Dual curve
Open-knowledge reference entry
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. Duality is an involution: the dual of the dual of C is the original curve C.
The construction of the dual curve is the geometrical underpinning for the Legendre transformation in the context of Hamiltonian mechanics.
Equations
Let f(x, y, z) = 0 be the equation of a curve in homogeneous coordinates on the projective plane. Let Xx + Yy + Zz = 0 be the equation of a line, with (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 F(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
x
∂
f
∂
x
(
p
,
q
,
r
)
+
y
∂
f
∂
y
(
p
,
q
,
r
)
+
z
∂
f
∂
z
(
p
,
q
,
r
)
=
0.
{\displaystyle 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.}
So Xx + Yy + Zz = 0 is a tangent to the curve if
X
=
λ
∂
f
∂
x
(
p
,
q
,
r
)
,
Y
=
λ
∂
f
∂
y
(
p
,
q
,
r
)
,
Z
=
λ
∂
f
∂
z
(
p
,
q
,
r
)
.
{\displaystyle {\begin{aligned}X&=\lambda {\frac {\partial f}{\partial x}}(p,q,r),\\Y&=\lambda {\frac {\partial f}{\partial y}}(p,q,r),\\Z&=\lambda {\frac {\partial f}{\partial z}}(p,q,r).\end{aligned}}}
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.
Conic
For example, let C be the conic ax2 + by2 + cz2 = 0.
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