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}}}

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]

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.^[3]

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:^[4]

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
¹⁄₂	Cardioid	Cusp	p²a = r³
− ¹⁄₂	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^[5]

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

or^[6]

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

with

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	$p^{2}={\frac {9}{8}}(r^{2}-a^{2})$
2, − ²⁄₃	Nephroid	$p^{2}={\frac {4}{3}}(r^{2}-a^{2})$
−3, − ³⁄₂	Deltoid	$p^{2}=-{\frac {1}{8}}(r^{2}-a^{2})$
−4, − ⁴⁄₃	Astroid	$p^{2}=-{\frac {1}{3}}(r^{2}-a^{2})$

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	${\frac {a^{2}b^{2}}{p^{2}}}+r^{2}=a^{2}+b^{2}$
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	$r=ae^{\theta \cot \alpha }\,$	Pole	$p=r\sin \alpha$

References

↑ Yates §1
↑ Edwards p. 161
↑ Yates p. 166, Edwards p. 162
↑ Yates p. 168, Edwards p. 162
↑ Edwards p. 163
↑ Yates p. 163
↑ 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.

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