Polar formula MP^T
- Dear all,
Can anyone tell me why polar of point P with respect to conic M is MP^T?
[Non-text portions of this message have been removed]
- 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.
> Dear all,
> Can anyone tell me why polar of point P with respect to
> conic M is MP^T?
> Sung Hyun