# Definitions of associated curves

**Caustic curves :** When light reflects off a curve then the envelope of the reflected rays is a caustic by reflection or a *catacaustic*. When light is refracted by a curve then the envelope of the refracted rays is a caustic by refraction or a *diacaustic*.

They were first studied by Huygens and Tschirnhaus around 1678. Johann Bernoulli, Jacob Bernoulli, de L'HÃ´pital and Lagrange all studied caustic curves.

**Evolute :** The envelope of the normals to a given curve.

This can also be thought of as the locus of the centres of curvature.

The idea appears in an early form in Apollonius's *Conics* Book V. It appears in its present form in Huygens work from around 1673.

**Inverse curves :** Given a circle *C* centre *O* radius *r* then two points *P* and *Q* are inverse with respect to *C* if *OP*.*OQ* = *r*^{2}. If *P* describes a curve *C*_{1} then *Q* describes a curve *C*_{2} called the inverse of *C*_{1} with respect to the circle *C*.

Although it does not make much geometric sense to take the circle *C* having negative radius, it makes no difference to the definition of the inverse of a point, except in this case *P* and *Q* are on opposite sides of *O* whereas when *r* is positive *P* and *Q* are on the same side of *O*.

**Involute :** If *C* is a curve and *C*' is its evolute, then *C* is called an involute of *C*'.

Any parallel curve to *C* is also an involute of *C*'. Hence a curve has a unique evolute but infinitely many involutes.

Alternatively an involute can be thought of as any curve orthogonal to all the tangents to a given curve.

**Negative pedal :** Given a curve *C* and *O* a fixed point then for a point *P* on *C* draw a line perpendicular to *OP*. The envelope of these lines as *P* describes the curve *C* is the *negative pedal* of *C*.

The ellipse is the negative pedal of a circle if the fixed point is inside the circle while the negative pedal of a circle from a point outside is a hyperbola.

**Pedal curve :** Given a curve *C* then the pedal curve of *C* with respect to a fixed point *O* (called the *pedal point*) is the locus of the point *P* of intersection of the perpendicular from *O* to a tangent to *C*.