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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]] in a dual projective plane. The condition that the line is tangent to the curve can be expressed in the form {{math|''F''(''X'', ''Y'', ''Z'') {{=}} 0}} which is the tangential equation of the curve.
:<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}}.
:<math>\begin{align}X&= 2\lambda ap, \\ Y&=2\lambda bq, \\ Z&=2\lambda cr,\end{align} \qquad Xp+Yq+Zr=0.</math>▼
▲
Xp+Yq+Zr=0.
\end{array}</math>
The first three equations are easily solved for {{mvar|p}}, {{mvar|q}}, {{mvar|r}}, and substituting in the last equation produces
:<math>\frac{X^2}{2\lambda a}+\frac{Y^2}{2\lambda b}+\frac{Z^2}{2\lambda c}=0.</math>
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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)
▲:<math>\begin{align}
= (-y',x',xy'-yx'),</math></blockquote>which in affine coordinates <math>(X,Y,1)</math> is:
Y=\
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===
==Degree==▼
From the projective description, one may compute the dual of the dual:<blockquote><math>(x(x'y''-y'x''),\, y(x'y''-y'x''),\, x'y''-y'x'') = (x, \, y,\, 1)(x'y''-y'x''),</math></blockquote>which is projectively equivalent to the original curve <math>(x(t),y(t),1)</math>.
If {{mvar|X}} 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 {{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)}}.▼
==Properties of dual curve==▼
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.▼
Properties of the original curve correspond to dual properties on the dual curve. In the Introduction image
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.▼
If {{mvar|X}} is smooth, i.e. there are no [[Singular point of a curve|singular points]] then the dual of {{mvar|X}} has the maximum degree {{math|''d''(''d'' − 1)}}. If {{mvar|X}} is a conic this implies its dual is also a conic. This can also be seen geometrically: the map from a conic to its dual is [[one-to-one correspondence|one-to-one]] (since no line is tangent to two points of a conic, as that requires degree 4), and tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires degree 3).▼
Further, both curves above have a reflectional symmetry
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 degree of the dual given in terms of the ''d'' and the number and types of singular points of {{mvar|X}} is one of the [[Plücker formula]]s.▼
▲==Degree==
==Polar reciprocal==▼
▲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 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 {{mvar|Q}} as the locus of the poles of the tangent lines of the curve {{mvar|C}}.<ref>{{cite book | first=J. | last=Edwards | title=Differential Calculus▼
| publisher= MacMillan | location=London | pages=176| year=1892▼
|url=https://books.google.com/books?id=unltAAAAMAAJ&pg=PA176#v=onepage&q&f=false}}</ref> The conic {{mvar|Q}} is nearly always taken to be a circle and this case the polar reciprocal is the [[Inverse curve|inverse]] of the [[Pedal curve|pedal]] of {{mvar|C}}.▼
▲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.
▲==Properties of dual curve==
▲Properties of the original curve correspond to dual properties on the dual curve. In the image at right, the red curve has three singularities – a node in the center, and two cusps at the lower right and lower left. The black curve has no singularities, but has four distinguished points: the two top-most points have the same tangent line (a horizontal line), while there are two inflection points on the upper curve. 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 inflection points correspond to the cusps, corresponding to the tangent lines first going one way, then the other (slope increasing, then decreasing).
▲If {{mvar|X}} is smooth
▲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.
▲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 formula]]s give the degree of the dual
▲Further, both curves have a reflectional symmetry, corresponding to the fact that symmetries of a projective space correspond to symmetries of the dual space, and that duality of curves is preserved by this, 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.
▲==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 {{mvar|Q}} as the locus of the poles of the tangent lines of the curve {{mvar|C}}.<ref>{{cite book | first=J. | last=Edwards | title=Differential Calculus
▲| publisher= MacMillan | location=London | pages=[https://archive.org/details/in.ernet.dli.2015.109607/page/n194 176]| year=1892
▲|url=https://
==Generalizations==
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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>
===Dual polygon===
{{Main article|Dual polygon|Convex conjugate}}
The dual curve construction works even if the curve is [[piecewise linear curve|piecewise linear]]
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 [[Dual cone and polar cone|polyhedral dual]], and any convex set ''X'' with boundary hypersurface ''H'' has a [[convex conjugate]] ''X*'' whose boundary is the dual variety ''H*''.
==See also==
*[[Hough transform]]
*[[Gauss map]]
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