Duality of Plane Curves
نویسنده
چکیده
The dual of an algebraic curve C in RP defined by the polynomial equation f(x, y, z) = 0 is the locus of points ( ∂f ∂x (a, b, c) : ∂f ∂y (a, b, c) : ∂f ∂z (a, b, c) ) where (a : b : c) ∈ C. The dual can alternatively be defined geometrically as the image under reciprocation of the envelope of tangent lines to the curve. It is known that the dual of an algebraic curve is also an algebraic curve, and that taking the dual twice results in the original curve. We use reciprocation to obtain an elementary geometric proof of the latter fact, and we use our methods to gain intuition about the relationship between a curve and its dual.
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