Definitions of associated curves


Some definitions are below. A full set of definitions of terms used is available.

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'Hôpital 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 = r2. If P describes a curve C1 then Q describes a curve C2 called the inverse of C1 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.


Main index Famous curves index
Biographical Index Timelines
History Topics Index Birthplace Maps
Mathematicians of the day Anniversaries for the year
Societies, honours, etc Search Form

JOC/EFR/BS January 1997

The URL of this page is:
http://www-history.mcs.st-andrews.ac.uk/Curves/Definitions.html