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PEDAL OF A CURVE

Notion studied by Roberval in 1693, Maclaurin in 1718,
Steiner in 1840, Terquem, who named it, in 1847.
From the Greek, pous, podos "foot". |

If M_{0} is the current
point on ,
then the current point M on the pedal is defined by ,
which gives:
in Cartesian coordinates, in complex parametrization, and in polar coordinates, with ; we thus have (see the notations). If the tangential equation of is f(u, v, w) = 0 (which means f(u, v, w) =
0 is a condition for the line
to be tangent to (G_{0})),
then the polar equation of the pedal with respect to O is
and its Cartesian equation is .
If the polar equation of the curve is , then the polar equation of its pedal is . |

The *pedal* of a curve
with respect to a point *O* (or with pole *O*) is the locus of
the feet of the lines passing by *O* perpendicular to the tangents
to the curve .

Therefore, it is also the envelope of the circles with
diameter [OM], when _{0}Mdescribes
(this property provides a construction of the normal, and therefore of
the tangent to the pedal)._{0} |
Prove that ! |

Finally, it is the inverse,
with any circle with centre *O* as reference circle, of the polar
of with
respect to this circle.

The pedal is the homothetic image of the orthotomic.

The curve the pedal of which is a given curve is called the negative pedal.

Examples:

- The *pedals of parabolas* with respect to a point
different from the focus, the cissoids
of a circle and a line with respect to a point on the circle, and the set
of the rational
circular cubics coincide. More precisely: the pedal with respect to
*O*
of the parabola with focus *F* and tangent at the vertex (*T*)
is the cissoid with pole *O* of the circle with diameter [*OF*]
and the line (*D*), image of (*T*) by the translation of .

- The *pedals of centred conics*, the cissoids
of two circles with respect to a point on one of them, and the set of the
rational
bicircular quartics coincide.

Other examples regrouped in a table:

negative pedal
(or orthocaustic) |
pole (position with respect to the negative pedal) | pole (position with respect to the pedal) | pedal |

line | any point | any point | point (projection of the pole on the line) |

parabola | focus | outside the line | line (tangent at the vertex of the parabola) |

" | different from the focus | singularity | rational circular cubic |

" | inside the parabola | isolated point | acnodal rational circular cubic |

" | on the internal part of the axis of the parabola | Sluze cubic | |

" | at the middle of the segment line [SF] | isolated point | visiera |

" | on the parabola | cuspidal point | cissoid |

" | at the vertex | cuspidal point | cissoid of Diocles |

" | outside the parabola | double point | crunodal rational circular cubic |

" | on the tangent at the vertex | double point | ophiuroid |

" | on the directrix | double point | strophoid |

" | foot of the directrix | double point | right strophoid |

" | symmetric image of the focus about the directrix | double point | Maclaurin trisectrix |

centred conic | focus | outside the circle | (principal) circle (of the conic) |

" | different from the focus | real singularity | rational bicircular quartic |

" | centre | real singularity | Booth curve |

circle | outside the circle | double point | limaçon of Pascal with a loop |

" | on the circle | cuspidal point | cardioid |

" | inside the circle | isolated point | limaçon of Pascal without a loop |

" | centre | centre | same circle |

rectangular hyperbola | centre | double point | lemniscate of Bernoulli |

Tschirnhausen cubic | focus (at the 8/9-th of the segment line [double point, vertex]) | focus | parabola |

cissoid of Diocles | point with coordinates (4a, 0) |
vertex | cardioid |

cardioid | cuspidal point | vertex of the loop | Cayley sextic |

cardioid | centre of the conchoidal circle | vertex of the loop | trisectrix limaçon |

cardioid | point with coordinates (-a,0) |
triple point | nephroid of Freeth |

deltoid | any point | folium | |

deltoid | inside the deltoid | trifolium | |

deltoid | on a symmetry axis of the deltoid | right folium | |

deltoid | on the deltoid | bifolium | |

deltoid | centre | regular trifolium | |

deltoid | cuspidal point | simple folium | |

deltoid | vertex | regular bifolium | |

centred cycloid | centre | centre | rose |

astroid | any point | beetle | |

astroid | centre | centre | rose with four branches |

paracycloid | centre of the frame | pole | spiral |

hypercycloid | centre of the frame | pole | spiral |

sinusoidal
spiral with parameter
n = –1/m |
centre | centre | sinusoidal
spiral with parameter n/(n+1) = –1/(m–1) |

logarithmic spiral | centre | centre | logarithmic spiral |

involute of a circle | centre | centre | Archimedean spiral |

hyperbolic spiral | centre | centre | tractrix spiral |

Norwich spiral | centre | centre | Galilean spiral |

Maltese cross | centre | centre | double egg |

Talbot curve | centre | centre | ellipse |

evolute of a centred conic | focus | Jerabek curve |

Let us state the beautiful Steiner-Habich theorem:
If a curve ( C) rolls on a line (D), and
if (R) is the roulette described
by a point M of the plane of this curve, then a copy of the pedal
(P) of (C) with respect to M can roll on (R),
in such a way that the point M describes the line (D).
Then, the couple (( R), (P)) is a wheel-road
couple. See many examples on the latter link. |

The *contrapedal* of (G_{0})
with respect to *O* can be defined as the locus of the feet of the
lines passing by *O* perpendicular to the *normals* to .
The contrapedal is then none other than the evolute.

See also the pedal
surfaces.

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© Robert FERRÉOL 2017