algorithm - Dispersing n points uniformly on a sphere -

algorithm - Dispersing n points uniformly on a sphere -

i trying disperse n points on sphere such each point has "same" area "around" it. basically, i'm trying integrate function on sphere evaluating @ n points , assuming each area element same (and equal 4pi r^2/n).

my question related this one, can't seem agree code presented in "accepted" reply works desired (see attached photo, generated choosing r = 1000, nx = ny = 40). clearly, points much more concentrated @ poles , un-concentrated along equator.

any suggestions?

edit: reference, did find some software generates mesh such each point has equal "area" around (scroll downwards see uniform area distribution on sphere), rather implement code went less-time consuming approach: iterated on azimuthal , polar angles ([0,2pi] , [0,pi]) , computed ''infinitesimal'' area of each patch (da = r^2 sin theta dtheta dphi). need integration on sphere, hoping uniform-area distribution wouldn't hard implement.

background information:

there 4 pi steradians in sphere, that's total 'degrees' in sphere, utilize term in relative sense because steradians different regular radians in circle, one, 3 dimensional , hence solid. consider them water ice cream shaped angles in sphere.

http://en.wikipedia.org/wiki/steradian provides great illustration of them.

they have direct relationship radius, radians in circle. 1 steradian = 1 unit of radius squared.

so, first find out how many items need plotted on sphere. allow number n. sr = steradians (unit of measure) = r^2 (radius squared)

4 pi / n sr = x

x how many steradians allocated each point.

let's 4 points.

4 pi / 4 sr = x

pi sr = x each point allocated space of pi sr.

now consider this... since plotting points, consider each point placed in middle of allocated space... say, in middle of conal shaped area sr is. need consider moment, possible fill area circles? seriously, think this... it's not it? solid circles leave room in between in spots. think soccer ball moment. constructed shapes can come provide distribution. point of thought realize dots cannot distance apart--like how circle has radius. yet, center of soccer ball squares comes close , uniform.

what if you, seek , write algorithm identify efficient 'shape' set each of these 'chunks' of allocated spherical space in... soccer ball. otherwise, think might best reply you're going get... 4 pi / n sr = x... , there's no way each point plotted same distance each other, (except in configurations, i.e. possible special number of points), there may algorithm out there find special cases.

i editing reply elaborate on special cases, little info here think. special cases points equidistance apart may form vertices of platonic solids. there 5 basic platonic solid shapes, others made these.

read page more info , proof of https://www.uwgb.edu/dutchs/symmetry/platonic.htm

now can't take credit, did quick research , found similar post http://math.stackexchange.com/questions/279544/return-an-array-of-evenly-distributed-points-on-a-sphere-give-radius-and-origin

using euler's polyhedron formula http://plus.maths.org/content/eulers-polyhedron-formula

and fact 3 basic shapes exist on polyhedrons, 'triangles, squares, , hexagons' can create algorithm round number of points want plot, nearest polyhedron shape , plot each 1 evenly.

oh, , take @ great article, explains steradians , 3-dimensional 'degrees' much improve i. http://mathforum.org/library/drmath/view/55358.html

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