Pedal Curve

above: Family of Pedals of a Sinusoid.

Mathematica Notebook for This Page.

History

From Robert Yates:

The idea of positive and negative pedal curves occurred first to Colin Maclaurin↗ in 1718; the name “pedal” is due to Terquem. The theory of Caustic Curves includes Pedals in an important role: the orthotomic is an enlargement of the pedal of the reflecting curve with respect to the point source of light (Quetelet↗, 1822). (See Caustics.) The notion may be enlarged upon to include loci formed by dropping perpendiculars upon a line making a constant angle with the tangent — viz., pedals formed upon the normals to a curve.

Description

Pedal and negative pedal are methods of deriving a new curve based on a given curve and a point.

Step by step description for positive pedal:

1. Given a curve and a fixed point O.
2. Draw a tangent at any point P on the curve.
3. Mark a point Q on this tangent so that line PQ and line OQ are penpendicular.
4. Repeat step (2) and (3) for every point P on the curve. The locus of Q is the pedal of the given curve with respect to point O.

above: A pedal of a sinusoid with respect to a point below the curve. This is one of the curves shown in the figure at the top of this page.

Pedal and negative pedal are inverse concepts. Negative pedal of a curve C can be defined as a curve C' such that the pedal of C is C'.

Step by step description for negative pedal:

1. Give a curve and a fixed point O.
2. Draw a line from O to any point P on the curve.
3. Draw a line perpendicular to line OP and passing P.
4. Repeat step (2) and (3) for every point P on the curve. The envelope of lines is the negative pedal of the given curve with respect to point O.

above: The pedal of a parabola with respect to its focus is a line. The negative pedal of a line is a parabola.

Formulas

The pedal of a parametric curve {xf[t],yf[t]} with respect to point {a,b} is:

{(xf'[t]^2 a + yf'[t]^2 xf[t] + xf'[t] yf'[t] (b - yf[t]))/(xf'[t]^2 + yf'[t]^2),
 (yf'[t]^2 b + xf'[t]^2 yf[t] + xf'[t] yf'[t] (a - xf[t]))/(xf'[t]^2 + yf'[t]^2)}

Properties

Curve relations by pedal

Related Web Sites

See: Websites on Plane Curves, Printed References On Plane Curves.

Robert Yates: Curves and Their Properties.