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Polar formula MP^T

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  • Sung Hyun Lim
    Dear all, Can anyone tell me why polar of point P with respect to conic M is MP^T? Regards, Sung Hyun [Non-text portions of this message have been removed]
    Message 1 of 2 , Mar 2, 2011
      Dear all,

      Can anyone tell me why polar of point P with respect to conic M is MP^T?

      Regards,
      Sung Hyun


      [Non-text portions of this message have been removed]
    • Nikolaos Dergiades
      Dear Sung Hyun, Let P = (p q r) as row matrix in barycentrics and a line passing through P meets the conic at the points P1 = (p1 q1 r1) and P2 = (p2 q2 r2).
      Message 2 of 2 , Mar 2, 2011
        Dear Sung Hyun,
        Let P = (p q r) as row matrix in barycentrics
        and a line passing through P meets the conic at the
        points P1 = (p1 q1 r1) and P2 = (p2 q2 r2).
        The tangent at P1 is the line (x y z)MP1^t = 0
        The tangent at P2 is the line (x y z)MP2^t = 0
        and let Q = (p' q' r') be their intersection point.
        We must have
        (p' q' r')MP1^t = 0 and (p' q' r')MP2^t = 0.
        The line (p' q' r')MR^t = 0 where R = (x y z)
        is satisfied by P1, P2 and hence must be satisfied
        by P also.This means that (p' q' r')MP^t = 0
        and since the locus of Q is the polar of P
        this locus must be the line (x y z)MP^t = 0
        which means that the coefficients of the polar are MP^t.
        Best regards
        Nikos Dergiades

        > Dear all,
        >
        > Can anyone tell me why polar of point P with respect to
        > conic M is MP^T?
        >
        > Regards,
        > Sung Hyun
        >
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