Arc

There are a number of meanings for the word "arc" in mathematics. In general, an arc is any smooth curve joining two points. The length of an arc is known as its arc length.

In a graph, a graph arc is an ordered pair of adjacent vertices.

In particular, an arc is any portion (other than the entire curve) of the circumference of a circle. An arc corresponding to the central angle is denoted . Similarly, the size of the central angle subtended by this arc (i.e., the measure of the arc) is sometimes (e.g., Rhoad et al. 1984, p. 421) but not always (e.g., Jurgensen 1963) denoted .

The center of an arc is the center of the circle of which the arc is a part.

An arc whose endpoints lie on a diameter of a circle is called a semicircle.

For a circle of radius , the arc length subtended by a central angle is proportional to , and if is measured in radians, then the constant of proportionality is 1, i.e.,

(1)

The length of the chord connecting the arc's endpoints is

(2)

As Archimedes proved, for chords and which are perpendicular to each other,

(3)

(Wells 1991).

An arc of a topological space is a homeomorphism , where is a subspace of . Every arc is a path, but not conversely. Very often, the name arc is given to the image of .

The prefix "arc" is also used to denote the inverse functions of trigonometric functions and hyperbolic functions. Finally, any path through a graph which passes through no vertex twice is called an arc (Gardner 1984, p. 96).