Pedal curve
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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).
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 |
|---|---|---|---|
| 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:
- <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