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Math Help - [SOLVED] quadric surfaces

  1. #1
    AK713
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    [SOLVED] quadric surfaces

    How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?
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  2. #2
    MHF Contributor
    Joined
    Apr 2005
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    Quote Originally Posted by AK713
    How can you tell which axis the graph is oriented on? Obviously this is very easy for a hyperbolic paraboloid, but what about for the equation x^2+y^2-z^2=0?
    x^2 +y^2 -z^2 = 0 --------(i)
    Rearranging that,
    x^2 +y^2 = z^2 ----------(ii)
    The LHS of (ii) is a circle.
    The RHS is a square of a z-coordinate. It could be the square of the radius of the LHS at the z-coordinate.

    When z=0, the circle at the LHS is zero too, or no circle. It is (0,0,0), or the origin.

    When z = 1, or z = -1, the circle at the LHS has a radius of 1.
    When z = 2, or z = -2, the circle at the LHS has a radius of 2.
    When z = 3, or z = -3, the circle at the LHS has a radius of 3.
    Etc...

    The figure then is that of a cone whose apex is at the (0,0,0) and opening upwards, and that of its mirror-shape that opens downward.
    That is if the z-axis is the usual up-down or vertical axis in the x,y,z axes setup.
    Last edited by ticbol; January 25th 2006 at 11:01 AM.
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