Cartesian coordinates

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]

${\displaystyle r={\sqrt {x^{2}+y^{2}}}}$

${\displaystyle p={\frac {x{\frac {\partial f}{\partial x}}+y{\frac {\partial f}{\partial y}}}{\sqrt {\left({\frac {\partial f}{\partial x}}\right)^{2}+\left({\frac {\partial f}{\partial y}}\right)^{2}}}}.}$

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]

${\displaystyle p={\frac {\frac {\partial g}{\partial z}}{\sqrt {\left({\frac {\partial g}{\partial x}}\right)^{2}+\left({\frac {\partial g}{\partial y}}\right)^{2}}}}}$

where the result is evaluated at z=1

Polar coordinates

For C given in polar coordinates by r = f(θ), then

${\displaystyle p=r\sin \phi }$

where ${\displaystyle \phi }$ is the polar tangential angle given by

${\displaystyle r={\frac {dr}{d\theta }}\tan \phi .}$

The pedal equation can be found by eliminating θ from these equations.^[3]

Alternatively, from the above we can find that

${\displaystyle \left|{\frac {dr}{d\theta }}\right|={\frac {rp_{c}}{p}},}$

where ${\displaystyle p_{c}:={\sqrt {r^{2}-p^{2}}}}$ is the "contrapedal" coordinate, i.e. distance to the normal. This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form:

${\displaystyle f\left(r,\left|{\frac {dr}{d\theta }}\right|\right)=0,}$

its pedal equation becomes

${\displaystyle f\left(r,{\frac {rp_{c}}{p}}\right)=0.}$

Force problems

Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.

Consider a dynamical system:

${\displaystyle {\ddot {x}}=F^{\prime }(|x|^{2})x+2G^{\prime }(|x|^{2}){\dot {x}}^{\perp },}$

describing an evolution of a test particle (with position ${\displaystyle x}$ and velocity ${\displaystyle {\dot {x}}}$) in the plane in the presence of central ${\displaystyle F}$ and Lorentz like ${\displaystyle G}$ potential. The quantities:

${\displaystyle L=x\cdot {\dot {x}}^{\perp }+G(|x|^{2}),\qquad c=|{\dot {x}}|^{2}-F(|x|^{2}),}$

are conserved in this system.

Then the curve traced by ${\displaystyle x}$ is given in pedal coordinates by

${\displaystyle {\frac {\left(L-G(r^{2})\right)^{2}}{p^{2}}}=F(r^{2})+c,}$

with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.^[5]

Sinusoidal spirals

For a sinusoidal spiral written in the form

${\displaystyle r^{n}=a^{n}\sin(n\theta )}$

the polar tangential angle is

${\displaystyle \psi =n\theta }$

which produces the pedal equation

${\displaystyle pa^{n}=r^{n+1}.}$

The pedal equation for a number of familiar curves can be obtained setting n to specific values:^[6]

Spirals

A spiral shaped curve of the form

${\displaystyle r=c\theta ^{\alpha },}$

satisfies the equation

${\displaystyle {\frac {dr}{d\theta }}=\alpha r^{\frac {\alpha -1}{\alpha }},}$

and thus can be easily converted into pedal coordinates as

${\displaystyle {\frac {1}{p^{2}}}={\frac {\alpha ^{2}c^{\frac {2}{\alpha }}}{r^{2+{\frac {2}{\alpha }}}}}+{\frac {1}{r^{2}}}.}$

Special cases include:

${\displaystyle \alpha }$	Curve	Pedal point	Pedal eq.
1	Spiral of Archimedes	Origin	${\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{2}}{r^{4}}}}$
−1	Hyperbolic spiral	Origin	${\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {1}{c^{2}}}}$
1⁄2	Fermat's spiral	Origin	${\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {c^{4}}{4r^{6}}}}$
−1⁄2	Lituus	Origin	${\displaystyle {\frac {1}{p^{2}}}={\frac {1}{r^{2}}}+{\frac {r^{2}}{4c^{4}}}}$

Epi- and hypocycloids

For an epi- or hypocycloid given by parametric equations

${\displaystyle x(\theta )=(a+b)\cos \theta -b\cos \left({\frac {a+b}{b}}\theta \right)}$

${\displaystyle y(\theta )=(a+b)\sin \theta -b\sin \left({\frac {a+b}{b}}\theta \right),}$

the pedal equation with respect to the origin is^[7]

${\displaystyle r^{2}=a^{2}+{\frac {4(a+b)b}{(a+2b)^{2}}}p^{2}}$

or^[8]

${\displaystyle p^{2}=A(r^{2}-a^{2})}$

with

${\displaystyle A={\frac {(a+2b)^{2}}{4(a+b)b}}.}$

Special cases obtained by setting b=a⁄n for specific values of n include:

n	Curve	Pedal eq.
1, −1⁄2	Cardioid	${\displaystyle p^{2}={\frac {9}{8}}(r^{2}-a^{2})}$
2, −2⁄3	Nephroid	${\displaystyle p^{2}={\frac {4}{3}}(r^{2}-a^{2})}$
−3, −3⁄2	Deltoid	${\displaystyle p^{2}=-{\frac {1}{8}}(r^{2}-a^{2})}$
−4, −4⁄3	Astroid	${\displaystyle p^{2}=-{\frac {1}{3}}(r^{2}-a^{2})}$

Other curves

Other pedal equations are:,^[9]

Curve	Equation	Pedal point	Pedal eq.
Line	${\displaystyle ax+by+c=0}$	Origin	${\displaystyle p={\frac {|c|}{\sqrt {a^{2}+b^{2}}}}}$
Point	${\displaystyle (x_{0},y_{0})}$	Origin	${\displaystyle r={\sqrt {x_{0}^{2}+y_{0}^{2}}}}$
Circle	${\displaystyle |x-a|=R}$	Origin	${\displaystyle 2pR=r^{2}+R^{2}-|a|^{2}}$
Involute of a circle	${\displaystyle r={\frac {a}{\cos \alpha }},\ \theta =\tan \alpha -\alpha }$	Origin	${\displaystyle p_{c}=|a|}$
Ellipse	${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$	Center	${\displaystyle {\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}+b^{2}}$
Hyperbola	${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}$	Center	${\displaystyle -{\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}-b^{2}}$
Ellipse	${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1}$	Focus	${\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}-1}$
Hyperbola	${\displaystyle {\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1}$	Focus	${\displaystyle {\frac {b^{2}}{p^{2}}}={\frac {2a}{r}}+1}$
Logarithmic spiral	${\displaystyle r=ae^{\theta \cot \alpha }}$	Pole	${\displaystyle p=r\sin \alpha }$
Cartesian oval	${\displaystyle |x|+\alpha |x-a|=C,}$	Focus	${\displaystyle {\frac {(b-(1-\alpha ^{2})r^{2})^{2}}{4p^{2}}}={\frac {Cb}{r}}+(1-\alpha ^{2})Cr-((1-\alpha ^{2})C^{2}+b),\ b:=C^{2}-\alpha ^{2}|a|^{2}}$
Cassini oval	${\displaystyle |x||x-a|=C,}$	Focus	${\displaystyle {\frac {(3C^{2}+r^{4}-|a|^{2}r^{2})^{2}}{p^{2}}}=4C^{2}\left({\frac {2C^{2}}{r^{2}}}+2r^{2}-|a|^{2}\right).}$
Cassini oval	${\displaystyle |x-a||x+a|=C,}$	Center	${\displaystyle 2Rpr=r^{4}+R^{2}-|a|^{2}.}$