Hyperbolic Paraboloid and Isometry

[BrainHurts]

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?

 

[tiny-tim]

Hi BrainHurts!

(try using the X² button just above the Reply box )

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

 

[BrainHurts]

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!