What is "pedal curve"

Pedal \Pe"dal\, a. [L. pedalis, fr. pes, pedis, foot. See Foot, and cf. Pew.]

1. Of or pertaining to the foot, or to feet, literally or figuratively; specifically (Zo["o]l.), pertaining to the foot of a mollusk; as, the pedal ganglion.

2. Of or pertaining to a pedal; having pedals.

   Pedal curve or Pedal surface (Geom.), the curve or surface which is the locus of the feet of perpendiculars let fall from a fixed point upon the straight lines tangent to a given curve, or upon the planes tangent to a given surface.

   Pedal note (Mus.), the note which is held or sustained through an organ point. See Organ point, under Organ.

   Pedal organ (Mus.), an organ which has pedals or a range of keys moved by the feet; that portion of a full organ which is played with the feet.

n. (context geometry English) For a plane curve ''C'' and a given fixed point ''P'', the locus of points ''X'' such that ''PX'' is perpendicular to a tangent to the curve passing through ''X''.

The pedal curve results from the orthogonal projection of a fixed point on the tangent lines of a given curve. More precisely, for a plane curve C and a given fixed pedal point P, the pedal curve of C is the locus of points X so that the 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, C, C, etc., where C is the pedal of C, C is the pedal of C, and so on. In this scheme, C is known as the first positive pedal of C, C is the second positive pedal of C, and so on. Going the other direction, C is the first negative pedal of C, the second negative pedal of C, etc.