Arc -- From Wolfram MathWorld

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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.

Arc

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 ∠AOC is denoted arcAC. 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 marcAC.

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.

ArcAngle

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

l=rtheta. (1)

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

a=2rsin(1/2theta). (2)
ArcTheorem

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

marcAB+marcCD=marcBC+marcDA (3)

(Wells 1991).

An arc of a topological space X is a homeomorphism f:[0,1]->S, where S is a subspace of X. Every arc is a path, but not conversely. Very often, the name arc is given to the image S of f.

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).

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