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 formula_1
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"("x", "y") = 0, and with "O" taken to be the origin, the pedal coordinates of the point ("x", "y") are given by:
formula_2
formula_3
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
formula_4
where the result is evaluated at "z"=1
Polar coordinates.
For "C" given in polar coordinates by "r" = "f"(θ), then
formula_5
where formula_6 is the polar tangential angle given by
formula_7
The pedal equation can be found by eliminating θ from these equations.
Alternatively, from the above we can find that
formula_8
where formula_9 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:
formula_10
its pedal equation becomes
formula_11
Example.
As an example take the logarithmic spiral with the spiral angle α:
formula_12
Differentiating with respect to formula_13 we obtain
formula_14
hence 
formula_15
and thus in pedal coordinates we get
formula_16
or using the fact that formula_17 we obtain
formula_18
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 (formula_19) in polar coordinates
formula_20
is the pedal curve of a curve given in pedal coordinates by
formula_21
where the differentiation is done with respect to formula_22.
Force problems.
Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.
Consider a dynamical system:
formula_23
describing an evolution of a test particle (with position formula_24 and velocity formula_25) in the plane in the presence of central formula_26 and Lorentz like formula_27 potential. The quantities:
formula_28
are conserved in this system.
Then the curve traced by formula_24 is given in pedal coordinates by
formula_30
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:
 ;math ; \ddot{x}=-\frac{M}
References.
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