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{{short description|Curve in the dual projective plane made from all lines tangent to a given curve}}
[[Image:Dual curve.svg|thumb|right|300px|Curves, dual to each other; see below for [[#Properties of dual curve|properties]].]]
In [[projective geometry]], a '''dual curve''' of a given [[plane curve]] {{mvar|C}} is a curve in the [[Duality (projective geometry)|dual projective plane]] consisting of the set of lines [[tangent]] to {{mvar|C}}. There is a [[Map (mathematics)|map]] from a curve to its dual, sending each point to the point dual to its tangent line. If {{mvar|C}} is [[algebraic curve|algebraic]] then so is its dual and the degree of the dual is known as the ''class'' of the original curve. The equation of the dual of {{mvar|C}}, given in [[line coordinates]], is known as the ''tangential equation'' of {{mvar|C}}. Duality is an [[Involution (mathematics)|involution]]: the dual of the dual of {{mvar|C}} is the original curve {{mvar|C}}.
The construction of the dual curve is the geometrical underpinning for the [[Legendre transformation]] in the context of [[Hamiltonian mechanics]].<ref>See {{harv|Arnold|1988}}</ref>
==Equations==
Let {{math|''f''(''x'', ''y'', ''z'') {{=}} 0}} be the equation of a curve in [[homogeneous coordinates]] on the [[projective plane]]. Let {{math|''Xx'' + ''Yy'' + ''Zz'' {{=}} 0}} be the equation of a line, with {{math|(''X'', ''Y'', ''Z'')}} being designated its [[line coordinates]]
At a point {{math|(''p'', ''q'', ''r'')}} on the curve, the tangent is given by
:<math>x\frac{\partial f}{\partial x}(p, q, r)+y\frac{\partial f}{\partial y}(p, q, r)+z\frac{\partial f}{\partial z}(p, q, r)=0.</math>
So {{math|''Xx'' + ''Yy'' + ''Zz'' {{=}} 0}} is a tangent to the curve if
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[[File:Dual.webm|thumb|thumbtime=0.5|458px|On the left: the ellipse {{math|({{sfrac|''x''|2}}){{su|p=2}} + ({{sfrac|''y''|3}}){{su|p=2}} {{=}} 1}} with tangent lines {{math|''xX'' + ''yY'' {{=}} 1}} for any {{mvar|X}}, {{mvar|Y}}, such that {{math|(2''X'')<sup>2</sup> + (3''Y'')<sup>2</sup> {{=}} 1}}.<br>On the right: the dual ellipse {{math|(2''X'')<sup>2</sup> + (3''Y'')<sup>2</sup> {{=}} 1}}. Each tangent to the first ellipse corresponds to a point on the second one (marked with the same color).]]
===Conic===
For example, let {{mvar|C}} be the [[Conic section|conic]] {{math|''ax''<sup>2</sup> + ''by''<sup>2</sup> + ''cz''<sup>2</sup> {{=}} 0}}. The dual is found by eliminating {{mvar|p}}, {{mvar|q}}, {{mvar|r}}, and {{mvar|λ}} from the equations
:<math>\begin{array}{c}
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:<math>\frac{X^2}{a}+\frac{Y^2}{b}+\frac{Z^2}{c}=0.</math>
===General algebraic curve===
Consider a [[parametric equation|parametrically defined curve]] <math>(x,y) = (x(t),y(t)),</math> in projective coordinates <math>(x,y,z)=(x(t),y(t),1)</math>. Its projective tangent line is a linear plane spanned by the point of tangency and the tangent vector, with linear equation coefficients given by the [[cross product]]:<blockquote><math>(X,Y,Z) = (x,y,1)\times (x',y',0)
= (-y',x',
:<math>X=\frac{-y'}{xy'-yx'} , \quad
Y=\frac{x'}{
The dual of an [[inflection point]] will give a [[Cusp (singularity)|cusp]] and two points sharing the same tangent line will give a self
===Dual of the dual===
From the projective description, one may compute the dual of the dual:<blockquote><math>(x(
==Properties of dual curve==
Properties of the original curve correspond to dual properties on the dual curve. In the Introduction image, the red curve has three singularities – a [[Crunode|node]] in the center, and two [[Cusp (singularity)|cusps]] at the lower right and lower left. The black curve has no singularities
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
==Degree==
If {{mvar|X}} is a plane algebraic curve, then the degree of the dual is the number of points in the 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 {{mvar|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 {{mvar|d}} then the degree of the polar is {{math|''d'' − 1}} and so the number of tangents that can be drawn through the given point is at most {{math|''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.
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::<math>x=(x_0,\ldots, x_n)\mapsto \left(\frac{\partial F}{\partial x_0}(x),\ldots, \frac{\partial F}{\partial x_n}(x)\right)</math>
:which lands in the dual projective space.
* The dual variety of a point {{math|(''a''<sub>0</sub> : ...
::<math>a_0x_0+\ldots +a_nx_n=0.</math>
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