Optimal illumination of a target sphere: Change record
Copyright (C) Hugo Pfoertner 2001-2014
Updates to the list of configurations and major changes of page content:
2014 Sep 13 Updated links to J.M. Palmer's Radiometry and photometry FAQ,
Halsted's article, ICF web pages, OEIS, Neil Sloane's home page,
Anton Sherwood's home page
2011 Jul 23 "Minimize difference" N=13,...,20
2010 Aug 3 "Minimize difference" N=16 best, with cyclic symmetry order 3
2010 Aug 2 "Minimize difference" N=16, first config with difference < 0.22
Updates during last month indicated by blinking
2010 Jul 30 "Minimize difference" N=11, 13; 17 with mirror symmetry
Updated links to NIF
2006/12/10 Updated links: Jim Buddenhagen, mathpages
2005/02/04 Updated links: Jim Buddenhagen, short form OEIS, Mersenne
Twister, new links to maximum volume visualizations and
Math. Annalen paper of Schuette and v.d. Waerden.
2004/04/21 Updated links: Jente Goslinga, LLNL NIF facility
2003/03/15 Link to symmetry groups for Tammes problem in Sloane's OEIS,
link to Goslinga's solutions of Thomson problem
2002/10/06 Some ICF links updated
2002/07/08 "Minimize difference" N=18 local opt with mirror symmetry
2002/05/30 "Minimize difference" N=7 local opt with cyclic symmetry order 3
2002/05/26 "Maximize minimum" N=13 local opt with mirror symmetry
2002/05/11 "Minimize maximum" N=8 local opt with cyclic symmetry order 4
2002/05/09 "Minimize difference" N=11 new best config with mirror symmetry
2002/05/05 Picture of triangle mesh and links to corresponding data
added in the "Visualization" paragraph. Link to pdf file with
optimal triangle meshes with 1082, 3002, 4322 and 5072 points.
Anton Sherwood's page with links added.
2002/04/17 Broken link to LLNL's Nova page corrected
2002/03/05 "Mimimize maximum" N=14 local opt with mirror symmetry
2002/02/10 Wording corrected - thanks to Anton Sherwood
2002/02/01 Example pictures (N=6) for the 3 optimality criteria created
with POVRAY. New file optcrit.html. Link to this file added
in introduction of illum.html
2002/01/27 Increased number of contour lines for "Minimize maximum"
pictures N=10,12,13 and local opt pictures DMMM09L1,
DMMM09L2, MAMI10L1, MIMA12L1. Dead link to pdf from intro
picture corrected.
2002/01/26 Increased number of contour lines for "Minimize difference"
pictures for N=9..13 and for N=9 with cos beam profile
2002/01/20 Table with links to result files introduced. This table is sorted
by date of change. If a new best config is found, it will no longer
be listed here (in update.txt), unless it has some special properties
(e.g. symmetry). The new best config N=16 "Minimize difference" thus
will be the last one explicitly being announced here.
2002/01/17 N=13,14 "Minimize difference", new best configs
2002/01/14 N=19 "Maximize minimum", new best config,
N=20 "Minimize difference, new best config
2002/01/10 N=18,19 "Minimize maximum", new best configs
2002/01/08 N=14,20 "Minimize maximum", new best configs
2001/12/21 N=11 "Minimize maximum", new best config with no symmetry
symmetric config moved to ILLOKTAB
N=10 "Minimize maximum" local opt with cyclic symmetry order 3
2001/12/19 N=15,16 "Minimize difference", new best configs
2001/12/14 N=12 "Maximize minimum", new best config with dihedral symmetry O8
N=12 "Minimize difference" new best config with dihedral symmetry O8
Both configs produce a highly symmetric illuminance distribution
on the target sphere, with exceptionally good uniformity.
