Editing Pedal curve (section)

{{Short description|Curve generated by the projections of a fixed point on the tangents of another curve}}
[[Image:PedalConstruction.svg|200px|right|thumb|Geometric construction of the pedal of ''C'' with respect to ''P'']]
In mathematics, a '''pedal curve''' of a given curve results from the [[orthogonal projection]] of a fixed point on the [[tangent line]]s of this curve. More precisely, for a [[plane curve]] ''C'' and a given fixed ''pedal point'' ''P'', the '''pedal curve''' of ''C'' is the [[locus (mathematics)|locus]] of points ''X'' so that the [[line (geometry)|line]] ''PX'' is perpendicular to a [[tangent]] ''T'' to the curve passing through the point ''X''. Conversely, at any point ''R'' on the curve ''C'', let ''T'' be the tangent line at that point ''R''; then there is a unique point ''X'' on the tangent ''T'' which forms with the pedal point ''P'' a line [[perpendicular]] to the tangent ''T'' (for the special case when the fixed point ''P'' lies on the tangent ''T'', the points ''X'' and ''P'' coincide) – the pedal curve is the set of such points ''X'', called the ''foot'' of the perpendicular to the tangent ''T'' from the fixed point ''P'', as the variable point ''R'' ranges over the curve ''C''.

Complementing the pedal curve, there is a unique point ''Y'' on the line normal to ''C'' at ''R'' so that ''PY'' is perpendicular to the normal, so ''PXRY'' is a (possibly degenerate) rectangle. The locus of points ''Y'' is called the '''contrapedal curve.'''

The '''orthotomic''' of a curve is its pedal magnified by a factor of 2 so that the [[center of similarity]] is ''P''. This is locus of the reflection of ''P'' through the tangent line ''T''.

The pedal curve is the first in a series of curves ''C''<sub>1</sub>, ''C''<sub>2</sub>, ''C''<sub>3</sub>, etc., where ''C''<sub>1</sub> is the pedal of ''C'', ''C''<sub>2</sub> is the pedal of ''C''<sub>1</sub>, and so on. In this scheme, ''C''<sub>1</sub> is known as the ''first positive pedal'' of ''C'', ''C''<sub>2</sub> is the ''second positive pedal'' of ''C'', and so on. Going the other direction, ''C'' is the ''first negative pedal'' of ''C''<sub>1</sub>, the ''second negative pedal'' of ''C''<sub>2</sub>, etc.<ref>Edwards p. 165</ref>