Pedal equation

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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 (the contrapedal coordinate) 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. This coordinates are also well suited for solving certain type of force problems in classical mechanics and celestial mechanics.

Equations[edit]

Cartesian coordinates[edit]

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[edit]

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

where ψ 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

Example[edit]

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:[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[edit]

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]

Example[edit]

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[edit]

Sinusoidal spirals[edit]

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.
1 Circle with radius 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

Spirals[edit]

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[edit]

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=​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[edit]

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[edit]

References[edit]

  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. 
  • P. Blaschke (2017). "Pedal coordinates, dark Kepler and other force problems". Journal of Mathematical Physics. 58/6. doi:10.1063/1.4984905. 

External links[edit]