In the differential geometry of curves, a pedal curve is a curve derived by construction from a given curve (as is, for example, the involute).

Take a curve and a fixed point P (called the pedal point). On any line T is a unique point X which is either P or forms with P a line perpendicular to T. The pedal curve is the set of all X for which T is a tangent of the curve.

The pedal "curve" may be disconnected; indeed, for a polygon, it is simply isolated points.

Analytically, if P is the pedal point and c a parametrisation of the curve then

parametrises the pedal curve (disregarding points where c' is zero or undefined).

The contrapedal curve is the set of all X for which T is perpendicular to the curve.

With the same pedal point, it happens to be the pedal curve of the evolute.

In the plane, for every point X other than P there is a unique line through X perpendicular to XP. The negative pedal curve is the envelope of the lines for which X lies on the given curve. The negative pedal curve of a pedal curve with the same pedal point is the original curve.

Example

Pedal curves of unit circle:

c(t) = (cos(t),sin(t))

c'(t) = ( - sin(t),cos(t))   and   | c'(t) | = 1

thus, the pedal curve with pedal point (x,y) is:

(cos(t) - ycos(t)sin(t) + xsin(t)²,sin(t) - xsin(t)cos(t) + ycos(t)²)

If the pedal point is at the center (i.e. (0,0)), the circle is its own pedal curve. If the pedal point is (1,0) the pedal curve is

(cos(t) + sin(t)²,sin(t) - sin(t)cos(t)) = (1,0) + (1 - cos(t))c(t)

i.e. a pedal point on the circumference gives a cardioid.

External links

• Pedal and Contrapedal on MathWorld