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).

File:Pedal-curve-1.mng
Hypocycloid (black)
generates rose (red),
one cusp "swept" by tangent (blue)

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

$t\mapsto c(t)+{\langle c'(t),P-c(t)\rangle \over |c'(t)|^{2}}c'(t)$

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.

$t\mapsto P-{\langle c'(t),P-c(t)\rangle \over |c'(t)|^{2}}c'(t)$

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.

given curve	pedal point	pedal curve	contrapedal curve
line	any	point	parallel line
circle	on circumference	cardioid	—
parabola	on axis	conchoid of de Sluze	—
parabola	on tangent of vertex	ophiuride	—
parabola	focus	line	—
other conic section	focus	circle	—
logarithmic spiral	pole	congruent log spiral	congruent log spiral
epicycloid hypocycloid	center	rose	rose
involute of circle	center of circle	Archimedean spiral	the circle

Example

Pedal curves of unit circle:

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

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

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

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

$(\cos(t)-y\cos(t)\sin(t)+x\sin(t)^{2},\sin(t)-x\sin(t)\cos(t)+y\cos(t)^{2})$

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