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.
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
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.
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] where the result is evaluated at z=1 For C given in polar coordinates by r = f(θ), then where
This implies that if a curve satisfies an autonomous differential equation in polar coordinates of the form: its pedal equation becomes As an example take the logarithmic spiral with the spiral angle α: Differentiating with respect to
we obtain This approach can be generalized to include autonomous differential equations of any order as follows:[4] 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
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 (with position
[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: we can arrive at the solution immediately in pedal coordinates where
corresponds to the particle's angular momentum and
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.
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:[6] A spiral shaped curve of the form satisfies the equation and thus can be easily converted into pedal coordinates as Special cases include: For an epi- or hypocycloid given by parametric equations the pedal equation with respect to the origin is[7] or[8] with Special cases obtained by setting b=a⁄n for specific values of n include: Other pedal equations are:,[9]