pedal curve


Given a curve C1 and a (pedal) point O, construct for each tangent l of C1 a point P, for which OP is perpendicular to the tangent. The collection of points P forms a curve C2, the (positive) pedal of C1 (with respect to the pedal point).
When C1 is given by (x, y) = (f(t), g(t)), and we translate C1 in such a way that the pedal point is the origin, then C2 has the form:


Two curves are invariant for making a pedal:

The pedal of the parabola is the curve given by the equation:


a pedal point pedal of the parabola
0 vertex cissoid (of Diocles)
1 foot of - intersection of axis and - directrix (right) strophoid
3 reflection of focus in directrix trisectrix of MacLaurin
- on directrix oblique strophoid
- focus line

Some other pedals are:

curve pedal point pedal
astroid center quadrifolium
cardioid cusp Cayley's sextic
circle any point limašon
  on circle cardioid
cissoid (of Diocles) focus cardioid
deltoid cusp simple folium
  vertex regular bifolium
  center trifolium
  on the curve bifolium
ellipse focus circle
epicycloid center rhodonea
equiangular spiral pole equiangular spiral
hyperbola focus circle
rectangular hyperbola center lemniscate
hypocycloid center rhodonea
involute of a circle center Archimedes' spiral
line any point point
parabola focus line
  on directrix strophoid
  center of directrix right strophoid
  reflection of focus by directrix trisectrix of Maclaurin
  vertex cissoid
sinusoidal spiral pole sinusoidal spiral
Talbot's curve center ellipse
Tschirnhausen's cubic focus parabola

The reverse operation of making a pedal is to construct from each point P of C2 a line l that is perpendicular to OP. The lines l together form an envelope of the curve C1. Now we call C1 the negative pedal 1) of C2. When C1 is a pedal of C2, then C2 is the negative pedal of C1.
Because of this definition, the curve is in fact also an orthocaustic: the orthocaustic of a curve C1 (with respect to a point O) is the envelope of the perpendiculars of P on OP (P on C1).

Instead of tangents to a curve we can consider normals to that curve. This pedal curve is called the normal pedal curve.

MacLaurin was the first author to investigate pedal curves (1718).


1) In French: antipodaire. In German: Gegenfusspunktskurve.