Pedal curve

From Academic Kids

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).

Missing image
Pedal-curve-1.mng
Image: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

<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).

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

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

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
lineanypointparallel line
circleon circumferencecardioid
parabolaon axisconchoid of de Sluze
parabolaon tangent
of vertex
ophiuride
parabolafocusline
other conic sectionfocuscircle
logarithmic spiralpolecongruent log spiralcongruent log spiral
epicycloid
hypocycloid
centerroserose
involute of circlecenter of circleArchimedean spiralthe circle

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 (http://mathworld.wolfram.com/PedalCurve.html) and Contrapedal (http://mathworld.wolfram.com/ContrapedalCurve.html) on MathWorld
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