Integer coordinate set of points that is a member of sphere surface

Question

I have a graphic application to develop which involve many spheres. I should determine then on run time.

Supposing that I have a sphere of radius r, how can I determine the sub set of the sphere surface points that are integer?

E.g., $r = 10$ I can have $(10,0,0), (8,6,0),$ etc.

(Obs.: I really think this is not a programming question, that's why I not posted In stack overflow. If I am wrong, please fell free to warn me that :)

Pedro

You could just check the integer "grid" the sphere is in by using the formula of the sphere to see which points are on the surface of the sphere. — Listing, Dec 06 '11 at 09:59
Ok, thats a alternative. But it results in a solution O(n³), which I am trying to avoid... — Pedro Dusso, Dec 06 '11 at 10:17
@PedroDusso by solving it for the third coordinate and only looking at points who have certain distance to the center you can easily make it quite faster than $\mathcal{O}(n^2)$. — Listing, Dec 06 '11 at 10:33
Obviously, there are no solutions if $r^2$ is not an integer. Are you assuming that $r$ is an integer? — Gerry Myerson, Dec 06 '11 at 11:54

score 1 · Answer 1 · edited Dec 08 '11 at 19:01

1

You could use Euclid's formula: For $m > n, a = m^2 - n^2, b = 2mn, c = m^2 + n^2$ are solutions to $a^2 + b^2 = c^2$.

edited Dec 08 '11 at 19:01

Quixotic

22,431

answered Dec 08 '11 at 17:57

Sam

11

1

The question's about spheres, not circles. – J. M. ain't a mathematician Dec 08 '11 at 18:11
And then there is the Gauss-Legendre theorem that an integer is the sum of three integer squares if and only if it is not of the form $4^m(8n+7)$. – marty cohen Apr 07 '12 at 04:34

Integer coordinate set of points that is a member of sphere surface

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