N=8 "Minimize difference", accuracy of result improved
2001/12/12 N=19 "Maximize minimum", new best config,
reference to article by Dave Kottwitz added
2001/12/11 N=9 "Minimize difference", new best config with cyclic symmetry O6,
N=8 "Minimize difference", accuracy of result improved
2001/11/30 Link to visualization of results for the problem "maximize
minimal distance between N points on the surface of a sphere"
(hostile neighbor problem)
added: http://www.enginemonitoring.org/sphere/index.htm
2001/11/24 Tables of symmetry groups of best known results of point
placement problems on sphere on NJA Sloane's web page added,
sorted list of edge lengths of convex hull of hostile neighbor
problem, result file for N=14
2001/11/23 Link to SPHERES FAQ in Dave Rusin's known-math pages
2001/11/17 N=18 "Maximize minimum", new best config,
N=10 "Minimize difference", local opt with mirror symmetry O2
Link to Mersenne Twister PRNG added in page
2001/11/16 N=11 "Minimize maximum", new best config with cyclic symmetry O2
2001/11/10 N=7 "Maximize minimum", local opt with cyclic symmetry order 3,
local opt with dihedral symmetry order 4 (nice)
N=16 "Minimize difference", new best config
2001/11/08 N=11,20 "Maximize minimum", new best config
2001/11/07 N=12 "Maximize minimum", new best config with dihedral symm. O6
N=14 "Maximize minimum", new best config
2001/11/03 N=8,9 "Minimize difference", accuracy of result improved
N=12, "Maximize minimium", new best config
N=12,14,15 "Minimize maximum", accuracy of result improved
N=6,7 "Maximize minimum", accuracy of result improved
N=9, "Minimize difference", local opt with cyclic symmetry O3
2001/11/01 N=10, "Maximize minimum", new best config with cyclic symm. O2
N=17, "Maximize minimum", new best config
2001/10/31 N=15, "Maximize minimium", accuracy of result improved,
N=18,19 "Maximize minimum", new best config
2001/10/29 N=15, "Minimize difference", accuracy of result improved
2001/10/27 N=12, "Minimize difference", new best config with cyclic symm. O3
2001/10/20 N=10, "Minimize maximum", accuracy of result improved
N=13, "Minimize maximum", new best config
N=17, "Minimize difference", new best config
N=13, "Maximize minimum", local opt with mirror symmetry
2001/10/19 N=20, "Minimize difference", new best config
2001/10/14 N=8, "Maximize minimum", new best config with mirror symmetry
N=12, "Maximize minimum", new best config
2001/10/13 N=14, "Minimize difference", new best config
2001/10/06 N=19, "Minimize maximum", new best config
2001/10/05 N=17, 18, "Minimize maximum", new best configs
2001/10/04 N=16, "Minimize maximum", significantly improved best config
2001/09/22 N=19, "Minimize maximum" new best config,
N=18, "Minimize difference" new best config, lowest difference
for all N<=20
N=11, "Maximize minimum" new best config
The increased number of improved arrangements found in the
last weeks is the result of changes in the optimization
process, that is now started with a very coarse mesh of test
points on the target sphere with an angular resolution of
only 4 degrees (approximately 2600 points) instead of 1
degree (approx. 41000 points). The search is now started with
100000 random configs. The Mersenne twister PRNG is used
within the Knuth method for generating random directions on
the sphere. The PRNG is seeded using seconds since start of
the year. The subsequent optimization runs with FMINSI use
angular resolutions of 4,2,1,0.5,0.3333..,0.25 degrees, that
are restarted until no improvement in the objective function
is found. When convergence has been achieved for 0.5 degrees
angular resolution, the source arrangement is rotated into a
standard orientation and the optimization with 0.5 deg
resolution is repeated and then continued until final
convergence with the finest test mesh is reached. For one
given N the initial phase (random configs with subsequent
multiple restarts of FMINSI until convergence) is run for 24
hours. If the value of the objective function on the coarse
mesh is better than the best result found so far on the
finest test mesh, the refinement process is started. This
refinement typically takes another 12 hours of CPU time.
Example: N=14, 100000 random configs plus 136 restarts of
FMINSI (223000 evaluations of the objective function) take 6
minutes of CPU time on an Athlon 800 MHz. In 24 hours this
can be repeated approximately 240 times, thus producing
77*10^6 evaluations of the objective function.
2001/09/19 N=20, "Minimize maximum" new best config
2001/09/18 N=20, "Maximize minimum" new best config
2001/09/16 N=14, "Maximize minimum" new best config, discussion of character
of best solutions added in bestlis.html, contour plots replaced
for DMMM10L1 and DMMM11L1
2001/09/15 Illumination contours replaced for some of the suboptimal symmetric
arrangements; increased number of contour lines
(N=12, Dodekahedron). It is now visible, that the dodekahedron
arrangement produces 60 illuminance minima + 20 maxima
First example picture replaced.
