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$f(x,y)=x^2-y^2$ is your friendly neighbourhood hyperbolic paraboloid. $f(x, y) = |x| - |y|$ naturally has similar appearance. Do shapes of the latter form have a name?

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Looks like a non-smooth saddle. Wolfram Alpha plot – user13838 Sep 13 '11 at 1:36
You could call $z=|x|+|y|$ a "circular cone" with circle here defined under the 1-norm $\|\cdot\|_1$ instead of the usual 2-norm $\|\cdot\|_2$, but I can't think of anything standard for $z=|x|-|y|$. I submit the term "four-plane saddle" for any shape given by $$z=ax+b|x|+cy+d|y|,$$ or any invertible affine transformation thereof, with $$\mathrm{sgn}(a+b)=\mathrm{sgn}(-a+b)=\pm1,$$ $$\mathrm{sgn}(c+d)=\mathrm{sgn}(-c+d)=\mp1.$$ – anon Sep 13 '11 at 1:41
up vote 4 down vote accepted

What about hyperbolic superparaboloid analogous to quadric - superquadric?

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I'll go with it. Mine's just $f(x, y) = |x|^m - |y|^n$ for $m = n = 1$. – Jon Purdy Sep 18 '11 at 17:20

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