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PEDAL OF A CURVE
En bleu l'antipodaire, en rouge la podaire

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 M0 is the current point on (G0), 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 (G0) is f(u, v, w) = 0 (which means f(u, v, w) = 0 is a condition for the line  to be tangent to (G0)), then the polar equation of the pedal with respect to O is  and its Cartesian equation is .
If the polar equation of the curve (G0) is , then the polar equation of its pedal is .

The pedal of a curve (G0) 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 (G0).
 
 
Therefore, it is also the envelope of the circles with diameter [OM0], when M0 describes (G0) (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 (G0) 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/(m1)
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 (G0) with respect to O can be defined as the locus of the feet of the lines passing by O perpendicular to the normals to (G0). The contrapedal is then none other than the evolute.

See also the pedal surfaces.
 
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© Robert FERRÉOL 2017