pedal equation

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

:$r=\backslash sqrt\{x^2+y^2\}$

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

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 byEdwards p. 161

:$p=\backslash frac\{\backslash frac\{\backslash partial\; g\}\{\backslash partial\; z\}\}\{\backslash sqrt\{\backslash left(\backslash frac\{\backslash partial\; g\}\{\backslash partial\; x\}\backslash right)^2+\backslash left(\backslash frac\{\backslash partial\; g\}\{\backslash partial\; y\}\backslash right)^2\}\}$

where the result is evaluated at z=1

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

:$p=r\backslash sin\; \backslash phi$

where $\backslash phi$ is the polar tangential angle given by

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

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

:$\backslash left|\backslash frac\{dr\}\{d\backslash theta\}\backslash right|=\backslash frac\{r\; p\_c\}\{p\},$

where $p\_c:=\backslash sqrt\{r^2-p^2\}$ 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:

:$f\backslash left(r,\backslash left|\backslash frac\{dr\}\{d\backslash theta\}\backslash right|\backslash right)=0,$

its pedal equation becomes

:$f\backslash left(r,\backslash frac\{rp\_c\}\{p\}\backslash right)=0.$

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 $\backslash theta$ 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

:$\backslash left|\backslash frac\{d\; r\}\{d\; \backslash theta\}\backslash right|=\backslash left|\backslash frac\{\backslash cos\backslash alpha\}\{\backslash sin\backslash alpha\}\backslash right|\; r,$

and thus in pedal coordinates we get

:$\backslash frac\{r\}\{p\}p\_c=\backslash left|\backslash frac\{\backslash cos\backslash alpha\}\{\backslash sin\backslash alpha\}\backslash right|\; r,\; \backslash qquad\; \backslash Rightarrow\; \backslash qquad\; |\backslash sin\backslash alpha|\; p\_c=|\backslash cos\backslash alpha|\; p,$

or using the fact that $p\_c^2=r^2-p^2$ we obtain

:$p=|\backslash sin\backslash alpha|r.$

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 ($n\backslash geq\; 1$) in polar coordinates

:$f\backslash left(r,|r\text{'}\_\{\backslash theta\}|,r$_{\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

:$f(p,p\_c,\; p\_c\; p\_c\text{'},p\_c\; (p\_c\; p\_c\text{'})\text{'},\backslash dots,\; (p\_c\backslash partial\_p)^n\; p)=0,$

where the differentiation is done with respect to $p$.

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

Consider a dynamical system:

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

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

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

are conserved in this system.

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

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

with the pedal point at the origin. This fact was discovered by P. Blaschke in 2017.Blaschke Theorem 2

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:

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

we can arrive at the solution immediately in pedal coordinates

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

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.

For a sinusoidal spiral written in the form

:$r^n\; =\; a^n\; \backslash sin(n\; \backslash theta)$

the polar tangential angle is

:$\backslash psi\; =\; n\backslash theta$

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:Yates p. 168, Edwards p. 162

A spiral shaped curve of the form

:$r\; =\; c\; \backslash theta^\backslash alpha,$

satisfies the equation

:$\backslash frac\{dr\}\{d\backslash theta\}=\backslash alpha\; r^\{\backslash frac\{\backslash alpha-1\}\{\backslash alpha\}\},$

and thus can be easily converted into pedal coordinates as

:$\backslash frac\{1\}\{p^2\}=\backslash frac\{\backslash alpha^2\; c^\{\backslash frac\{2\}\{\backslash alpha\}\}\}\{r^\{2+\backslash frac\{2\}\{\backslash alpha\}\}\}+\backslash frac\{1\}\{r^2\}.$

Special cases include:

For an epi- or hypocycloid given by parametric equations

:$x\; (\backslash theta)\; =\; (a\; +\; b)\; \backslash cos\; \backslash theta\; -\; b\; \backslash cos\; \backslash left(\; \backslash frac\{a\; +\; b\}\{b\}\; \backslash theta\; \backslash right)$

