# Hyperbolic Paraboloid and Isometry

by BrainHurts
Tags: hyperbolic, isometry, paraboloid
 P: 84 If the hyperbolic paraboloid z=(x/a)^2 - (y/b)^2 is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface z=(1/2)(x^2 + y^2) ((1/a^2)-((1/b^2)) + xy((1/a^2)-((1/b^2)) and if a= b then this simplifies to z=2/(a^2) (xy) suppose z= x^2 - y^2 does this mean that z=2xy ? if so can someone tell me how to put z=x^2 - y^2 into it's quadric form? Also the rotation by the angle of π/4 is that just the typical rotation matrix Rz?
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P: 26,148
Hi BrainHurts!

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 Quote by BrainHurts suppose z= x^2 - y^2 does this mean that z=2xy ?
yes, x2 + y2 = 2* (x - y)/√2 * (x + y)/√2 = 2(Rx)(Ry)
 if so can someone tell me how to put z=x^2 - y^2 into it's quadric form?
not following you
 Also the rotation by the angle of π/4 is that just the typical rotation matrix Rz?
yes
 P: 84 i actually got it, a quadric is an equation that can be written in the form: v'Av + bv + c = 0 so z=x^2 - y^2 by the above equation: let v' is the transpose of v and v = [x y z] A= [ 1 0 0; 0 -1 0; 0 0 0] and b = [0 0 -1] so rotating v by the rotation matrix where the angle is pi/4 about the z axis gave me the results I was looking for. Thank you very much sorry i don't see the button over the reply box but i'll try to do it next time!

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