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 M0 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 (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 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 [OM0], when M0 describes (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 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 (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 . The contrapedal is then none other than the evolute.