Plane curve constructed from a given curve and fixed point
In Euclidean geometry , for a plane curve C and a given fixed point O , the pedal equation of the curve is a relation between r and p where r is the distance from O to a point on C and p is the perpendicular distance from O to the tangent line to C at the point. The point O is called the pedal point and the values r and p are sometimes called the pedal coordinates of a point relative to the curve and the pedal point. It is also useful to measure the distance of O to the normal pc (the contrapedal coordinate ) even though it is not an independent quantity and it relates to (r , p ) as
p
c
:=
r
2
−
p
2
.
{\textstyle p_{c}:={\sqrt {r^{2}-p^{2}}}.}
Some curves have particularly simple pedal equations and knowing the pedal equation of a curve may simplify the calculation of certain of its properties such as curvature . These coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics .
Cartesian coordinates [ edit ] For C given in rectangular coordinates by f (x , y ) = 0, and with O taken to be the origin, the pedal coordinates of the point (x , y ) are given by:[ 1]
r
=
x
2
+
y
2
{\displaystyle r={\sqrt {x^{2}+y^{2}}}}
p
=
x
∂
f
∂
x
+
y
∂
f
∂
y
(
∂
f
∂
x
)
2
+
(
∂
f
∂
y
)
2
.
{\displaystyle p={\frac {x{\frac {\partial f}{\partial x}}+y{\frac {\partial f}{\partial y}}}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}}}}.}
The pedal equation can be found by eliminating x and y from these equations and the equation of the curve.
The expression for p may be simplified if the equation of the curve is written in homogeneous coordinates by introducing a variable z , so that the equation of the curve is g (x , y , z ) = 0. The value of p is then given by[ 2]
p
=
∂
g
∂
z
(
∂
g
∂
x
)
2
+
(
∂
g
∂
y
)
2
{\displaystyle p={\frac {\frac {\partial g}{\partial z}}{\sqrt {\left({\frac {\partial g}{\partial x}}\right)^{2}+\left({\frac {\partial g}{\partial y}}\right)^{2}}}}}
where the result is evaluated at z =1
For C given in polar coordinates by r = f (θ), then
p
=
r
sin
ϕ
{\displaystyle p=r\sin \phi }
where
ϕ
{\displaystyle \phi }
polar tangential angle given by
r
=
d
r
d
θ
tan
ϕ
.
{\displaystyle r={\frac {dr}{d\theta }}\tan \phi .}
The pedal equation can be found by eliminating θ from these equations.[ 3]
Alternatively, from the above we can find that
|
d
r
d
θ
|
=
r
p
c
p
,
{\displaystyle \left|{\frac {dr}{d\theta }}\right|={\frac {rp_{c}}{p}},}
where
p
c
:=
r
2
−
p
2
{\displaystyle p_{c}:={\sqrt {r^{2}-p^{2}}}}
f
(
r
,
|
d
r
d
θ
|
)
=
0
,
{\displaystyle f\left(r,\left|{\frac {dr}{d\theta }}\right|\right)=0,}
its pedal equation becomes
f
(
r
,
r
p
c
p
)
=
0.
{\displaystyle f\left(r,{\frac {rp_{c}}{p}}\right)=0.}
As an example take the logarithmic spiral with the spiral angle α:
r
=
a
e
cos
α
sin
α
θ
.
{\displaystyle r=ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }.}
Differentiating with respect to
θ
{\displaystyle \theta }
d
r
d
θ
=
cos
α
sin
α
a
e
cos
α
sin
α
θ
=
cos
α
sin
α
r
,
{\displaystyle {\frac {dr}{d\theta }}={\frac {\cos \alpha }{\sin \alpha }}ae^{{\frac {\cos \alpha }{\sin \alpha }}\theta }={\frac {\cos \alpha }{\sin \alpha }}r,}
hence
|
d
r
d
θ
|
=
|
cos
α
sin
α
|
r
,
{\displaystyle \left|{\frac {dr}{d\theta }}\right|=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,}
and thus in pedal coordinates we get
r
p
p
c
=
|
cos
α
sin
α
|
r
,
⇒
|
sin
α
|
p
c
=
|
cos
α
|
p
,
{\displaystyle {\frac {r}{p}}p_{c}=\left|{\frac {\cos \alpha }{\sin \alpha }}\right|r,\qquad \Rightarrow \qquad |\sin \alpha |p_{c}=|\cos \alpha |p,}
or using the fact that
p
c
2
=
r
2
−
p
2
{\displaystyle p_{c}^{2}=r^{2}-p^{2}}
p
=
|
sin
α
|
r
.
