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Feb 6, 2007, 7:27:42 AM2/6/07

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In 1994 Rakhmanov, Saff and Zhou (1) suggested a simple and elegant

spiral set of n points on the surface of the unit sphere, which for

increasing n has a total 'potential energy' that converges nicely to

the global extremum value for n 'evenly distributed' points on the

sphere.

spiral set of n points on the surface of the unit sphere, which for

increasing n has a total 'potential energy' that converges nicely to

the global extremum value for n 'evenly distributed' points on the

sphere.

It seems that the small modification of the spiral algorithm suggested

below leads to a slight improvement in the potential energy.

We shall restrict ourselves to three types of potential energy E(s,n)

which are discussed in detail by the above authors:

E(-1,n) = sum(n>=j>i>=1) 1/dij

E( 0,n) = sum(n>=j>i>=1) ln(1/dij)

E(+1,n) = sum(n>=j>i>=1) -dij

dij being the Euclidean distance between points i and j.

Note that in this discussion the sign of E(+1,n) is chosen so that the

global extremum is a minimum, as for the other two.

The spiral point set of Rakhmanov et al. can be defined, in spherical

coordinates (pi>=theta>=0, 2pi>=phi>=0) and for point index 1<=k<=n:

h(k) = -1 + 2(k-1)/(n-1)

theta(k) = arccos(h(k))

r(k) = sqrt(1-h(k)^2)

0, k=1

phi(k) = [phi(k-1) + 3.6/sqrt(n) 1/r(k)] (mod 2pi), 2<k<n-1

0, k=n

Note that the horizontal radius r(k) is used to define the increment

in phi from point k-1 to point k.

An interesting idea is to use instead a kind of midway radius in the

calculation of that phi increment, namely (r(k-1)+r(k))/2, and this

actually results in a small decrease in the potential energy.

Let us denote the energies measured for Rakhmanov et al.'s original

algorithm as E(s,n), and those measured for the suggested modification

as E'(s,n).

Fortunately, the authors also created good approximations to the

minimum energy for n points, here denoted f(s,n), and we shall use

these approximations to estimate the fractional improvement towards

the minimum energy:

dE(s,n) = (E(s,n)-E'(s,n)) / (E(s,n)-f(s,n))

The table below gives the fractional improvement for selected n:

n dE(-1,n) dE( 0,n) dE(+1,n)

---------------------------------

2 -0.000000 -0.000000 -0.000000

3 0.000000 0.000000 0.000000

4 0.000000 -0.000000 0.000000

5 0.002477 -0.000791 -0.008293

6 0.018490 0.025285 0.046053

7 0.035677 0.054766 0.110411

8 0.043506 0.062189 0.109718

9 0.048726 0.060885 0.083121

10 0.044847 0.051442 0.053930

20 0.066426 0.070708 0.068975

50 0.065120 0.067782 0.073075

100 0.064206 0.061789 0.049650

200 0.056133 0.052754 0.036536

500 0.034383 0.033912 0.024079

1000 0.021504 0.022263 0.015808

2000 0.012785 0.013871 0.010400

5000 0.006038 0.006875 0.005531

10000 0.003248 0.003796 0.003402

20000 0.001626 0.001934 0.001959

1. Rakhmanov, Saff and Zhou: Minimal Discrete Energy on the Sphere,

Mathematical Research Letters, Vol. 1 (1994), pp. 647-662.

www.math.vanderbilt.edu/~esaff/texts/155.pdf

Best regards,

Knud Thomsen

Geologist

Feb 6, 2007, 7:42:06 AM2/6/07

to

In the definition of phi(k), please read 2<=k<=n-1 rather than

2<k<n-1.

2<k<n-1.

Knud Thomsen

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