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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>
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