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 Plücker formulas give the degree of the dual in terms of d and the number and types of singular points of . Polar reciprocal The dual can be visualized as a locus in the plane in the form of the polar reciprocal. This is defined with reference to a fixed conic as the locus of the poles of the tangent lines of the curve . The conic is nearly always taken to be a circle, so the polar reciprocal is the inverse of the pedal of . Generalizations Higher dimensions Similarly, generalizing to higher dimensions, given a hypersurface, the tangent space at each point gives a family of hyperplanes, and thus defines a dual hypersurface in the dual space. For any closed subvariety in a projective space, the set of all hyperplanes tangent to some point of is a closed subvariety of the dual of the projective space, called the dual variety of . Examples If is a hypersurface defined by a homogeneous polynomial , then the dual variety of is the image of by the gradient map which lands in the dual projective space. The dual variety of a point is the hyperplane Dual polygon The dual curve construction works even if the curve is piecewise linear or piecewise differentiable, but the resulting map is degenerate (if there are linear components) or ill-defined (if there are singular points). In the case of a polygon, all points on each edge share the same tangent line, and thus map to the same vertex of the dual, while the tangent line of a vertex is ill-defined, and can be interpreted as all the lines passing through it with angle between the two edges. This accords both with projective duality (lines map to points, and points to lines), and with the limit of smooth curves with no linear component: as a curve flattens to an edge, its tangent lines map to closer and closer points; as a curve sharpens to a vertex, its tangent lines spread further apart. More generally, any convex polyhedron or cone has a polyhedral dual, and any convex set X with boundary hypersurface H has a convex conjugate X*