the curve C′. 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. Then when the curves touch at R′ the point corresponding to P on the moving plane is X, and so the roulette is the pedal curve. Equivalently, the orthotomic of a curve is the roulette of the curve on its mirror image. Example When C is a circle the above discussion shows that the following definitions of a limaçon are equivalent: It is the pedal of a circle. It is the envelope of circles whose diameters have one endpoint on a fixed point and another endpoint which follow a circle. It is the envelope of circles through a fixed point whose centers follow a circle. It is the roulette formed by a circle rolling around a circle with the same radius. We also have shown that the catacaustic of a circle is the evolute of a limaçon. Pedals of specific curves Pedals of some specific curves are: See also List of curves References Notes Sources Further reading Differential and integral calculus: with applications by George Greenhill (1891) p326 ff. (Internet Archive) "Note on the Problem of Pedal Curves" by Arthur Cayley External links Differential geometry Curves