{\displaystyle p=|\sin \alpha |r.}
This approach can be generalized to include autonomous differential equations of any order as follows:[ 4] C which a solution of an n -th order autonomous differential equation (
n
≥
1
{\displaystyle n\geq 1}
f
(
r
,
|
r
θ
′
|
,
r
θ
″
,
|
r
θ
‴
|
…
,
r
θ
(
2
j
)
,
|
r
θ
(
2
j
+
1
)
|
,
…
,
r
θ
(
n
)
)
=
0
,
{\displaystyle f\left(r,|r'_{\theta }|,r''_{\theta },|r'''_{\theta }|\dots ,r_{\theta }^{(2j)},|r_{\theta }^{(2j+1)}|,\dots ,r_{\theta }^{(n)}\right)=0,}
is the pedal curve of a curve given in pedal coordinates by
f
(
p
,
p
c
,
p
c
p
c
′
,
p
c
(
p
c
p
c
′
)
′
,
…
,
(
p
c
∂
p
)
n
p
)
=
0
,
{\displaystyle f(p,p_{c},p_{c}p_{c}',p_{c}(p_{c}p_{c}')',\dots ,(p_{c}\partial _{p})^{n}p)=0,}
where the differentiation is done with respect to
p
{\displaystyle p}
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
x
¨
=
F
′
(
|
x
|
2
)
x
+
2
G
′
(
|
x
|
2
)
x
˙
⊥
,
{\displaystyle {\ddot {x}}=F^{\prime }(|x|^{2})x+2G^{\prime }(|x|^{2}){\dot {x}}^{\perp },}
describing an evolution of a test particle (with position
x
{\displaystyle x}
x
˙
{\displaystyle {\dot {x}}}
F
{\displaystyle F}
G
{\displaystyle G}
L
=
x
⋅
x
˙
⊥
+
G
(
|
x
|
2
)
,
c
=
|
x
˙
|
2
−
F
(
|
x
|
2
)
,
{\displaystyle L=x\cdot {\dot {x}}^{\perp }+G(|x|^{2}),\qquad c=|{\dot {x}}|^{2}-F(|x|^{2}),}
are conserved in this system.
Then the curve traced by
x
{\displaystyle x}
(
L
−
G
(
r
2
)
)
2
p
2
=
F
(
r
2
)
+
c
,
{\displaystyle {\frac {\left(L-G(r^{2})\right)^{2}}{p^{2}}}=F(r^{2})+c,}
with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.[ 5]
As an example consider the so-called Kepler problem , i.e. central force problem, where the force varies inversely as a square of the distance:
x
¨
=
−
M
|
x
|
3
x
,
{\displaystyle {\ddot {x}}=-{\frac {M}{|x|^{3}}}x,}
we can arrive at the solution immediately in pedal coordinates
L
2
2
p
2
=
M
r
+
c
,
{\displaystyle {\frac {L^{2}}{2p^{2}}}={\frac {M}{r}}+c,}
where
L
{\displaystyle L}
c
{\displaystyle c}
Inversely, for a given curve C , we can easily deduce what forces do we have to impose on a test particle to move along it.
Pedal equations for specific curves [ edit ] For a sinusoidal spiral written in the form
r
n
=
a
n
sin
(
n
θ
)
{\displaystyle r^{n}=a^{n}\sin(n\theta )}
the polar tangential angle is
ψ
=
n
θ
{\displaystyle \psi =n\theta }
which produces the pedal equation
p
a
n
=
r
n
+
1
.
{\displaystyle pa^{n}=r^{n+1}.}
The pedal equation for a number of familiar curves can be obtained setting n to specific values:[ 6]
n
Curve
Pedal point
Pedal eq.
