How to tile a sphere with points at an even density?











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I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon coordinates. The fact that this is lat/lon coordinates is irrelevant, since this portion is done in cartesian coordinates.



I need to tile the earth (a sphere in this model) with m points, which will serve as the centre of a search radius. The location of each point (x y z) is subject to the constraint that it cannot be within m km of any other point (x' y' z'), so that the search radii do not overlap. Any thoughts? It seems like the kind of problem that has a perfect solution.






geometry tiling







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    up vote
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    favorite
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    I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon coordinates. The fact that this is lat/lon coordinates is irrelevant, since this portion is done in cartesian coordinates.



    I need to tile the earth (a sphere in this model) with m points, which will serve as the centre of a search radius. The location of each point (x y z) is subject to the constraint that it cannot be within m km of any other point (x' y' z'), so that the search radii do not overlap. Any thoughts? It seems like the kind of problem that has a perfect solution.






    geometry tiling







    share|cite|improve this question
























      up vote
      4
      down vote

      favorite
      4









      up vote
      4
      down vote

      favorite
      4






      4





      I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon coordinates. The fact that this is lat/lon coordinates is irrelevant, since this portion is done in cartesian coordinates.



      I need to tile the earth (a sphere in this model) with m points, which will serve as the centre of a search radius. The location of each point (x y z) is subject to the constraint that it cannot be within m km of any other point (x' y' z'), so that the search radii do not overlap. Any thoughts? It seems like the kind of problem that has a perfect solution.






      geometry tiling







      share|cite|improve this question













      I'm writing a bit of code to plot twitter usage across the globe. To do this, I'm searching for users within n km of a certain longitude/latitude (a circular area), at many different lat/lon coordinates. The fact that this is lat/lon coordinates is irrelevant, since this portion is done in cartesian coordinates.



      I need to tile the earth (a sphere in this model) with m points, which will serve as the centre of a search radius. The location of each point (x y z) is subject to the constraint that it cannot be within m km of any other point (x' y' z'), so that the search radii do not overlap. Any thoughts? It seems like the kind of problem that has a perfect solution.






      geometry tiling




      geometry tiling






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      asked Jul 2 '12 at 19:05









      RyanGrannell

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          2 Answers
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          The problem of distributing points on a sphere (as opposed to a circle) does not have a neat solution. There is a lot of information online, but you'll have to see for yourself what you can use. (There is also a common confusion between uniform random distribution and evenly-spaced distribution, which are very different things).




          • Spherical codes

          • Tammes problem

          • Thomson problem

          • nice pictures

          • some code






          share|cite|improve this answer





















          • thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
            – RyanGrannell
            Jul 23 '12 at 18:23




















          up vote
          2
          down vote













          Your question is a bit ambiguous. You seem to have the answer for uniform random distribution which are mostly approximations.



          But if you want exact, evenly-spaced distributions (without including the malevolent varient of n=1), there are only (and exactly only) 6 solutions for a 3-sphere.



          These correspond to the vertices of the 5 Platonic solids + the solution of 2 points on opposite sides of each other.






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            2 Answers
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            active

            oldest

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            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            8
            down vote













            The problem of distributing points on a sphere (as opposed to a circle) does not have a neat solution. There is a lot of information online, but you'll have to see for yourself what you can use. (There is also a common confusion between uniform random distribution and evenly-spaced distribution, which are very different things).




            • Spherical codes

            • Tammes problem

            • Thomson problem

            • nice pictures

            • some code






            share|cite|improve this answer





















            • thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
              – RyanGrannell
              Jul 23 '12 at 18:23

















            up vote
            8
            down vote













            The problem of distributing points on a sphere (as opposed to a circle) does not have a neat solution. There is a lot of information online, but you'll have to see for yourself what you can use. (There is also a common confusion between uniform random distribution and evenly-spaced distribution, which are very different things).




            • Spherical codes

            • Tammes problem

            • Thomson problem

            • nice pictures

            • some code






            share|cite|improve this answer





















            • thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
              – RyanGrannell
              Jul 23 '12 at 18:23















            up vote
            8
            down vote










            up vote
            8
            down vote









            The problem of distributing points on a sphere (as opposed to a circle) does not have a neat solution. There is a lot of information online, but you'll have to see for yourself what you can use. (There is also a common confusion between uniform random distribution and evenly-spaced distribution, which are very different things).




            • Spherical codes

            • Tammes problem

            • Thomson problem

            • nice pictures

            • some code






            share|cite|improve this answer












            The problem of distributing points on a sphere (as opposed to a circle) does not have a neat solution. There is a lot of information online, but you'll have to see for yourself what you can use. (There is also a common confusion between uniform random distribution and evenly-spaced distribution, which are very different things).




            • Spherical codes

            • Tammes problem

            • Thomson problem

            • nice pictures

            • some code







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Jul 2 '12 at 20:49







            user31373



















            • thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
              – RyanGrannell
              Jul 23 '12 at 18:23




















            • thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
              – RyanGrannell
              Jul 23 '12 at 18:23


















            thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
            – RyanGrannell
            Jul 23 '12 at 18:23






            thanks for the answer, the links were really useful. after playing around with a few techniques, the most efficient method seems to be the iterative method in the last link. Thanks again :)
            – RyanGrannell
            Jul 23 '12 at 18:23












            up vote
            2
            down vote













            Your question is a bit ambiguous. You seem to have the answer for uniform random distribution which are mostly approximations.



            But if you want exact, evenly-spaced distributions (without including the malevolent varient of n=1), there are only (and exactly only) 6 solutions for a 3-sphere.



            These correspond to the vertices of the 5 Platonic solids + the solution of 2 points on opposite sides of each other.






            share|cite|improve this answer



























              up vote
              2
              down vote













              Your question is a bit ambiguous. You seem to have the answer for uniform random distribution which are mostly approximations.



              But if you want exact, evenly-spaced distributions (without including the malevolent varient of n=1), there are only (and exactly only) 6 solutions for a 3-sphere.



              These correspond to the vertices of the 5 Platonic solids + the solution of 2 points on opposite sides of each other.






              share|cite|improve this answer

























                up vote
                2
                down vote










                up vote
                2
                down vote









                Your question is a bit ambiguous. You seem to have the answer for uniform random distribution which are mostly approximations.



                But if you want exact, evenly-spaced distributions (without including the malevolent varient of n=1), there are only (and exactly only) 6 solutions for a 3-sphere.



                These correspond to the vertices of the 5 Platonic solids + the solution of 2 points on opposite sides of each other.






                share|cite|improve this answer














                Your question is a bit ambiguous. You seem to have the answer for uniform random distribution which are mostly approximations.



                But if you want exact, evenly-spaced distributions (without including the malevolent varient of n=1), there are only (and exactly only) 6 solutions for a 3-sphere.



                These correspond to the vertices of the 5 Platonic solids + the solution of 2 points on opposite sides of each other.







                share|cite|improve this answer














                share|cite|improve this answer



                share|cite|improve this answer








                edited 2 days ago

























                answered May 12 '16 at 22:37









                theDoctor

                215213




                215213






























                     

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