Polar curve
![The elliptic curve E : 4Y2Z = X3 − XZ2 in blue, and its polar curve (E) : 4Y2 = 2.7X2 − 2XZ − 0.9Z2 for the point Q = (0.9, 0) in red. The black lines show the tangents to E at the intersection points of E and its first polar with respect to Q meeting at Q.](/Images/godic/202502/03/PolarEllipticCurvePlot.svg3114.png")
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.