Pedal equation

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:

:r=\sqrt{x^2+y^2} :p=\frac{x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}}{\sqrt{\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial f}{\partial y}\right)^2}}.

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 :p=\frac{\frac{\partial g}{\partial z}}{\sqrt{\left(\frac{\partial g}{\partial x}\right)^2+\left(\frac{\partial g}{\partial y}\right)^2}} where the result is evaluated at z=1

Polar coordinates

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

:p=r\sin \phi

where \phi is the polar tangential angle given by

:r=\frac{dr}{d\theta}\tan \phi.

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

Alternatively, from the above we can find that

:\left|\frac{dr}{d\theta}\right|=\frac{r p_c}{p},

where p_c:=\sqrt{r^2-p^2} 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:

:f\left(r,\left|\frac{dr}{d\theta}\right|\right)=0, its pedal equation becomes

:f\left(r,\frac{rp_c}{p}\right)=0.

Example

As an example take the logarithmic spiral with the spiral angle α: : r=a e^{\frac{\cos\alpha}{\sin\alpha} \theta}. Differentiating with respect to \theta we obtain : \frac{dr}{d\theta}= \frac{\cos\alpha}{\sin\alpha} a e^{\frac{\cos\alpha}{\sin\alpha} \theta}=\frac{\cos\alpha}{\sin\alpha} r, hence : \left|\frac{d r}{d \theta}\right|=\left|\frac{\cos\alpha}{\sin\alpha}\right| r, and thus in pedal coordinates we get : \frac{r}{p}p_c=\left|\frac{\cos\alpha}{\sin\alpha}\right| r, \qquad \Rightarrow \qquad |\sin\alpha| p_c=|\cos\alpha| p, or using the fact that p_c^2=r^2-p^2 we obtain : p=|\sin\alpha|r.

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 (n\geq 1) in polar coordinates

: f\left(r,|r'{\theta}|,r''{\theta},|r'''{\theta}|\dots,r\theta^{(2j)},|r_\theta^{(2j+1)}|,\dots, r_\theta^{(n)}\right)=0,

is the pedal curve of a curve given in pedal coordinates by

: f(p,p_c, p_c p_c',p_c (p_c p_c')',\dots, (p_c\partial_p)^n p)=0,

where the differentiation is done with respect to p.

Force problems

Solutions to some force problems of classical mechanics can be surprisingly easily obtained in pedal coordinates.

Consider a dynamical system:

: \ddot x=F^\prime(|x|^2)x+2 G^\prime(|x|^2){\dot x}^\perp,

describing an evolution of a test particle (with position x and velocity \dot x) in the plane in the presence of central F and Lorentz like G potential. The quantities:

: L=x\cdot \dot x^\perp+G(|x|^2), \qquad c=|\dot x|^2-F(|x|^2),

are conserved in this system.

Then the curve traced by x is given in pedal coordinates by

: \frac{\left(L-G(r^2)\right)^2}{p^2}=F(r^2)+c,

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:

: \ddot{x}=-\frac{M}{|x|^{3}}x,

we can arrive at the solution immediately in pedal coordinates

:\frac{L^2}{2p^2}=\frac{M}{r}+c, ,

where L corresponds to the particle's angular momentum and c 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 :r^n = a^n \sin(n \theta) the polar tangential angle is :\psi = n\theta which produces the pedal equation :pa^n=r^{n+1}. The pedal equation for a number of familiar curves can be obtained setting n to specific values:

Spirals

A spiral shaped curve of the form :r = c \theta^\alpha, satisfies the equation : \frac{dr}{d\theta}=\alpha r^{\frac{\alpha-1}{\alpha}}, and thus can be easily converted into pedal coordinates as :\frac{1}{p^2}=\frac{\alpha^2 c^{\frac{2}{\alpha}}}{r^{2+\frac{2}{\alpha}}}+\frac{1}{r^2}.

Special cases include:

Epi- and hypocycloids

For an epi- or hypocycloid given by parametric equations :x (\theta) = (a + b) \cos \theta - b \cos \left( \frac{a + b}{b} \theta \right) :y (\theta) = (a + b) \sin \theta - b \sin \left( \frac{a + b}{b} \theta \right), the pedal equation with respect to the origin is :r^2=a^2+\frac{4(a+b)b}{(a+2b)^2}p^2 or :p^2=A(r^2-a^2) with :A=\frac{(a+2b)^2}{4(a+b)b}. Special cases obtained by setting b= for specific values of n include:

Other curves

Other pedal equations are:,

References

• {{Cite book | author=J. Edwards | title=Differential Calculus

References

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