d=5 (3592K text file)
- Covering base (10208 points) for d=6 (2085K text file)
- Covering base (3195 points) for d=7 (701K text file)
- Covering base (1122 points) for d=8 (258K text file)
- Covering base (400 points) for d=9 (92K text file)
- Covering base (150 points) for d=10 (30K text file)