:$y\; (\backslash theta)\; =\; (a\; +\; b)\; \backslash sin\; \backslash theta\; -\; b\; \backslash sin\; \backslash left(\; \backslash frac\{a\; +\; b\}\{b\}\; \backslash theta\; \backslash right),$

the pedal equation with respect to the origin isEdwards p. 163

:$r^2=a^2+\backslash frac\{4(a+b)b\}\{(a+2b)^2\}p^2$

orYates p. 163

:$p^2=A(r^2-a^2)$

with

:$A=\backslash frac\{(a+2b)^2\}\{4(a+b)b\}.$

Special cases obtained by setting b={{frac|a|n}} for specific values of n include:

Other pedal equations are:,Yates p. 169, Edwards p. 163, Blaschke sec. 2.1

|-

| Point

| $(x\_0,y\_0)$

| Origin

| $r=\backslash sqrt\{x\_0^2+y\_0^2\}$

|-

| Circle

| $|x-a|=R$

| Origin

| $2pR=r^2+R^2-|a|^2$

|-

| Involute of a circle

| $r=\backslash frac\{a\}\{\backslash cos\backslash alpha\},\backslash \; \backslash theta=\backslash tan\backslash alpha-\backslash alpha$

| Origin

| $p\_c=|a|$

|-

| Ellipse

| $\backslash frac\{x^2\}\{a^2\}+\backslash frac\{y^2\}\{b^2\}=1$

| Center

| $\backslash frac\{a^2b^2\}\{p^2\}+r^2=a^2+b^2$

|-

| Hyperbola

| $\backslash frac\{x^2\}\{a^2\}-\backslash frac\{y^2\}\{b^2\}=1$

| Center

| $-\backslash frac\{a^2b^2\}\{p^2\}+r^2=a^2-b^2$

|-

| Ellipse

| $\backslash frac\{x^2\}\{a^2\}+\backslash frac\{y^2\}\{b^2\}=1$

| Focus

| $\backslash frac\{b^2\}\{p^2\}=\backslash frac\{2a\}\{r\}-1$

|-

| Hyperbola

| $\backslash frac\{x^2\}\{a^2\}-\backslash frac\{y^2\}\{b^2\}=1$

| Focus

| $\backslash frac\{b^2\}\{p^2\}=\backslash frac\{2a\}\{r\}+1$

|-

| Logarithmic spiral

| $r\; =\; ae^\{\backslash theta\; \backslash cot\; \backslash alpha\}$

| Pole

| $p=r\; \backslash sin\; \backslash alpha$

|-

| Cartesian oval

| $|x|+\backslash alpha|x-a|=C,$

| Focus

| $\backslash frac\{(b-(1-\backslash alpha^2)r^2\; )^2\}\{4p^2\}=\backslash frac\{Cb\}\{r\}+(1-\backslash alpha^2)C\; r\; -((1-\backslash alpha^2)C^2+b),\backslash \; b:=C^2-\backslash alpha^2|a|^2$

|-

| Cassini oval

| $|x||x-a|=C,$

| Focus

| $\backslash frac\{(3C^2+r^4-|a|^2\; r^2)^2\; \}\{p^2\}=4C^2\backslash left(\backslash frac\{2C^2\}\{r^2\}+2r^2-|a|^2\backslash right).$

|-

| Cassini oval

| $|x-a||x+a|=C,$

| Center

| $2R\; pr=r^\{4\}+R^2-|a|^2.$

|}

• Pedal curve

{{Reflist}}

• {{Cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | year=1952 | chapter=Pedal Equations| pages=166 ff }}
• {{Cite book | author=J. Edwards | title=Differential Calculus

| publisher= MacMillan and Co.| location=London | pages=[https://archive.org/details/in.ernet.dli.2015.109607/page/n169 161] ff| year=1892

|url=https://archive.org/details/in.ernet.dli.2015.109607}}

• {{cite journal| author = P. Blaschke| year = 2017| title = Pedal coordinates, dark Kepler and other force problems| journal = Journal of Mathematical Physics| volume = 58/6| issue = 6| doi = 10.1063/1.4984905| url = https://zenodo.org/record/897629| arxiv = 1704.00897| bibcode = 2017JMP....58f3505B}}

• {{MathWorld|title=Pedal coordinates|urlname=PedalCoordinates}}

{{DEFAULTSORT:Pedal Coordinates}}

Category:Coordinate systems