Pedal equation - Alchetron, The Free Social Encyclopedia

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.

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 = x 2 + y 2 p = x ∂ f ∂ x + y ∂ f ∂ y ( ∂ f ∂ x ) 2 + ( ∂ f ∂ 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 = ∂ g ∂ z ( ∂ g ∂ x ) 2 + ( ∂ g ∂ 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 ⁡ ψ

where ψ is the polar tangential angle given by

r = d r d θ tan ⁡ ψ .

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

Sinusoidal spirals

For a sinusoidal spiral written in the form

r n = a n sin ⁡ ( n θ )

the polar tangential angle is

ψ = n θ

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:

Epi- and hypocycloids

For a epi- or hypocycloid given by parametric equations

x ( θ ) = ( a + b ) cos ⁡ θ − b cos ⁡ ( a + b b θ ) y ( θ ) = ( a + b ) sin ⁡ θ − b sin ⁡ ( a + b b θ ) ,

the pedal equation with respect to the origin is

r 2 = a 2 + 4 ( a + b ) b ( a + 2 b ) 2 p 2

or

p 2 = A ( r 2 − a 2 )

with

A = ( a + 2 b ) 2 4 ( a + b ) b .

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

Other curves

Other pedal equations are:

References

Pedal equation Wikipedia

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