Pedal equation

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:^[1]

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

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

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^[2]

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

where the result is evaluated at z=1

Polar coordinates

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

${\displaystyle p=r\sin \psi \,}$

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.^[3]

Pedal equations for specific curves

Sinusoidal spirals

For a sinusoidal spiral written in the form

${\displaystyle r^{n}=a^{n}\sin(n\theta )\,}$

the polar tangential angle is

${\displaystyle \psi =n\theta }$

which produces the pedal equation

${\displaystyle pa^{n}=r^{n+1}.}$

The pedal equation for a number of familiar curves can be obtained setting n to specific values:^[4]

n	Curve	Pedal point	Pedal eq.
1	Circle with radius a	Point on circumference	pa = r^2
−1	Line	Point distance a from line	p = a
^1⁄2	Cardioid	Cusp	p^2a = r^3
− ^1⁄2	Parabola	Focus	p^2 = ar
2	Lemniscate of Bernoulli	Center	pa^2 = r^3
−2	Rectangular hyperbola	Center	rp = a^2

Epi- and hypocycloids

For a epi- or hypocycloid given by parametric equations

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

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

the pedal equation with respect to the origin is^[5]

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

or^[6]

${\displaystyle p^{2}=A(r^{2}-a^{2})}$

with

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

Special cases obtained by setting b= ^a⁄_n for specific values of n include:

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

Other curves

Other pedal equations are:^[7]

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

See also

• Pedal curve

References

1. Yates §1
2. Edwards p. 161
3. Yates p. 166, Edwards p. 162
4. Yates p. 168, Edwards p. 162
5. Edwards p. 163
6. Yates p. 163
7. Yates p. 169, Edwards p. 163

• 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.

External links

• Weisstein, Eric W. "Pedal coordinates". MathWorld.