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Hyperbolic Paraboloid and Isometry

  1. Nov 7, 2012 #1
    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?
     
  2. jcsd
  3. Nov 8, 2012 #2

    tiny-tim

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    Hi BrainHurts! :smile:

    (try using the X2 button just above the Reply box :wink:)
    yes, x2 + y2 = 2* (x - y)/√2 * (x + y)/√2 = 2(Rx)(Ry) :wink:
    not following you :confused:
    yes
     
  4. Nov 8, 2012 #3
    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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