Dual Curve - Degree

Degree

If X is a plane algebraic curve then the degree of the dual is the number of points intersection with a line in the dual plane. Since a line in the dual plane corresponds to a point in the plane, the degree of the dual is the number of tangents to the X that can be drawn through a given point. The points where these tangents touch the curve are the points of intersection between the curve and the polar curve with respect to the given point. If the degree of the curve is d then the degree of the polar is d−1 and so the number of tangents that can be drawn through the given point is at most d(d−1).

The dual of a line (a curve of degree 1) is an exception to this and is taken to be a point in the dual space (namely the original line). The dual of a single point is taken to be the collection of lines though the point; this forms a line in the dual space which corresponds to the original point.

If X is smooth, i.e. there are no singular points then the dual of X has the maximum degree d(d − 1). If X is a conic this implies its dual is also a conic. This can also be seen geometrically: the map from a conic to its dual is 1-to-1 (since no line is tangent to two points of a conic, as that requires degree 4), and tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires degree 3).

For curves with singular points, these points will also lie on the intersection of the curve and its polar and this reduces the number of possible tangent lines. The degree of the dual given in terms of the d and the number and types of singular points of X is one of the Plücker formulas.