Polar Curve
In algebraic geometry, the first polar, or simply polar of an algebraic plane curve ''C'' of degree ''n'' with respect to a point ''Q'' is an algebraic curve of degree ''n''−1 which contains every point of ''C'' whose tangent line passes through ''Q''. It is used to investigate the relationship between the curve and its dual, for example in the derivation of the Plücker formulas. Definition Let ''C'' be defined in homogeneous coordinates by ''f''(''x, y, z'') = 0 where ''f'' is a homogeneous polynomial of degree ''n'', and let the homogeneous coordinates of ''Q'' be (''a'', ''b'', ''c''). Define the operator :\Delta_Q = a+b+c. Then Δ''Q''''f'' is a homogeneous polynomial of degree ''n''−1 and Δ''Q''''f''(''x, y, z'') = 0 defines a curve of degree ''n''−1 called the ''first polar'' of ''C'' with respect of ''Q''. If ''P''=(''p'', ''q'', ''r'') is a non-singular point on the curve ''C'' then the equation of the t ...
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