Circle

A closed plane curve all points of which are at the same distance from a given point (the centre of the circle) and lie in the same plane as the curve. Circles with a common centre are called concentric. A segment joining the centre of the circle to any point on it (as well as the length $R$ of this segment) is called a radius. The equation of the circle in Cartesian coordinates is

$$(x-a)^2+(y-b)^2=R^2,$$

where $a$ and $b$ are the coordinates of the centre.

A straight line passing through two points of the circle is called a secant; the segment of it which lies within the circle is called a chord. Chords which are equidistant from the centre are equal. A chord passing through the centre of the circle is called its diameter. The diameter perpendicular to a chord divides it in half.

The two parts into which two points of the circle divide the circle are called arcs.

An angle formed by two radii of the circle joining its centre to the ends of an arc is called a central angle, and the corresponding arc is the arc on which it depends. The angle formed by two chords with a common end is called an inscribed angle. An inscribed angle is equal to half the central angle depending on the arc confined between the ends of the inscribed angle. The length of the circle equals $C=2\pi R$, while the length of an arc is $l=(\pi Ra^\circ)/180^\circ=R\alpha$, where $a^\circ$ is the size (in degrees) of the relevant central angle and $\alpha$ is its radial measure.

If through any point of a plane several secants are drawn towards the circle, the product of the distances from that point to both points of intersection of each secant with the circle is a constant number (for the given point); in particular, it is equal to the square of the length of the segment touching the circle from that point (the power of the point). The totality of all circles in the plane in relation to which a given point has an identical power is a bundle of circles. The totality of all common circles of two bundles in one plane is called a pencil of circles.

The part of the plane bounded by a circle and containing its centre is called a disc. The part of the disc bounded by an arc of the circle and by the radii leading to the ends of this arc is called a sector. The part of the disc between an arc and its chord is called a segment.

The area of the disc is $S=\pi R^2$, the area of a sector is $S_1=\pi R^2(a^\circ/360^\circ)$, where $a^\circ$ is the measure in degrees of the appropriate central angle, while the area of a segment is $S_2=\pi R^2(a^\circ/360^\circ)\pm S_\Delta$, where $S_\Delta$ is the area of the triangle with vertices at the centre of the circle and at the ends of the radii bounding the relevant sector. The sign "-" is used if $a^\circ<180^\circ$, and the sign "+" if $a^\circ>180^\circ$.

A circle on a convex surface is locally almost isometric to the boundary of a cone of the convex surface (Zalgaller's theorem). A circle on a manifold of bounded curvature can have a highly complex structure (i.e. corner and multiple points may exist, the circle may have several components, etc.). Nevertheless, the points of a circle on a manifold of bounded curvature can be ordered naturally, turning the curve into a cyclically ordered set (see [1]).

For circles in more general spaces, such as Banach, Finsler and other spaces, see Sphere.

References

[1]	, Enzyklopaedie der Elementarmathematik , Deutsch. Verlag Wissenschaft. (1969) (Translated from Russian)
[2]	Yu.D. Burago, M.B. Stratilatova, "Circumferences on a surface" Proc. Steklov. Inst. Math. , 76 (1967) pp. 109–141 Trudy Mat. Inst. Steklov. , 76 (1965) pp. 88–114

Comments

For circles in more general spaces, see also [a1], Chapt. 10.

References

[a1]	M. Berger, "Geometry" , I , Springer (1987) pp. Chapt. 4
[a2]	Yu.D. Burago, V.A. Zalgaller, "Geometric inequalities" , Springer (1988) (Translated from Russian)
[a3]	J.L. Coolidge, "A treatise on the circle and the sphere" , Chelsea, reprint (1971)