Circle

This article is about the geometric curve. For other uses, see Circle (disambiguation).

A circle is the locus of points in a plane that are a given distance from a given point, called the center of the circle. Equivalently, it is a two-dimensional cog.

The distance from the center of the circle to a point on the circle is called the circle's radius, and is frequently abbreviated r. Twice the radius—the maximum distance between two points on the circle—is called the diameter (sometimes abbreviated d). These same words are also used respectively for a line segment connecting the center and a point on the circle, and for a segment connecting two points on the circle and passing through the center—the lengths of these segments being of course the distances of the corresponding names. A circle with a radius of 1 is called a unit circle, and has some special applications—it can be used, for instance, to define the basic trigonometric functions. A circle of radius zero comprises a single point.

In parabolic geometry, given any set of three noncollinear points, there exists a unique circle that passes through those points. A line can be considered to be a degenerate circle of infinite radius, in which case the noncollinearity condition is unnecessary. This rule also holds in elliptic geometry (sans the restriction to noncollinearity), but in not in hyperbolic; in hyperbolic geometry any three noncollinear points determine a unique cycle—that is, either a unique circle, a unique horocycle, or a unique hypercycle, but each particular set of three points determines only one of these three curves.

The curvature of a circle of radius [math]\displaystyle{ r }[/math] is [math]\displaystyle{ \frac{1}{r} }[/math].

In topological contexts, a circle is a compact, connected, one-dimensional manifold—and in fact is the only compact, connected, one-dimensional manifold. Any simple closed curve is homeomorphic to a circle. This includes any simple polygons, as well as more exotic shapes such as real projective lines.

Circles often pay an important role in the physics of two-dimensional worlds. Force isodynamics about point sources are usually circular in shape; radiation propagates in circular fronts; etc. Two-dimensional mounds are often close to circular in outline.

Equations

In Cartesian coordinates, the circle can be described by the equation [math]\displaystyle{ (x-h)^2 + (y-k)^2 = r^2 }[/math], where [math]\displaystyle{ (h, k) }[/math] is the center of the circle and [math]\displaystyle{ r }[/math] is the radius. While this is considered the "standard form" of the equation of the circle, it can also be written in a "general form" where each term is a constant times a power of x or y; the general form of the circle is [math]\displaystyle{ x^2 + y^2 - 2hx - 2ky + \left(h^2 + k^2 - r^2\right) = 0 }[/math]. For instance, for a circle with radius 3 centered at the point [math]\displaystyle{ (1, -2) }[/math], the standard form of the equation is [math]\displaystyle{ (x-1)^2 + (y+2)^2 = 9 }[/math], and the general form is [math]\displaystyle{ x^2 + y^2 - 2x + 4y - 4 = 0 }[/math]. For a circle centered at the origin, the standard form reduces to [math]\displaystyle{ x^2 + y^2 = r^2 }[/math] and the general form to the trivially equivalent [math]\displaystyle{ x^2 + y^2 - r^2 = 0 }[/math]. (Converting from the standard form of a circle to the general form is a simple matter of expanding the squares of binomials and combining like terms; converting from the general form to the standard form is a bit trickier but can be done with techniques such as completing the square.)

In polar coordinates, a circle centered on the origin has the simple equation [math]\displaystyle{ \rho = r }[/math]. If the circle is not centered at the origin, the equation can be written as [math]\displaystyle{ \rho^2 = 2 h \rho \cos \theta + 2 k \rho \sin \theta + (r^2 - h^2 - k^2) }[/math]. In log-polar coordinates, a circle centered on the origin has the equation [math]\displaystyle{ \rho = \ln r }[/math]; the equation for a circle centered elsewhere is rather more complicated.

A circle can also be defined parametrically by the equations [math]\displaystyle{ x = h + r \cos t }[/math] and [math]\displaystyle{ y = k + r \sin t }[/math], among other equivalent formulations.

Circumference and area formulas

The length of the perimeter of the circle is called the circle's circumference. In parabolic geometry, the circumference of a circle is equal to the length of the diameter times the transcendental number pi (π)—or, equivalently, to [math]\displaystyle{ \pi d }[/math], or [math]\displaystyle{ 2\pi r }[/math]. This was, in fact, the original basis for the definition of pi, though it has since been defined in many other ways that don't rely on parabolic geometry. The area of the circle, in parabolic geometry, is equal to [math]\displaystyle{ \pi r^2 }[/math].

