Dual curve
Encyclopedia
In projective geometry
, a dual curve of a given plane curve C is a curve in the dual projective plane
consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is 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 C, given in line coordinates
, is known as the tangential equation of C.
The construction of the dual curve is the geometrical underpinning for the Legendre transformation
in the context of Hamiltonian mechanics
.
. Let Xx+Yy+Zz=0 be the equation of a line, with (X, Y, Z) being designated its line coordinates. The condition that the line is tangent to the curve can be expressed in the form F(X, Y, Z)=0 which is the tangential equation of the curve.
Let (p, q, r) be the point on the curve, then the equation of the tangent at this point is given by
So Xx+Yy+Zz=0 is a tangent to the curve if
Eliminating p, q, r, and λ from these equations, along with Xp+Yq+Zr=0, gives the equation in X, Y and Z of the dual curve.
For example, let C be the conic
ax2+by2+cz2=0. Then dual is found by eliminating p, q, r, and λ from the equations
The first three equations are easily solved for p, q, r, and substituting in the last equation produces
Clearing 2λ from the denominators, the equation of the dual is
For a parametrically defined curve its dual curve is defined by the following parametric equations:
The dual of an inflection point
will give a cusp
and two points sharing the same tangent line will give a self intersection point on the dual.
If d > 2, then d−1 > 1 so d(d − 1) > d, and thus the dual curve must be singular, by duality, otherwise the bidual would have higher degree than the original curve.
For a smooth curve of degree d = 2, the degree of the dual is also 2: the dual of a conic is a conic. This can also be seen geometrically: the map from a conic to its dual is 1-to-1 (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).
Finally, the dual of a line (a curve of degree 1) is a point (the tangent line is the same at all points, and agrees with the line itself).
, where 
is the number of singularities of X
counted with certain multiplicities: each node is counted with multiplicity 2 and each
cusp with multiplicity 3 (cf. Fulton, Ex. 4.4.4).
It is known that the bi-dual curve to an algebraic curve X is isomorphic to X in characteristics 0.
, there is a beautiful classical construction of a dual curve 
(cf., for example, [Brieskorn, Knorrer]). It uses the notion of inversion and polar curve
.
Let S be a (real) unit circle
, and assume one is given a point p inside S. Let l be a line through p orthogonal to the radius through p. The line l intersects S at two points, say, a and b. Let 
be the intersection point of tangents to S at the points a and b. Then p and 
are said to be inverse
to each other with respect to the circle S. Let
be a line through 
parallel to l. Then the line 
is said to be polar to the circle S with respect to the point p, and l is said to be polar to S with respect to the point 
Now, if p is on a curve C, and l is tangent to C at p, then one can see that
is a point of the dual curve 
The converse is also true:
if
is tangent to C at 
then 
is a point of 
Algebraically, if a is the distance from 0 to p, and
is a distance from 0 to 
then 
,
as vectors,
and, if
, then 
has an equation
From the last formula one can see that
,
i.e., p is the class of the line l if we identify the plane
and its dual.
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 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 symmetric 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.
, the tangent space
at each point gives a family of hyperplane
s, and thus defines a dual hypersurface in the dual space. For any closed subvariety X in a projective space, the set of all hyperplanes tangent to some point of X is a closed subvariety of the dual of the projective projective, called the dual variety of X.
Examples
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 – regardless, the map from the vertex is ill-defined. 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.
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
, a dual curve of a given plane curve C is a curve in the dual projective plane
Duality (projective geometry)
A striking feature of projective planes is the "symmetry" of the roles played by points and lines in the definitions and theorems, and duality is the formalization of this metamathematical concept. There are two approaches to the subject of duality, one through language and the other a more...
consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point dual to its tangent line. If C is algebraic
Algebraic
Algebraic may refer to any subject within the algebra branch of mathematics and related branches like algebraic geometry and algebraic topology.Algebraic may also refer to:...
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 C, given in line coordinates
Line coordinates
In geometry, line coordinates are used to specify the position of a line just as point coordinates are used to specify the position of a point.-Lines in the plane:...
, is known as the tangential equation of C.
The construction of the dual curve is the geometrical underpinning for the Legendre transformation
Legendre transformation
In mathematics, the Legendre transformation or Legendre transform, named after Adrien-Marie Legendre, is an operation that transforms one real-valued function of a real variable into another...
in the context of Hamiltonian mechanics
Hamiltonian mechanics
Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without...
.
Equations
Let f(x, y, z)=0 be the equation of a curve in homogeneous coordinatesHomogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
. Let Xx+Yy+Zz=0 be the equation of a line, with (X, Y, Z) being designated its line coordinates. The condition that the line is tangent to the curve can be expressed in the form F(X, Y, Z)=0 which is the tangential equation of the curve.
