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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".
Video about the history of pedals. |

If M_{0} is the current point on (G_{0}), 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 (G _{0}) 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 (G _{0}) is , then the polar equation of its pedal is . |

The *pedal* of a curve (G_{0}) 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 (G_{0}).

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

Finally, it is the inverse, with any circle with centre *O* as reference circle, of the polar of (G_{0}) 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 *contra-pedal* 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 (G_{0}). The contra-pedal is then none other than the evolute.

See also the pedal surfaces.

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© Robert FERRÉOL, Jacques MANDONNET 2017