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

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


For C given in rectangular coordinates by f(xy) = 0, and with O taken to be the origin, the pedal coordinates of the point (xy) are given by:[1]


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(xyz) = 0. The value of p is then given by[2]


where the result is evaluated at z=1

Polar coordinates


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


where   is the polar tangential angle given by


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

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




As an example take the logarithmic spiral with the spiral angle α:


Differentiating with respect to   we obtain




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:[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  .

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 (with position   and velocity  ) 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.[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   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:[6]

n Curve Pedal point Pedal eq.
All Circle with radius a Center  
1 Circle with diameter a Point on circumference pa = r2
−1 Line Point distance a from line p = a
12 Cardioid Cusp p2a = r3
12 Parabola Focus p2 = ar
2 Lemniscate of Bernoulli Center pa2 = r3
−2 Rectangular hyperbola Center rp = a2



A spiral shaped curve of the form


satisfies the equation


and thus can be easily converted into pedal coordinates as


Special cases include:

  Curve Pedal point Pedal eq.
1 Spiral of Archimedes Origin  
−1 Hyperbolic spiral Origin  
12 Fermat's spiral Origin  
12 Lituus Origin  

Epi- and hypocycloids


For an epi- or hypocycloid given by parametric equations


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






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

n Curve Pedal eq.
1, −12 Cardioid  
2, −23 Nephroid  
−3, −32 Deltoid  
−4, −43 Astroid  

Other curves


Other pedal equations are:,[9]

Curve Equation Pedal point Pedal eq.
Line   Origin  
Point   Origin  
Circle   Origin  
Involute of a circle   Origin  
Ellipse   Center  
Hyperbola   Center  
Ellipse   Focus  
Hyperbola   Focus  
Logarithmic spiral   Pole  
Cartesian oval   Focus  
Cassini oval   Focus  
Cassini oval   Center  

See also



  1. ^ Yates §1
  2. ^ Edwards p. 161
  3. ^ Yates p. 166, Edwards p. 162
  4. ^ Blaschke Proposition 1
  5. ^ Blaschke Theorem 2
  6. ^ Yates p. 168, Edwards p. 162
  7. ^ Edwards p. 163
  8. ^ Yates p. 163
  9. ^ 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. London: MacMillan and Co. pp. 161 ff.