1.5: Pedal Curves

A point of the pedal curve ZC(t) of a plane curve X(t) is defined to be the foot of the perpendicular from C to the tangent line to the curve at X(t). We may express this by the following formula:

ZC(t) = C + ((X(t)-C)·U(t))U(t)

Pedal curves express various properties of the original curve, and lead to some intereseting global theorems for closed plane curves.


This demo allows you to examine the pedal curve ZC(t) of a plane curve X(t). The point C is indicated by a white hotspot which you can move around in the plane. You can see both the tangent line to the curve at X(t) (in cyan) and the perpendicular (in pink) from C to the tangent line. The intersection of these two lines is ZC(t0) which is shown as a red dot. This point traces out the magenta pedal curve as you scroll through the tapedeck controlling t0.



1. What happens to the curve ZC(t) as C changes? What is the nature of the pedal curve if the original curve has an inflection point? What if the original curve has an ordinary or rhamphoid cusp?

2. What is the condition for the pedal curve to intersect the original curve?

3. What familty of curves arise as the pedal curves of a circle?

4. What types of cusps does the pedal curve to (t2, t3 - t/2) have for various positions of the pedal point?
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