In Euclidean geometry, for a plane curve and a given fixed point , the pedal equation of the curve is a relation between and where is the distance from to a point on and is the perpendicular distance from to the tangent line to at the point. The point is called the pedal point and the values and 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 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. 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:Yates §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(x, y, z) = 0. The value of p is then given byEdwards p. 161
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.Yates p. 166, Edwards p. 162
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:Blaschke Proposition 1 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.Blaschke Theorem 2
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:Yates p. 168, Edwards p. 162 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 Cardioid Cusp p2a = r3 − Parabola Focus p2 = ar 2 Lemniscate of Bernoulli Center pa2 = r3 −2 Rectangular hyperbola Center rp = 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: Curve Pedal point Pedal eq. 1 Spiral of Archimedes Origin −1 Hyperbolic spiral Origin Fermat's spiral Origin − Lituus Origin
Epi- and hypocycloids For an epi- or hypocycloid given by parametric equations
the pedal equation with respect to the origin isEdwards p. 163
orYates p. 163
with
Special cases obtained by setting b= for specific values of n include:
n Curve Pedal eq. 1, − Cardioid 2, − Nephroid −3, − Deltoid −4, − Astroid
Other curves Other pedal equations are:,Yates p. 169, Edwards p. 163, Blaschke sec. 2.1 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 Pedal curve
References
External links