Pedal Equation 2c3k3k

Pedal equations for specific curves[edit] Sinusoidal spirals[edit] For a sinusoidal spiral written in the form

the polar tangential angle is

which produces the pedal equation

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

n

1

Curve

Circle with radius a

Pedal point

Point on circumference

Pedal eq.

pa = r2

−1 Line

Point distance a from line p = a

⁄2

Cusp

p2a = r3

Focus

p2 = ar

Lemniscate of Bernoulli Center

pa2 = r3

−2 Rectangular hyperbola Center

rp = a2

1

Cardioid

−1⁄2 Parabola

2

Epi- and hypocycloids[edit] For a epi- or hypocycloid given by parametric equations

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

or[6]

with

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

n

Curve

1, −1⁄2

Cardioid

2, −2⁄3

Nephroid

Pedal eq.

−3, −3⁄2 Deltoid

−4, −4⁄3 Astroid

Other curves[edit] Other pedal equations are:[7]

Curve

Ellipse

Equation

Pedal point

Cente r

Pedal eq.

Hyperbola

Cente r

Ellipse

Focus

Hyperbola

Focus

Logarithmi c spiral

Pole