In nonparabolic geometry, of course, these values vary. In uniform elliptic geometry with curvature K, a circle of radius [math]\displaystyle{ r }[/math] has a circumference of [math]\displaystyle{ \frac{2\pi}{\sqrt{K}}\sin\left(r\sqrt{K}\right) }[/math] and an area of [math]\displaystyle{ \frac{4\pi}{K} \sin^2 \left(\frac{r\sqrt{K}}{2}\right) }[/math]; in uniform hyperbolic geometry with curvature K, such a circle has a circumference of [math]\displaystyle{ \frac{2\pi}{\sqrt{-K}}\sinh\left(r\sqrt{-K}\right) }[/math] and an area of [math]\displaystyle{ -\frac{4\pi}{K} \sinh^2 \left(\frac{r\sqrt{-K}}{2}\right) }[/math], where "sinh" represents the hyperbolic sine. In terms of the radius of curvature [math]\displaystyle{ R = \frac{1}{\sqrt{|K|}} }[/math], these formulæ simplify to [math]\displaystyle{ C = 2 \pi R \sin \frac{r}{R} }[/math] and [math]\displaystyle{ A = 4 \pi R^2 \sin^2 \frac{r}{2 R} }[/math] for elliptic geometry and [math]\displaystyle{ C = 2\pi R \sinh \frac{r}{R} }[/math] and [math]\displaystyle{ A = 4 \pi R^2 \sinh^2 \frac{r}{2 R} }[/math] for hyperbolic.

Associated curves and manifolds

Because circles are such a simple and fundamental geometric shape, there are many other curves and manifolds that have been defined in relation to it—as parts of the circle, or resulting from transformations of the circle, or as the envelope of lines or locus of points generated by processes performed with or on the circle.

The piece of a circle between two points is called a circular arc—or just an "arc" if from context it's clear that it's the arc of a circle that's referred to. More formally, an arc is the part of the circle subtended by an angle with its vertex at the circle's center, which is called a central angle; the measure of the arc is defined as the same as the measure of the central angle that subtends it. The length of the arc is simply [math]\displaystyle{ \theta d }[/math] or [math]\displaystyle{ 2 \theta r }[/math], where [math]\displaystyle{ \theta }[/math] is the measure of the arc in radians. The line segment connecting the two endpoints of an arc is called a chord; a line that includes a chord is called a secant. The sagitta of an arc is a measurement equal to the distance from the arc's midpoint to the center of the corresponding chord, and is equal to [math]\displaystyle{ 2 r \sin^2 \frac{\theta}{2} }[/math], where [math]\displaystyle{ r }[/math] is the radius of the circle and [math]\displaystyle{ \theta }[/math] is the measure of the arc. An arc with a measure of \pi radians (180°) is a semicircle; its chord is a diameter of the circle, and its sagitta is equal to the circle's radius.

A trochoid is the roulette traced by a point fixed relative to a circle as that circle rolls along a line. If the point is outside the circle, it is a prolate trochoid; if inside the circle, a curtate trochoid; and if the point is on the circle itself, then the traced curve is a cycloid. If instead of a line, the circle rolls around the outside of another circle, then the roulette produced is an epitrochoid—or an epicycloid, if the point lies directly on the rolling circle—; if the circle rolls around the inside of another circle, the roulette produced is a hypotrochoid—or hypocycloid, if the point is on the rolling circle.

Angles and polygons

An angle is said to be an inscribed angle of the circle if the angle's vertex lies on the circle and each of the angle's rays intersects the circle in an additional point. An angle is said to be a circumscribed angle of the circle if its vertex is exterior to the circle and both its rays are tangent to the circle. If every vertex of a polygon lies on a circle, then every angle of the polygon is an inscribed angle of that circle, and the polygon is said to be inscribed in the circle. Conversly, it can also be said that the circle is circumscribed around the polygon. If each side of a polygon is tangent to the circle, then every angle of the polygon is a circumscribed angle of the circle, and the polygon itself is said to be circumscribed around the circle, or, equivalently, the circle is inscribed in the polygon. Not all polygons can have circles inscribed in them, or have circles circumscribed about them. If a circle can be inscribed in a polygon, then that circle is unique, and is called the polygon's incircle. A polygon that has an incircle is called a tangential polygon. Likewise, if a circle can be circumscribed around a polygon, then that circle is unique, and is called the polygon's circumcircle. A polygon that has a circumcircle is called a cyclic polygon. All tangential polygons are convex, and all cyclic polygons are strictly convex; however, not all convex polygons are tangential or cyclic.

