Wente torus

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. 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.

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:

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

$p=\frac{x\frac{\partial f}{\partial x}+y\frac{\partial f}{\partial y}}{\sqrt{(\frac{\partial f}{\partial x}{)}^2+(\frac{\partial f}{\partial y}{)}^2}}.$

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

$p=\frac{\frac{\partial g}{\partial z}}{\sqrt{(\frac{\partial g}{\partial x}{)}^2+(\frac{\partial g}{\partial y}{)}^2}}$

where the result is evaluated at z=1

Polar coordinates

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

$p=r\sin \psi$

where ψ is the polar tangential angle given by

$r=\frac{dr}{d\theta }\tan \psi .$

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

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

$p{a}^n=r^{n+1}.$

The pedal equation for a number of familiar curves can be obtained setting n to specific values:

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
Template:Frac	Cardioid	Cusp	p2a = r3
−Template:Frac	Parabola	Focus	p2 = ar
2	Lemniscate of Bernoulli	Center	pa2 = r3
−2	Rectangular hyperbola	Center	rp = a2

Epi- and hypocycloids

For a 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=Template:Frac for specific values of n include:

n	Curve	Pedal eq.
1, −Template:Frac	Cardioid	$p^2=\frac{9}{8}(r^2-a^2)$
2, −Template:Frac	Nephroid	$p^2=\frac{4}{3}(r^2-a^2)$
−3, −Template:Frac	Deltoid	$p^2=-\frac{1}{8}(r^2-a^2)$
−4, −Template:Frac	Astroid	$p^2=-\frac{1}{3}(r^2-a^2)$

Other curves

Other pedal equations are:

Curve	Equation	Pedal point	Pedal eq.
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Center	$\frac{a^2{b}^2}{p^2}+r^2=a^2+b^2$
Hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	Center	$-\frac{a^2{b}^2}{p^2}+r^2=a^2-b^2$
Ellipse	$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$	Focus	$\frac{b^2}{p^2}=\frac{2a}{r}-1$
Hyperbola	$\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$	Focus	$\frac{b^2}{p^2}=\frac{2a}{r}+1$
Logarithmic spiral	$r=a{e}^{\theta \cot \alpha }$	Pole	$p=r\sin \alpha$

See also

• Pedal curve

References

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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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