How can I generate 3-d random points with minimum distance between each of them?

Question

I am going to generate 10^6 random points in matlab with this particular characters. the points should be inside a sphere with radius 25, the are 3-D so we have x, y, z or r, theta, phi. there is a minimum distance between each points. First, I decided to generate points and then check the distances, then omit points with do not have these condition. but, it may omit many of points. Another way is to use RSA (Random Sequential Addition), it means generate points one by one with this minimum distance between them. For example generate first point, then generate second randomly out of the minimum distance from point 1. And go on till achieving 10^6 points. but it takes lots of time and I can not reach 10^6 points, since the speed of searching appropriate position for new points will take long time.

Right now I am using this program:

Nmax=10000; 
R=25; 
P=rand(1,3); 
k=1; 
while k<Nmax 
theta=2*pi*rand(1); 
phi=pi*rand(1); 
r = R*sqrt(rand(1)); 
% convert to cartesian 
x=r.*sin(theta).*cos(phi); 
y=r.*sin(theta).*sin(phi); 
z=r.*cos(theta); 
P1=[x y z]; 
r=sqrt((x-0)^2+(y-0)^2+(z-0)^2); 
D = pdist2(P1,P,'euclidean'); 
% euclidean distance 

    if D>0.146*r^(2/3) 
        P=[P;P1]; 
        k=k+1;
    end 
    i=i+1; 
end 
x=P(:,1);y=P(:,2);z=P(:,3); plot3(x,y,z,'.');

How can I efficiently generate points by these condition?

Answer 1

I took a closer look at your algorithm, and concluded there is NO WAY it will ever work - at least not if you really want to get a million points in that sphere. There is a simple picture that explains why not - this is a plot of the number of points that you need to test (using your technique of RSA) to get one additional "good" point. As you can see, this goes asymptotic at just a few thousand points (I ran a slightly faster algorithm against 200k points to produce this):

I don't know if you ever tried to compute the theoretical number of points you could fit in your sphere when you have them perfectly arranged, but I'm beginning to suspect the number is a good deal smaller than 1E6.

The complete code I used to investigate this, plus the output it generated, can be found here. I never got as far as the technique I described in my earlier answer... there was just too much else going on in the setup you described.

EDIT: I started to think it might not be possible, even with "perfect" arrangement, to get to 1M points. I made a simple model for myself as follows:

Imagine you start on the "outer shell" (r=25), and try to fit points at equal distances. If you divide the area of the "shell" by the area of one "exclusion disk" (of radius r_sub_crit), you get a (high) estimate of the number of points at that distance:

numpoints = 4*pi*r^2 / (pi*(0.146 * r^(2/3))^2) ~ 188 * r^(2/3)

The next "shell" in should be at a radius that is 0.146*r^(2/3) less - but if you think of the points as being very carefully arranged, you might be able to get a tiny bit closer. Again, let's be generous and say the shells can be just 1/sqrt(3) closer than the criteria. You can then start at the outer shell and work your way in, using a simple python script:

import scipy as sc
r = 25
npts = 0
def rc(r):
  return 0.146*sc.power(r, 2./3.)
while (r > rc(r)):  
  morePts = sc.floor(4/(0.146*0.146)*sc.power(r, 2./3.))
  npts = npts + morePts
  print morePts, ' more points at r = ', r
  r = r - rc(r)/sc.sqrt(3)

print 'total number of points fitted in sphere: ', npts

The output of this is:

1604.0  more points at r =  25
1573.0  more points at r =  24.2793037966
1542.0  more points at r =  23.5725257555
1512.0  more points at r =  22.8795314897
1482.0  more points at r =  22.2001865995
1452.0  more points at r =  21.5343566722
1422.0  more points at r =  20.8819072818
1393.0  more points at r =  20.2427039885
1364.0  more points at r =  19.6166123391
1336.0  more points at r =  19.0034978659
1308.0  more points at r =  18.4032260869
1280.0  more points at r =  17.8156625053
1252.0  more points at r =  17.2406726094
1224.0  more points at r =  16.6781218719
1197.0  more points at r =  16.1278757499
1171.0  more points at r =  15.5897996844
1144.0  more points at r =  15.0637590998
1118.0  more points at r =  14.549619404
1092.0  more points at r =  14.0472459873
1066.0  more points at r =  13.5565042228
1041.0  more points at r =  13.0772594652
1016.0  more points at r =  12.6093770509
991.0  more points at r =  12.1527222975
967.0  more points at r =  11.707160503
943.0  more points at r =  11.2725569457
919.0  more points at r =  10.8487768835
896.0  more points at r =  10.4356855535
872.0  more points at r =  10.0331481711
850.0  more points at r =  9.64102993012
827.0  more points at r =  9.25919600154
805.0  more points at r =  8.88751153329
783.0  more points at r =  8.52584164948
761.0  more points at r =  8.17405144976
740.0  more points at r =  7.83200600865
718.0  more points at r =  7.49957037478
698.0  more points at r =  7.17660957023
677.0  more points at r =  6.86298858965
657.0  more points at r =  6.55857239952
637.0  more points at r =  6.26322593726
618.0  more points at r =  5.97681411037
598.0  more points at r =  5.69920179546
579.0  more points at r =  5.43025383729
561.0  more points at r =  5.16983504778
542.0  more points at r =  4.91781020487
524.0  more points at r =  4.67404405146
506.0  more points at r =  4.43840129415
489.0  more points at r =  4.21074660206
472.0  more points at r =  3.9909446055
455.0  more points at r =  3.77885989456
438.0  more points at r =  3.57435701766
422.0  more points at r =  3.37730048004
406.0  more points at r =  3.1875547421
390.0  more points at r =  3.00498421767
375.0  more points at r =  2.82945327223
360.0  more points at r =  2.66082622092
345.0  more points at r =  2.49896732654
331.0  more points at r =  2.34374079733
316.0  more points at r =  2.19501078464
303.0  more points at r =  2.05264138052
289.0  more points at r =  1.91649661498
276.0  more points at r =  1.78644045325
263.0  more points at r =  1.66233679273
250.0  more points at r =  1.54404945973
238.0  more points at r =  1.43144220603
226.0  more points at r =  1.32437870508
214.0  more points at r =  1.22272254805
203.0  more points at r =  1.1263372394
192.0  more points at r =  1.03508619218
181.0  more points at r =  0.94883272297
170.0  more points at r =  0.867440046252
160.0  more points at r =  0.790771268402
150.0  more points at r =  0.718689381062
140.0  more points at r =  0.65105725389
131.0  more points at r =  0.587737626612
122.0  more points at r =  0.528593100237
113.0  more points at r =  0.473486127367
105.0  more points at r =  0.422279001431
97.0  more points at r =  0.374833844693
89.0  more points at r =  0.331012594847
82.0  more points at r =  0.290676989951
75.0  more points at r =  0.253688551418
68.0  more points at r =  0.219908564725
61.0  more points at r =  0.189198057381
55.0  more points at r =  0.161417773651
49.0  more points at r =  0.136428145311
44.0  more points at r =  0.114089257597
38.0  more points at r =  0.0942608092113
33.0  more points at r =  0.0768020649149
29.0  more points at r =  0.0615717987589
24.0  more points at r =  0.0484282253244
20.0  more points at r =  0.0372289153633
17.0  more points at r =  0.0278306908104
13.0  more points at r =  0.0200894920319
10.0  more points at r =  0.013860207063
8.0  more points at r =  0.00899644813842
5.0  more points at r =  0.00535025545232 

total number of points fitted in sphere:  55600.0

This seems to confirm that you really can't get to a million, no matter how you try...