For example, all triangles are both tangential and cyclic. All regular polygons, including squares, are both tangential and cyclic, while non-square rectangles are cyclic but not tangential, and non-square rhombi are tangential but not cyclic. Parallelograms which are neither rectangles nor rhombi are neither cyclic nor tangential.

A circle can be thought of as a limit of a sequence of regular polygons as the number of sides approaches infinity and the length of each side approaches zero) while either the apothem or the perithem retains a fixed value. If the apothem is fixed, then all the polygons in the sequence are circumscribed around the limit circle; if the perithem is fixed, then all the polygons in the sequence are inscribed in the limit circle. As the number of sides increases, the perimeters and areas of the polygons in the two sequences both converge to those of the circle from opposite sides—indeed, this fact has been used since antiquity to estimate the circumference and/or area of the circle and hence the value of [math]\displaystyle{ \pi }[/math].

In the Poincaré disk model of hyperbolic geometry, lines, horocycles, and hypercycles all appear as circles or circular arcs (or as straight lines, which, again, may be considered circles of infinite radius). This does not, however, hold for other models of hyperbolic geometry, and does not imply that these curves are circles in any deeper sense.

Two-dimensional manifolds

The interior of a circle—the locus of points such that the distance from the center to the point is less than a given value—is called a disc. The area bounded by two radii and an arc that they intercept is called a sector. The area bounded by a chord of the circle and the arc that it intercepts is called a circular segment.

The trace of a circle as it moves along a line perpendicular to the circle is a right circular cylinder; if it moves along a line not perpendicular to the circle, the trace is an oblique circular cylinder.

Rotating a circle about a line that includes a diameter traces a sphere; rotating a circle about any other line coplanar to the circle traces a torus—a ring torus if the point is outside the circle; a spindle torus if the point is inside the circle; a horn torus if the point is on the circle itself. Rotating a circle about a line not in the plane of the circle produces a surface of revolution the meridianal sections of which are anacardioids, quartic curves satisfying the equation [math]\displaystyle{ \left(x^2+y^2-a^2\right)^2=b^2\left(x^2-c^2\right) }[/math]. (The torus can be considered a special case of this where [math]\displaystyle{ b=0 }[/math], in which case the anacardioid reduces to a pair of circles.)

Properties

The circle has a number of special properties. In parabolic geometry, it is the only two-dimensional manifold with constant nonzero curvature, and the only such manifold with infinite axes of rotational symmetry. It also has the smallest ratio of perimeter to area of any shape. Every circle is similar to every other circle.

There are also consistent relationships between the lengths of various angles and segments related to a circle in parabolic geometry. The measure of an inscribed angle of a circle is always equal to exactly half the measure of the arc it subtends, or of the central angle that subtends that same arc; a circumscribed angle of a circle is always [[supplementary angle|supplementary] to the central angle that subtends the same arc. As a trivial corollary of the former fact, any inscribed angle that subtends the diameter of a circle is always a right angle. Two chords of a circle are congruent if and only if they are the same distance from the circle's center. A radius of the circle that intersects a chord always bisects that chord—and, conversely, the perpendicular bisector of a chord always passes through the circle's center.

Through any point [math]\displaystyle{ P }[/math], if we draw a line that intersects a given circle, then the product of the distance from [math]\displaystyle{ P }[/math] to the nearer point of intersection and the distance from [math]\displaystyle{ P }[/math] to the farther point of intersection is a constant no matter where we draw the line. (If we draw a tangent line, then there is only one point of intersection and the two distances are equal, but the relationship still holds. Similarly, if [math]\displaystyle{ P }[/math] actually lies on the circle, then the distance from point [math]\displaystyle{ P }[/math] to the nearer point of intersection is zero and the relationship, again, still holds, with the constant in question being zero.) This constant is called the power of point [math]\displaystyle{ P }[/math] (relative to the circle in question), and can be written as [math]\displaystyle{ \Pi(P) = OP^2 - r^2 }[/math], where [math]\displaystyle{ O }[/math] is the circle's center and [math]\displaystyle{ r }[/math] its radius. Note that [math]\displaystyle{ \Pi(P) }[/math] is negative when [math]\displaystyle{ P }[/math] is outside the circle—and zero if the point lies on the circle itself.

