11442 - Linear Diophantine Tidbits
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11442 - Linear Diophantine Tidbits
Any hints for this one? It reduces to counting the number of lattice points in a triangle in 3-D space. How do you do that? Is there a generalisation of Pick's theorem which could be applied here?
Re: 11442 - Linear Diophantine Tidbits
Short answer is:
No, there is no known generalization of Pick's theorem for 3d space.
You probably know that the given triangle lies on a certain plane, so it lies in 2d space. All you need to do is to find a parametrization of the plane in such a way that the use of Pick's theorem on the parametrization gives the required answer.
In other words, find linear function F such that (x,y,z)=F(s,t) and (x0,y0,z0) is a solution to the linear diophantine equation iff (s0,t0) is in Z X Z where (x0,y0,z0)=F(s0,t0)
The fact that F is linear is important since it guarantees the transformation preserves straight lines.
No, there is no known generalization of Pick's theorem for 3d space.
You probably know that the given triangle lies on a certain plane, so it lies in 2d space. All you need to do is to find a parametrization of the plane in such a way that the use of Pick's theorem on the parametrization gives the required answer.
In other words, find linear function F such that (x,y,z)=F(s,t) and (x0,y0,z0) is a solution to the linear diophantine equation iff (s0,t0) is in Z X Z where (x0,y0,z0)=F(s0,t0)
The fact that F is linear is important since it guarantees the transformation preserves straight lines.