Let (p, q, r) be the point on the curve, then the equation of the tangent at this point is given by
So Xx+Yy+Zz=0 is a tangent to the curve if
Eliminating p, q, r, and λ from these equations, along with Xp+Yq+Zr=0, gives the equation in X, Y and Z of the dual curve.
For example, let C be the conic
Conic section
In mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
ax2+by2+cz2=0. Then dual is found by eliminating p, q, r, and λ from the equations
The first three equations are easily solved for p, q, r, and substituting in the last equation produces
Clearing 2λ from the denominators, the equation of the dual is
For a parametrically defined curve its dual curve is defined by the following parametric equations:
The dual of an inflection point
Inflection point
In differential calculus, an inflection point, point of inflection, or inflection is a point on a curve at which the curvature or concavity changes sign. The curve changes from being concave upwards to concave downwards , or vice versa...
will give a cusp
Cusp (singularity)
In the mathematical theory of singularities a cusp is a type of singular point of a curve. Cusps are local singularities in that they are not formed by self intersection points of the curve....
and two points sharing the same tangent line will give a self intersection point on the dual.
Smooth curves
If X is a smooth plane algebraic curve of degree d>1, then the dual of X is a (usually singular) plane curve of degree d(d − 1) (cf. Fulton, Ex. 3.2.21).If d > 2, then d−1 > 1 so d(d − 1) > d, and thus the dual curve must be singular, by duality, otherwise the bidual would have higher degree than the original curve.
For a smooth curve of degree d = 2, the degree of the dual is also 2: the dual of a conic is a conic. This can also be seen geometrically: the map from a conic to its dual is 1-to-1 (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).
Finally, the dual of a line (a curve of degree 1) is a point (the tangent line is the same at all points, and agrees with the line itself).
Singular curves
For an arbitrary plane algebraic curve X of degree d>1, its dual is a plane curve of degree, where is the number of singularities of Xcounted with certain multiplicities: each node is counted with multiplicity 2 and each
cusp with multiplicity 3 (cf. Fulton, Ex. 4.4.4).
It is known that the bi-dual curve to an algebraic curve X is isomorphic to X in characteristics 0.
Classical construction
If C is a curve in a real Euclidean plane, there is a beautiful classical construction of a dual curve (cf., for example, [Brieskorn, Knorrer]). It uses the notion of inversion and polar curvePolar curve
In algebraic geometry, the first polar, or simply polar of an algebraic plane curve C of degree n with respect to a point Q is an algebraic curve of degree n−1 which contains every point of C whose tangent line passes through Q...
.
Let S be a (real) unit circle
, and assume one is given a point p inside S. Let l be a line through p orthogonal to the radius through p. The line l intersects S at two points, say, a and b. Let be the intersection point of tangents to S at the points a and b. Then p and are said to be inverseInverse
Inverse may refer to:* Inverse , a type of immediate inference from a conditional sentence* Inverse , a program for solving inverse and optimization problems...
to each other with respect to the circle S. Let
be a line through parallel to l. Then the line is said to be polar to the circle S with respect to the point p, and l is said to be polar to S with respect to the point Now, if p is on a curve C, and l is tangent to C at p, then one can see that
is a point of the dual curve The converse is also true:if
is tangent to C at then is a point of Algebraically, if a is the distance from 0 to p, and
is a distance from 0 to then , as vectors,and, if
, then has an equation From the last formula one can see that
,i.e., p is the class of the line l if we identify the plane
and its dual.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 lines 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).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 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 symmetric 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.
Higher dimensions
Similarly, generalizing to higher dimensions, given a hypersurfaceHypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
, the tangent space
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other....
at each point gives a family of hyperplane
Hyperplane
A hyperplane is a concept in geometry. It is a generalization of the plane into a different number of dimensions.A hyperplane of an n-dimensional space is a flat subset with dimension n − 1...
s, and thus defines a dual hypersurface in the dual space. For any closed subvariety X in a projective space, the set of all hyperplanes tangent to some point of X is a closed subvariety of the dual of the projective projective, called the dual variety of X.
Examples
- If X is a hypersurface defined by a homogeneous polynomial
, then the dual variety of X is the image of X by the gradient map 
which lands in the dual projective space.
- The dual variety of a point
is the hyperplane 
.
Dual polygon
The dual curve construction works even if the curve is piecewise linear (or piecewise 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 – regardless, the map from the vertex is ill-defined. 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.
See also
- Dual polygonDual polygonIn geometry, polygons are associated into pairs called duals, where the vertices of one correspond to the edges of the other.-Properties:Regular polygons are self-dual....
- Plücker formulaPlücker formulaIn mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to...
- Hough transformHough transformThe Hough transform is a feature extraction technique used in image analysis, computer vision, and digital image processing. The purpose of the technique is to find imperfect instances of objects within a certain class of shapes by a voting procedure...
- Gauss mapGauss mapIn differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X → S2 such that N is a unit vector orthogonal to X at p, namely the normal vector to X at p.The Gauss map can be defined...