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From: Joe H. <hea...@gm...> - 2012-04-17 17:12:40
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Can someone direct me to an algorithm for covering a sphere with patches of equal area for the purposes of illustrating the surface integral in Gauss's law? I've tried three so far and can't get good results. The problem is mostly with the poles and in the foreshortening of the patches near the poles. Is there a standard algorithm for this somewhere? Rectangular surfaces are easy. Joe Heafner Sent from one of my Macs |
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From: Kadir H. <kha...@ya...> - 2012-04-17 17:45:56
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You may try something like a Geodesic Dome. http://en.wikipedia.org/wiki/Geodesic_dome Kadir ________________________________ From: Joe Heafner <hea...@gm...> To: Visualpython-users Visualpython-users <vis...@li...> Sent: Tuesday, April 17, 2012 8:12 PM Subject: [Visualpython-users] covering a sphere with patches Can someone direct me to an algorithm for covering a sphere with patches of equal area for the purposes of illustrating the surface integral in Gauss's law? I've tried three so far and can't get good results. The problem is mostly with the poles and in the foreshortening of the patches near the poles. Is there a standard algorithm for this somewhere? Rectangular surfaces are easy. Joe Heafner Sent from one of my Macs ------------------------------------------------------------------------------ Better than sec? Nothing is better than sec when it comes to monitoring Big Data applications. Try Boundary one-second resolution app monitoring today. Free. http://p.sf.net/sfu/Boundary-dev2dev _______________________________________________ Visualpython-users mailing list Vis...@li... https://lists.sourceforge.net/lists/listinfo/visualpython-users |
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From: Jerzy K. <jer...@un...> - 2012-04-17 17:49:46
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Joe Heafner: > Can someone direct me to an algorithm for covering a sphere with patches of equal area for the purposes of illustrating the surface integral in Gauss's law? I've tried three so far and can't get good results. The problem is mostly with the poles and in the foreshortening of the patches near the poles. Is there a standard algorithm for this somewhere? Rectangular surfaces are easy. > > Joe Heafner I don't think there is any "standard" algorithm. Perhaps you should have written what DID you try... And, what are your restrictions concerning patches? There are of course some trivial ways. Map into your sphere through the central projection one of the following: 1. A tetrahedron 2. A cube (this is known as the Cobe Sky Cube...) 3. A dodecahedron. Voilà. May I suggest one more? Thanks... 4. An icosahedron... (http://space.mit.edu/home/tegmark/icosahedron.gif) OK, I know, I forgot the octahedron... Then, the subdivision of the faces you obtain into smaller ones should not be too difficult. Jerzy Karczmarczuk PS. People who want always have one nice mathematical formula for some spherical problems too often forget that it is not necessary to have two singularities at the poles, just one is possible with the stereographic projection. But the distortions are awful... ** |
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From: kirby u. <kir...@gm...> - 2012-04-17 18:04:37
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> OK, I know, I forgot the octahedron... > Then, the subdivision of the faces you obtain into smaller ones should not > be too difficult. > > Jerzy Karczmarczuk > The triangles of a geodesic sphere e.g. subdivided icosahedron, are not equal area though, not exactly. Maybe exactness is not critical. There are lots of reflections so don't have to compute a full sphere. 1/120th is a common patch to reflect around, based on rhombic triacontahedron breaking into 4 triangles. There's a book coming out any day now, 'Divided Spheres' by Edward Popko that should have a lot of good info. Kirby http://www.amazon.com/Divided-Spheres-Geodesics-Orderly-Subdivision/dp/1466504293 > PS. People who want always have one nice mathematical formula for some > spherical problems too often forget that it is not necessary to have two > singularities at the poles, just one is possible with the stereographic > projection. But the distortions are awful... > > > ------------------------------------------------------------------------------ > Better than sec? Nothing is better than sec when it comes to > monitoring Big Data applications. Try Boundary one-second > resolution app monitoring today. Free. > http://p.sf.net/sfu/Boundary-dev2dev > _______________________________________________ > Visualpython-users mailing list > Vis...@li... > https://lists.sourceforge.net/lists/listinfo/visualpython-users > |
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From: Kevin K. <ka...@so...> - 2012-04-17 20:51:47
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Try http://www.cse.cuhk.edu.hk/~ttwong/papers/udpoint/udpoints.html http://local.wasp.uwa.edu.au/~pbourke/geometry/spherepoints/ http://mathforum.org/kb/thread.jspa?forumID=13&threadID=1248382&messageID=3985254 http://www.mathworks.com/matlabcentral/newsreader/view_thread/21747 Many of these use random points with a uniform distribution on the surface of the sphere, but at least one of them was deterministic. Most of the surface integral calculations I've seen have used sample points, rather than patches. |
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From: Anton S. <br...@po...> - 2012-04-18 04:27:08
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On 2012-4-17 10:12, Joe Heafner wrote: > Can someone direct me to an algorithm for covering a sphere with > patches of equal area for the purposes of illustrating the surface > integral in Gauss's law? I've tried three so far and can't get > good results. The problem is mostly with the poles and in the > foreshortening of the patches near the poles. Is there a standard > algorithm for this somewhere? Rectangular surfaces are easy. The key thing to remember is that equally spaced parallel planes cut the sphere into bands of equal area. http://maths.anu.edu.au/~leopardi/Leopardi-Sphere-PhD-Thesis.pdf (5.8MB) gives a partition of the sphere into relatively compact "rectangles" of equal area. I think it even covers spheres of higher dimension. -- Anton Sherwood *\\* www.bendwavy.org *\\* www.zazzle.com/tamfang |
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From: Anton S. <br...@po...> - 2012-04-18 04:31:55
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On 2012-4-17 21:26, Anton Sherwood wrote: > http://maths.anu.edu.au/~leopardi/Leopardi-Sphere-PhD-Thesis.pdf (5.8MB) > gives a partition of the sphere into relatively compact "rectangles" of > equal area. ... And now looking at Leopardi's homepage I see that he's made a Matlab 'toolbox' for the algo: http://eqsp.sourceforge.net/ -- maybe looking at the source will be useful even without Matlab. -- Anton Sherwood *\\* www.bendwavy.org *\\* www.zazzle.com/tamfang |
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