New diagram showing deviation of best solutions from uniformity:
bestlis.html
2001/09/09 Major update: Best configurations for N=17..20 added, illuminance
contours recomputed for N=12..16 with increased number of
contour lines (now minimum=5 contours), pictures look much better;
new best config N=16, "Maximize minimum"
2001/09/01 N=12, "Minimize difference" new best config, old config removed,
N=9, "Minimize Maximum" MAMI09L1.TXT, local opt with cyclic
symmetry: Illustrates typical behavior of "Minimize Maximum"
solutions with many maxima and a few (2) very dark spots, that do
not influence the objective function.
WebCounter included, preset with 472 hits counted by ISP Strato.
2001/08/20 Preview page for illumination results with cosine beam profiles
introduced, most CBDMxx results updated to fulfill the antipodal
sum=const condition (without degrading the objective function value)
2001/08/03 N=12, "Maximize Minimum" MIMA12L1.TXT local opt with cyclic
symmetry order 2. The illuminance distribution has mirror
symmetry w.r.t. the equator plane.
CBDM21 added: (equator 13-ring plus two 4-rings)
2001/08/01 N=12, "Minimize Difference" DMMM12L1.TXT local opt with mirror
symmetry order 2
2001/07/26 N=11, "Maximize Minimum" MIMA11L1.TXT, local opt with
dihedral symmetry order 4
2001/07/22 CBDM15, CBDM17 replaced by equator ring
plus 2 3-rings configurations, which do not fulfil the
antipodal conjecture, but seem to be the best one can do for
odd N. CBDM19 added (equator 11-ring plus two 4-rings).
CBDM13 replaced by a very surprising config: 2 nearly conincident
beams at both poles plus 3 3-rings, one at equator, 2 offest
and rotated by +-40Deg. This config beats the 7+2*3 config.
Improved solutions found for N=15: Maximize minimum,
minimize difference, previous MIMA15, DMMM15 moved to ILLOKTAB
2001/07/16 CBDM17.TXT and CBDM18.TXT added, CBDM18.TXT is a perfect solution
with a 10-ring at the equator, 2 pole sources and 2 3-rings
2001/07/05 N=8, "Minimize difference" DMMM08L2.TXT, local opt with
dihedral symmetry order 4, 2 points near N-pole + not far from
regular polygon with 6 edges below equator
2001/06/22 N=12, mirror symmetric config for "minimize maximum"
2001/06/17 CBDM11.TXT updated, also .pdf, axis of maximum symmetry now
= z axis, example gif included
2001/06/16 CBDM15.TXT added, conjecture for symmetry property of best odd N
solutions for cosine intensity beams.
2001/06/15 Inclusion of comments (many thanks!) by Andy J. Schmitt
into chapter "A related problem".
CBDM13.TXT replaced (now non-symmetric) and CBDM16.TXT added
CBDM16.TXT is a perfect solution consisting of a 10-ring at the
equator and a pair of 3-rings
2001/06/11 Result plot for regular dodekahedron.
2001/06/10 Result plots for N=7, 9, 11 for problem "Minimize difference"
with cosine beam intensity profile. Files CBDMXX.TXT,
XX=07,09,10,11,12,13,14 included in illumtab.zip.
2001/06/07 N=13, "Minimize difference", new best config, previous one
moved to ILLOKTAB
2001/06/01 N=11, "Minimize difference", mirror symmetry + local sym
2001/05/31 N=9, "Minimize difference", mirror symmetry + local sym
(see equator view 90Deg)
2001/05/29 2 Configs for N=10, "Minimize difference", cyclic sym O3
"Minimize Maximum", mirror sym O2
2001/05/27 Config with cyclic symmetry (Order 6) for N=11 (not far from
dihedral O12) "Maximize minimum"
2001/05/25 Cube pictures added
Symmetric config for N=8 with dihedral symmetry (Order 16)
for "Maximize mimimum"
Symmetric config for N=8 with dihedral symmetry (Order 8)
for "Minimize difference"
2001/05/24 New best config for "Minimize difference" N=14
previous config moved to LOKOPT
Regular octahedron pictures added
Symmetric config for N=8 (two rectangles)
Hugo Pfoertner (mailto:hugo@pfoertner.org)