All
Circle with radius a
Center
p
a
n
=
r
n
+
1
{\displaystyle pa^{n}=r^{n+1}}
1
Circle with diameter a
Point on circumference
pa = r 2
−1
Line
Point distance a from line
p = a
1 ⁄2 Cardioid
Cusp
p 2 a = r 3
−1 ⁄2
Parabola
Focus
p 2 = ar
2
Lemniscate of Bernoulli
Center
pa 2 = r 3
−2
Rectangular hyperbola
Center
rp = a 2
A spiral shaped curve of the form
r
=
c
θ
α
,
{\displaystyle r=c\theta ^{\alpha },}
satisfies the equation
d
r
d
θ
=
α
r
α
−
1
α
,
{\displaystyle {\frac {dr}{d\theta }}=\alpha r^{\frac {\alpha -1}{\alpha }},}
and thus can be easily converted into pedal coordinates as
1
p
2
=
α
2
c
2
α
r
2
+
2
α
+
1
r
2
.
{\displaystyle {\frac {1}{p^{2}}}={\frac {\alpha ^{2}c^{\frac {2}{\alpha }}}{r^{2+{\frac {2}{\alpha }}}}}+{\frac {1}{r^{2}}}.}
Special cases include:
α
{\displaystyle \alpha }
Curve
Pedal point
Pedal eq.
1
Spiral of Archimedes
Origin
1
p
2
=
1
r
2
+
c
2
r
4
{\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{2}}{r^{4}}}}
−1
Hyperbolic spiral
Origin
1
p
2
=
1
r
2
+
1
c
2
{\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {1}{c^{2}}}}
1 ⁄2 Fermat's spiral
Origin
1
p
2
=
1
r
2
+
c
4
4
r
6
{\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{4}}{4r^{6}}}}
−1 ⁄2
Lituus
Origin
1
p
2
=
1
r
2
+
r
2
4
c
4
{\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {r^{2}}{4c^{4}}}}
Epi- and hypocycloids [ edit ] For an epi- or hypocycloid given by parametric equations
x
(
θ
)
=
(
a
+
b
)
cos
θ
−
b
cos
(
a
+
b
b
θ
)
{\displaystyle x(\theta )=(a+b)\cos \theta -b\cos \left({\frac {a+b}{b}}\theta \right)}
y
(
θ
)
=
(
a
+
b
)
sin
θ
−
b
sin
(
a
+
b
b
θ
)
,
{\displaystyle y(\theta )=(a+b)\sin \theta -b\sin \left({\frac {a+b}{b}}\theta \right),}
the pedal equation with respect to the origin is[ 7]
r
2
=
a
2
+
4
(
a
+
b
)
b
(
a
+
2
b
)
2
p
2
{\displaystyle r^{2}=a^{2}+{\frac {4(a+b)b}{(a+2b)^{2}}}p^{2}}
or[ 8]
p
2
=
A
(
r
2
−
a
2
)
{\displaystyle p^{2}=A(r^{2}-a^{2})}
with
A
=
(
a
+
2
b
)
2
4
(
a
+
b
)
b
.
{\displaystyle A={\frac {(a+2b)^{2}}{4(a+b)b}}.}
Special cases obtained by setting b =a n n include:
n
Curve
Pedal eq.
1, −1 ⁄2
Cardioid
p
2
=
9
8
(
r
2
−
a
2
)
{\displaystyle p^{2}={\frac {9}{8}}(r^{2}-a^{2})}
2, −2 ⁄3
Nephroid
p
2
=
4
3
(
r
2
−
a
2
)
{\displaystyle p^{2}={\frac {4}{3}}(r^{2}-a^{2})}
−3, −3 ⁄2
Deltoid
p
2
=
−
1
8
(
r
2
−
a
2
)
{\displaystyle p^{2}=-{\frac {1}{8}}(r^{2}-a^{2})}
−4, −4 ⁄3
Astroid
p
2
=
−
1
3
(
r
2
−
a
2
)
{\displaystyle p^{2}=-{\frac {1}{3}}(r^{2}-a^{2})}
Other pedal equations are:,[ 9]
Curve
Equation
Pedal point
Pedal eq.
