Pedal curve

The locus of points Y is called the contrapedal curve.

For a curve given by the equation F(x, y)=0, if the equation of the tangent line at R=(x0, y0) is written in the form then the vector (cos α, sin α) is parallel to the segment PX, and the length of PX, which is the distance from the tangent line to the origin, is p. So X is represented by the polar coordinates (p, α) and replacing (p, α) by (r, θ) produces a polar equation for the pedal curve.

[2] For example,[3] for the ellipse the tangent line at R=(x0, y0) is and writing this in the form given above requires that The equation for the ellipse can be used to eliminate x0 and y0 giving and converting to (r, θ) gives as the polar equation for the pedal.

This is easily converted to a Cartesian equation as For P the origin and C given in polar coordinates by r = f(θ).

Then the vertex of this angle is X and traces out the pedal curve.

Hence the pedal is the envelope of the circles with diameters PR where R lies on the curve.

Let C′ be the curve obtained by shrinking C by a factor of 2 toward P. Then the point R′ corresponding to R is the center of the rectangle PXRY, and the tangent to C′ at R′ bisects this rectangle parallel to PY and XR.

The reflected ray, when extended, is the line XY which is perpendicular to the pedal of C. The envelope of lines perpendicular to the pedal is then the envelope of reflected rays or the catacaustic of C′.

This proves that the catacaustic of a curve is the evolute of its orthotomic.

As noted earlier, the circle with diameter PR is tangent to the pedal.

Let D′ be a curve congruent to C′ and let D′ roll without slipping, as in the definition of a roulette, on C′ so that D′ is always the reflection of C′ with respect to the line to which they are mutually tangent.