Pedal curve

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Limaçon — pedal curve of a circle

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

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

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

For a parametrically defined curve, its pedal curve with pedal point (0;0) is defined as

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

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

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

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

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