Line
a
x
+
b
y
+
c
=
0
{\displaystyle ax+by+c=0}
Origin
p
=
|
c
|
a
2
+
b
2
{\displaystyle p={\frac {|c|}{\sqrt {a^{2}+b^{2}}}}}
Point
(
x
0
,
y
0
)
{\displaystyle (x_{0},y_{0})}
Origin
r
=
x
0
2
+
y
0
2
{\displaystyle r={\sqrt {x_{0}^{2}+y_{0}^{2}}}}
Circle
|
x
−
a
|
=
R
{\displaystyle |x-a|=R}
Origin
2
p
R
=
r
2
+
R
2
−
|
a
|
2
{\displaystyle 2pR=r^{2}+R^{2}-|a|^{2}}
Involute of a circle
r
=
a
cos
α
,
θ
=
tan
α
−
α
{\displaystyle r={\frac {a}{\cos \alpha }},\ \theta =\tan \alpha -\alpha }
Origin
p
c
=
|
a
|
{\displaystyle p_{c}=|a|}
Ellipse
x
2
a
2
+
y
2
b
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}
Center
a
2
b
2
p
2
+
r
2
=
a
2
+
b
2
{\displaystyle {\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}+b^{2}}
Hyperbola
x
2
a
2
−
y
2
b
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
Center
−
a
2
b
2
p
2
+
r
2
=
a
2
−
b
2
{\displaystyle -{\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}-b^{2}}
Ellipse
x
2
a
2
+
y
2
b
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}
Focus
b
2
p
2
=
2
a
r
−
1
{\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}-1}
Hyperbola
x
2
a
2
−
y
2
b
2
=
1
{\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}
Focus
b
2
p
2
=
2
a
r
+
1
{\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}+1}
Logarithmic spiral
r
=
a
e
θ
cot
α
{\displaystyle r=ae^{\theta \cot \alpha }}
Pole
p
=
r
sin
α
{\displaystyle p=r\sin \alpha }
Cartesian oval
|
x
|
+
α
|
x
−
a
|
=
C
,
{\displaystyle |x|+\alpha |x-a|=C,}
Focus
(
b
−
(
1
−
α
2
)
r
2
)
2
4
p
2
=
C
b
r
+
(
1
−
α
2
)
C
r
−
(
(
1
−
α
2
)
C
2
+
b
)
,
b
:=
C
2
−
α
2
|
a
|
2
{\displaystyle {\frac {(b-(1-\alpha ^{2})r^{2})^{2}}{4p^{2}}}={\frac {Cb}{r}}+(1-\alpha ^{2})Cr-((1-\alpha ^{2})C^{2}+b),\ b:=C^{2}-\alpha ^{2}|a|^{2}}
Cassini oval
|
x
|
|
x
−
a
|
=
C
,
{\displaystyle |x||x-a|=C,}
Focus
(
3
C
2
+
r
4
−
|
a
|
2
r
2
)
2
p
2
=
4
C
2
(
2
C
2
r
2
+
2
r
2
−
|
a
|
2
)
.
{\displaystyle {\frac {(3C^{2}+r^{4}-|a|^{2}r^{2})^{2}}{p^{2}}}=4C^{2}\left({\frac {2C^{2}}{r^{2}}}+2r^{2}-|a|^{2}\right).}
Cassini oval
|
x
−
a
|
|
x
+
a
|
=
C
,
{\displaystyle |x-a||x+a|=C,}
Center
2
R
p
r
=
r
4
+
R
2
−
|
a
|
2
.
{\displaystyle 2Rpr=r^{4}+R^{2}-|a|^{2}.}
^ Yates §1
^ Edwards p. 161
^ Yates p. 166, Edwards p. 162
^ Blaschke Proposition 1
^ Blaschke Theorem 2
^ Yates p. 168, Edwards p. 162
^ Edwards p. 163
^ Yates p. 163
^ Yates p. 169, Edwards p. 163, Blaschke sec. 2.1
R.C. Yates (1952). "Pedal Equations". A Handbook on Curves and Their Properties . Ann Arbor, MI: J. W. Edwards. pp. 166 ff. J. Edwards (1892). Differential Calculus 161 ff. 
