Inprojective geometry, adual curve of a givenplane curveC is a curve in thedual projective plane consisting of the set of linestangent toC. There is amap from a curve to its dual, sending each point to the point dual to its tangent line. IfC isalgebraic then so is its dual and the degree of the dual is known as theclass of the original curve. The equation of the dual ofC, given inline coordinates, is known as thetangential equation ofC. Duality is aninvolution: the dual of the dual ofC is the original curveC.

The construction of the dual curve is the geometrical underpinning for theLegendre transformation in the context ofHamiltonian mechanics.^[1]

Letf(x,y,z) = 0 be the equation of a curve inhomogeneous coordinates on theprojective plane. LetXx +Yy +Zz = 0 be the equation of a line, with(X,Y,Z) being designated itsline coordinates in a dual projective plane. The condition that the line is tangent to the curve can be expressed in the formF(X,Y,Z) = 0 which is the tangential equation of the curve.

At a point(p,q,r) on the curve, the tangent is given by

${\displaystyle x\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(p,q,r)+y\frac{\mathrm{\partial }f}{\mathrm{\partial }y}(p,q,r)+z\frac{\mathrm{\partial }f}{\mathrm{\partial }z}(p,q,r)=0.}$

SoXx +Yy +Zz = 0 is a tangent to the curve if

Eliminatingp,q,r, andλ from these equations, along withXp +Yq +Zr = 0, gives the equation inX,Y andZ of the dual curve.

For example, letC be theconicax² +by² +cz² = 0. The dual is found by eliminatingp,q,r, andλ from the equations

${\displaystyle \begin{array}{c}X=2\lambda ap,Y=2\lambda bq,Z=2\lambda cr,\\ Xp+Yq+Zr=0.\end{array}}$

The first three equations are easily solved forp,q,r, and substituting in the last equation produces

${\displaystyle \frac{X^2}{2\lambda a}+\frac{Y^2}{2\lambda b}+\frac{Z^2}{2\lambda c}=0.}$

Clearing2λ from the denominators, the equation of the dual is

${\displaystyle \frac{X^2}{a}+\frac{Y^2}{b}+\frac{Z^2}{c}=0.}$

Consider aparametrically defined curve${\displaystyle (x,y)=(x(t),y(t)),}$ in projective coordinates${\displaystyle (x,y,z)=(x(t),y(t),1)}$. Its projective tangent line is a linear plane spanned by the point of tangency and the tangent vector, with linear equation coefficients given by thecross product:

${\displaystyle (X,Y,Z)=(x,y,1)\times (x^{\prime },y^{\prime },0)=(-y^{\prime },x^{\prime },x{y}^{\prime }-y{x}^{\prime }),}$

which in affine coordinates${\displaystyle (X,Y,1)}$ is:

${\displaystyle X=\frac{-y^{\prime }}{x{y}^{\prime }-y{x}^{\prime }},Y=\frac{x^{\prime }}{x{y}^{\prime }-y{x}^{\prime }}.}$

The dual of aninflection point will give acusp and two points sharing the same tangent line will give a self-intersection point on the dual.

From the projective description, one may compute the dual of the dual:

${\displaystyle (x(x^{\prime }{y}^{″}-y^{\prime }{x}^{″}),y(x^{\prime }{y}^{″}-y^{\prime }{x}^{″}),x^{\prime }{y}^{″}-y^{\prime }{x}^{″})=(x,y,1)(x^{\prime }{y}^{″}-y^{\prime }{x}^{″}),}$

which is projectively equivalent to the original curve${\displaystyle (x(t),y(t),1)}$.

Properties of the original curve correspond to dual properties on the dual curve. In the Introduction image, the red curve has three singularities – anode in the center, and twocusps at the lower right and lower left. The black curve has no singularities but has four distinguished points: 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 twoinflection points correspond to the cusps, since the tangent lines first go one way then the other (slope increasing, then decreasing).

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 symmetries a projective space, 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.

IfX 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 theX 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 thepolar curve with respect to the given point. If the degree of the curve isd then the degree of the polar isd − 1 and so the number of tangents that can be drawn through the given point is at mostd(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.

IfX is smooth (nosingular points) then the dual ofX has maximum degreed(d − 1). This implies the dual of a conic is also a conic. Geometrically, the map from a conic to its dual isone-to-one (since no line is tangent to two points of a conic, as that requires degree 4), and the 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).

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. ThePlücker formulas give the degree of the dual in terms ofd and the number and types of singular points ofX.

The dual can be visualized as a locus in the plane in the form of thepolar reciprocal. This is defined with reference to a fixed conicQ as the locus of the poles of the tangent lines of the curveC.^[2] The conicQ is nearly always taken to be a circle, so the polar reciprocal is theinverse of thepedal ofC.

Similarly, generalizing to higher dimensions, given ahypersurface, thetangent space at each point gives a family ofhyperplanes, and thus defines a dual hypersurface in the dual space. For any closed subvarietyX in a projective space, the set of all hyperplanes tangent to some point ofX is a closed subvariety of the dual of the projective space, called thedual variety ofX.

Examples

• IfX is a hypersurface defined by a homogeneous polynomialF(x₀, ...,x_n), then the dual variety ofX is the image ofX by the gradient map

${\displaystyle x=(x_0,\dots ,x_n)\mapsto \left(\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_0}(x),\dots ,\frac{\mathrm{\partial }F}{\mathrm{\partial }{x}_n}(x)\right)}$

which lands in the dual projective space.

• The dual variety of a point(a₀ : ... :a_n) is the hyperplane

${\displaystyle a_0{x}_0+\dots +a_n{x}_n=0.}$

The dual curve construction works even if the curve ispiecewise linear orpiecewise 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 apolyhedral dual, and any convex setX with boundary hypersurfaceH has aconvex conjugateX* whose boundary is the dual varietyH*.

• Hough transform
• Gauss map

• Arnold, Vladimir Igorevich (1988),Geometrical Methods in the Theory of Ordinary Differential Equations, Springer,ISBN 3-540-96649-8
• Hilton, Harold (1920), "Chapter IV: Tangential Equation and Polar Reciprocation",Plane Algebraic Curves, Oxford
• Fulton, William (1998),Intersection Theory, Springer-Verlag,ISBN 978-3-540-62046-4
• Walker, R. J. (1950),Algebraic Curves, Princeton
• Brieskorn, E.; Knorrer, H. (1986),Plane Algebraic Curves, Birkhäuser,ISBN 978-3-7643-1769-0

• Curves
• Projective geometry
• Differential geometry

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