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Need help on understanding cone geometry

  1. Sep 25, 2007 #1
    Hi all,

    1. The problem statement, all variables and given/known data

    Given a right circular cone with origin at the centre of the base, the positive z-axis pointing towards the apex, and the height is h and radius of base is r. What is the cartesian equation of the cone?

    2. Relevant equations



    3. The attempt at a solution

    The equation that I get is (h-z)^2 = (h/r)^2 (x^2+y^2). Can anyone confirm this?

    Assuming that my above equation is correct, how is it that the general equation of a cone is instead x^2 + y^2 = z^2? Where did the extra terms from the first equation go to?

    Also, what significance does it bring when the equation of a cone becomes ax^2 + by^2 = (h-cz)^2? If I compare it with the equation i obtained, I suppose that this should mean that the height of the cone is equal to its base radius? What about the constants a, b, c; what do they represent in the physical sense?

    Thanks much!
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 25, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    Your equation looks fine to me. x^2 + y^2 = z^2 is the equation a 45 degree cone with apex at the origin. Not very general. And for this one, ax^2 + by^2 = (h-cz)^2, if a!=b, that's a vertical cone (axis along z) with an elliptical cross-section.
     
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