In the
differential geometry of curves
Differential geometry of curves is the branch of geometry that deals with smooth curves in the plane and the Euclidean space by methods of differential and integral calculus.
Many specific curves have been thoroughly investigated using the ...
, the evolute of a
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
is the
locus of all its
centers of curvature. That is to say that when the center of curvature of each point on a curve is drawn, the resultant shape will be the evolute of that curve. The evolute of a circle is therefore a single point at its center. Equivalently, an evolute is the
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter or card.
Traditional envelopes are made from sheets of paper cut to one of three shapes: a rhombus, a ...
of the
normals to a curve.
The evolute of a curve, a surface, or more generally a
submanifold
In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...
, is the
caustic
Caustic most commonly refers to:
* Causticity, a property of various corrosive substances
** Sodium hydroxide, sometimes called ''caustic soda''
** Potassium hydroxide, sometimes called ''caustic potash''
** Calcium oxide, sometimes called ''caust ...
of the normal map. Let be a smooth, regular submanifold in . For each point in and each vector , based at and normal to , we associate the point . This defines a
Lagrangian map, called the normal map. The caustic of the normal map is the evolute of .
Evolutes are closely connected to
involute
In mathematics, an involute (also known as an evolvent) is a particular type of curve that is dependent on another shape or curve. An involute of a curve is the locus of a point on a piece of taut string as the string is either unwrapped from o ...
s: A curve is the evolute of any of its involutes.
History
Apollonius ( 200 BC) discussed evolutes in Book V of his ''Conics''. However,
Huygens is sometimes credited with being the first to study them (1673). Huygens formulated his theory of evolutes sometime around 1659 to help solve the problem of finding the
tautochrone curve, which in turn helped him construct an isochronous pendulum. This was because the tautochrone curve is a
cycloid
In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another cu ...
, and the cycloid has the unique property that its evolute is also a cycloid. The theory of evolutes, in fact, allowed Huygens to achieve many results that would later be found using calculus.
Evolute of a parametric curve
If