There are three special cases of this "power of a point theorem" that are often taught in basic geometry as separate theorems:

• The chord theorem, or intersecting chords theorem: When two chords of a circle intersect, the products of the measures of the two segments into which the point of intersection divides one chord is equal to the same product for the other. (That is, for example, if chord [math]\displaystyle{ \overline{AB} }[/math] and chord [math]\displaystyle{ \overline{CD} }[/math] intersect at point [math]\displaystyle{ X }[/math], then [math]\displaystyle{ AX \cdot XB = CX \cdot XD }[/math].)
• The secant theorem, or intersecting secants theorem: When two secants of a circle intersect outside the circle, the products of the measures of the point of intersection of the secants to each point where one of the secants intersects the circle are equal. (That is, for example, if two secants intersect each other at point [math]\displaystyle{ X }[/math], and one secant intersects the circle at points [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math] and the other at points [math]\displaystyle{ C }[/math] and [math]\displaystyle{ D }[/math], then [math]\displaystyle{ XA \cdot XB = XC \cdot XD }[/math].)
• The tangent-secant theorem: When a tangent and a secant of a circle intersect, then the product of the distances from the point of intersection to each point where the secant intersects the circle equals the square of the distance from the point of intersection to the point where the tangent intersects the circle. (That is, for example, if the secant and the tangent intersect at point [math]\displaystyle{ X }[/math], the secant intersects the circle at points [math]\displaystyle{ A }[/math] and [math]\displaystyle{ B }[/math], and the tangent intersects the circle at point [math]\displaystyle{ C }[/math], then [math]\displaystyle{ XA \cdot XB = XC^2 }[/math].)

Generalizations

There are several curves that could be considered "generalizations" of the circle—families of curves that include circles as a special case. Perhaps the best known is the ellipse, which can be seen as a circle that has been stretched or squashed along one axis. An ellipse aligned to the co&omul;rdinate axes can be described by the equation [math]\displaystyle{ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 }[/math], where [math]\displaystyle{ a }[/math] and b are the "semimajor" and "semiminor" axes (not necessarily in that order), essentially the ellipses' greatest and smallest width. Of course, when [math]\displaystyle{ a=b }[/math], this equation reduces to [math]\displaystyle{ x^2 + y^2 = a^2 }[/math], and the ellipse becomes a circle.

Another circle generalization is the supercircle, a shape described by the equation [math]\displaystyle{ |x|^n + |y|^n = |r|^n }[/math], for some positive value n. When [math]\displaystyle{ n = 2 }[/math], this of course reduces to the equation of a circle; as n increases toward infinity, the shape tends toward a square. If [math]\displaystyle{ n \lt 2 }[/math], then as [math]\displaystyle{ n }[/math] decreases, the quarters of the circle tend to "flatten out" and eventually bow inward; at [math]\displaystyle{ n = 1 }[/math] the supercircle becomes a square (rotated 45\deg; (and halved in area) from the limit square as [math]\displaystyle{ n -\gt \inf }[/math]), and then as [math]\displaystyle{ n }[/math] further decreases toward zero it becomes concave. Just as the circle can be generalized to an ellipse, the supercircle can be generalized to a superellipse with different semiminor and semimajor axes, described by the equation [math]\displaystyle{ \left|\frac{x}{a}\right|^n + \left|\frac{y}{b}\right|^n = 1 }[/math].

The basic definition of a circle—the locus of points the same distance from a given point—also yields different curves in metric spaces that use different metrics from the standard bowstring metric. In a metric using the taxicab distance, for instance, a circle is a square, its sides at half right angles relative to the coördinate axes. In one using Chebyshev distance, it is a square with the sides parallel to the axes. More exotic metrics may give rise to correspondingly exotic circles; in the discrete metric, for instance, a unit circle includes every point in the space except the circle's center, while a circle of any other nonzero radius is empty.