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Re: [EMHL] A line and a conic

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  • Chris Van Tienhoven
    Dear friends, Thanks Barry! This brings me to a new problem. I never got into this. Given an equation f(x,y,z). How to convert it to a point in time (t):
    Message 1 of 12 , Feb 24, 2013
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      Dear friends,

      Thanks Barry!
      This brings me to a new problem.
      I never got into this.
      Given an equation f(x,y,z).
      How to convert it to a point in time (t): (x(t) : y(t) : z(t))?

      To make it visual and more appealing. When we have a conical orbit of a planet or comet.
      How can we describe its location as a function of time in barycentric coordinates ?

      Any ideas?

      Chris van Tienhoven


      --- In Hyacinthos@yahoogroups.com, Barry Wolk <wolkbarry@...> wrote:
      >
      > > Dear friends
      > > if we know the line
      > > Px + Qy + Rz = 0
      > > and the conic
      > > fxx + gyy + hzz + 2pyz + 2qzx + 2rxy = 0
      > > is there a formula giving the barycentric
      > > coordinates of their intersection points?
      > > Best regards
      > > Nikos Dergiades
      >
      > There were several replies, including asking for cyclically symmetric coordinates for those intersection points.
      >  
      > The points on the line Px+Qy+Rz=0 can be parametrized as
      > (x, y, z)=((Q-R)(QR+t), (R-P)(RP+t), (P-Q)(PQ+t))
      >  
      > Substitute these formulas for x,y,z into the equation of the conic, to get a cyclically symmetric quadratic equation for the parameter t. Its two roots will give the intersection points in the requested form.
      > --
      > Barry Wolk
      >
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