curve has no singularities but has four distinguished points: the two top-most points correspond to the node (double point), as they both have the same tangent line, hence map to the same point in the dual curve, while the two inflection points correspond to the cusps, since the tangent lines first go one way then the other (slope increasing, then decreasing). By contrast, on a smooth, convex curve the angle of the tangent line changes monotonically, and the resulting dual curve is also smooth and convex. Further, both curves above have a reflectional symmetry: projective duality preserves symmetries a projective space, so dual curves have the same symmetry group. In this case both symmetries are realized as a left-right reflection; this is an artifact of how the space and the dual space have been identified – in general these are symmetries of different spaces. Degree If 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 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 then the degree of the polar is and so the number of tangents that can be drawn through the given point is at most . 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 is smooth (no singular points) then the dual of has maximum degree . This implies the dual of a conic is also a conic. Geometrically, the map from a conic to its dual is one-to-one (since no line is tangent to two points of a conic, as that requires degree 4), and the tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require