Pedal equation


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 even though it is not an independent quantity and it relates to as.
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.

Equations

Cartesian coordinates

For C given in rectangular coordinates by f = 0, and with O taken to be the origin, the pedal coordinates of the point are given by:
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 = 0. The value of p is then given by
where the result is evaluated at z=1

Polar coordinates

For C given in polar coordinates by r = f, then
where is the polar tangential angle given by
The pedal equation can be found by eliminating θ from these equations.
Alternatively, from the above we can find that
where 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:
its pedal equation becomes

Example

As an example take the logarithmic spiral with the spiral angle α:
Differentiating with respect to we obtain
hence
and thus in pedal coordinates we get
or using the fact that we obtain
This approach can be generalized to include autonomous differential equations of any order as follows: A curve C which a solution of an n-th order autonomous differential equation in polar coordinates
is the pedal curve of a curve given in pedal coordinates by
where the differentiation is done with respect to.

Force problems

Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
describing an evolution of a test particle in the plane in the presence of central and Lorentz like potential. The quantities:
are conserved in this system.
Then the curve traced by is given in pedal coordinates by
with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.

Example

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:
we can arrive at the solution immediately in pedal coordinates
where corresponds to the particle's angular momentum and to its energy. Thus we have obtained the equation of a conic section in pedal coordinates.
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

Sinusoidal spirals

For a sinusoidal spiral written in the form
the polar tangential angle is
which produces the pedal equation
The pedal equation for a number of familiar curves can be obtained setting n to specific values:
nCurvePedal pointPedal eq.
1Circle with radius aPoint on circumferencepa = r2
−1LinePoint distance a from linep = a
CardioidCuspp2a = r3
−ParabolaFocusp2 = ar
2Lemniscate of BernoulliCenterpa2 = r3
−2Rectangular hyperbolaCenterrp = a2

Spirals

A spiral shaped curve of the form
satisfies the equation
and thus can be easily converted into pedal coordinates as
Special cases include:
CurvePedal pointPedal eq.
1Spiral of ArchimedesOrigin
−1Hyperbolic spiralOrigin
Fermat's spiralOrigin
−LituusOrigin

Epi- and hypocycloids

For an epi- or hypocycloid given by parametric equations
the pedal equation with respect to the origin is
or
with
Special cases obtained by setting b= for specific values of n include:
nCurvePedal eq.
1, −Cardioid
2, −Nephroid
−3, −Deltoid
−4, −Astroid

Other curves

Other pedal equations are:,
CurveEquationPedal pointPedal eq.
LineOrigin
PointOrigin
CircleOrigin
Involute of a circleOrigin
EllipseCenter
HyperbolaCenter
EllipseFocus
HyperbolaFocus
Logarithmic spiralPole
Cartesian ovalFocus
Cassini ovalFocus
Cassini ovalCenter