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March 15th, 2011, 09:22 AM   #1
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Geometry help needed :)

Hey guys!

Can someone tell me how does this surface looks like -> S: abs(x-y+z)+abs(y-z+x) +abs(z-x+y)=1 ?? Maybe some ingenious substitution needed?

Thanks in advance.
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March 15th, 2011, 09:53 AM   #2
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Re: Geometry help needed :)

Looking at the cross-section z=0, you get 2|x-y|+|x+y| = 1. The substitution u=x-y, v=x+y (which involves a tilting of the axes by 45°) gives |v| = 1 - 2|u|, and this is a rhombus. By symmetry, the cross-sections through x=0 and y=0 are also rhombi, which suggests that the shape is maybe a rhombohedron, a kind of parallelepiped.
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March 15th, 2011, 10:54 AM   #3
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Re: Geometry help needed :)

Thanks a lot indeed!
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March 15th, 2011, 09:19 PM   #4
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Re: Geometry help needed :)

On second thoughts, this is a linear transformation of |x|+|y|+|z|=1, which suggests that it is an octahedron. Oops!
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