Pedal curve

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 there 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

<math>t\mapsto c(t)+{\langle c'(t),P-c(t)\rangle\over|c'(t)|^2} c'(t)</math>

parametrises the pedal curve (disregarding points where c' is zero or undefined). For a parametrically defined curve, its pedal curve with pedal point (0;0) is defined as

<math>X[x,y]=\frac{(xy'-yx')y'}{x'^2 + y'^2}</math>

<math>Y[x,y]=\frac{(yx'-xy')x'}{x'^2 + y'^2}</math>

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

<math>t\mapsto P-{\langle c'(t),P-c(t)\rangle\over|c'(t)|^2} c'(t)</math>

With the same pedal point, the contrapedal curve is the pedal curve of the evolute of the given curve.

Example

Pedal curves of unit circle:

<math>c(t)=(\cos(t),\sin(t))</math>

<math>c'(t)=(-\sin(t),\cos(t))</math>   and   <math>|c'(t)|=1</math>

<math>{\langle c'(t),(x,y)-c(t)\rangle\over|c'(t)|^2}=y\cos(t)-x\sin(t)</math>

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

<math>(\cos(t)-y\cos(t)\sin(t)+x\sin(t)^2,\sin(t)-x\sin(t)\cos(t)+y\cos(t)^2)</math>

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

<math>(\cos(t)+\sin(t)^2,\sin(t)-\sin(t)\cos(t))=(1,0)+(1-\cos(t))c(t)</math>

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

External links

• Pedal and Contrapedal on MathWorld