Caustic curves : When light reflects off a curve then the envelope of the reflected rays is a caustic by reflection or acatacaustic. When light is refracted by a curve then the envelope of the refracted rays is a caustic by refraction or adiacaustic.
They were first studied byHuygens andTschirnhaus around1678.Johann Bernoulli,Jacob Bernoulli,de L'Hôpital andLagrange 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 inApollonius'sConics Book V. It appears in its present form inHuygens work from around1673.

Inverse curves : Given a circleC centreO radiusr then two pointsP andQ are inverse with respect toC ifOP.OQ =r². IfP describes a curveC₁ thenQ describes a curveC₂ called the inverse ofC₁ with respect to the circleC.
Although it does not make much geometric sense to take the circleC having negative radius, it makes no difference to the definition of the inverse of a point, except in this caseP andQ are on opposite sides ofO whereas whenr is positiveP andQ are on the same side ofO.

Involute : IfC is a curve andC' is its evolute, thenC is called an involute ofC'.
Any parallel curve toC is also an involute ofC'. 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 curveC andO a fixed point then for a pointP onC draw a line perpendicular toOP. The envelope of these lines asP describes the curveC is thenegative pedal ofC.
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 curveC then the pedal curve ofC with respect to a fixed pointO(called thepedal point) is the locus of the pointP of intersection of the perpendicular fromO to a tangent toC.