Pedal curve

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Limaçon — pedal curve of a circle
Limaçon — pedal curve of a circle
Another example : pedal curve of an ellipse
Another example : pedal curve of an ellipse
Contrapedal of the same ellipse
Contrapedal of the same ellipse
Pedal of the evolute of the ellipse : same as the contrapedal
Pedal of the evolute of the ellipse : same as the contrapedal

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, tangent to the curve, 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

[edit] 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) − ycos(t)sin(t) + xsin(t)2,sin(t) − xsin(t)cos(t) + ycos(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.


[edit] External links

v • d • e
Differential transforms of plane curves