Circle

Circle illustration showing a radius, a diameter, the center and the circumference.

Tycho crater, one of many examples of circles that arise in nature. NASA photo.

A circle is a simple shape of Euclidean geometry consisting of those points in a plane which are at a constant distance, called the radius, from a fixed point, called the center. A circle with center A is sometimes denoted by the symbol A.

A chord of a circle is a line segment whose two endpoints lie on the circle. A diameter is a chord passing through the center and is the largest chord in a circle. The length of a diameter is twice the length of the radius.

Circles are simple closed curves which divide the plane into an interior and an exterior. The circumference of a circle is the perimeter of the circle, and the interior of the circle is called a disk. An arc of a circle is any connected part of a circle's perimeter.

A circle is a special ellipse in which the two foci are coincident. Circles are conic sections attained when a right circular cone is intersected with a plane perpendicular to the axis of the cone.

1 History
2 Analytic results
3 Properties
4 Apollonius circle
- 4.1 Cross-ratios
- 4.2 Generalized circles
5 See also
6 Notes
7 References
- 7.1 External links

History

Early science, particularly geometry and astronomy/astrology, was connected to the divine for most medieval scholars. Notice, even, the circular shape of the halo. The compass in this 13th century manuscript is a symbol of God's act of Creation, as many believed that there was something intrinsically "divine" or "perfect" that could be found in circles

The circle has been known since before the beginning of recorded history. It is the basis for the wheel which, with related inventions such as gears, makes much of modern civilization possible. In mathematics, the study of the circle has helped inspire the development of geometry and calculus. Some highlights in the history of the circle are:

1700 BC – The Rhind papyrus gives a method to find the area of a circular field. The result corresponds to 256/81 as an approximate value of $\pi$ .^[1]
300 BC – Book 3 of Euclid's Elements deals with the properties of circles.
1880 – Lindemann proves that $\pi$ is transcendental, effectively settling the millennia-old problem of squaring the circle.^[2]

Analytic results

Circle of radius r = 1, center (a, b) = (1.2, -0.5).

Chord, secant, tangent, and diameter.

In an x-y Cartesian coordinate system, the circle with center (a, b) and radius r is the set of all points (x, y) such that

$\left( x - a \right)^2 + \left( y - b \right)^2=r^2.$

This equation of the circle follows from the Pythagorean theorem applied to any point on the circle. If the circle is centered at the origin (0, 0), then the equation simplifies to

$x^2 + y^2 = r^2. \!\$

The equation can be written in parametric form using the trigonometric functions sine and cosine as

$x = a+r\,\cos t,\,\!$

$y = b+r\,\sin t\,\!$

where t is a parametric variable, interpreted geometrically as the angle that the ray from the origin to (x, y) makes with the x-axis. Alternatively, in stereographic coordinates, the circle has a parametrization

$x = a + r \frac{2t}{1+t^2}$

$y = b + r \frac{1-t^2}{1+t^2}.$

In homogeneous coordinates each conic section with equation of a circle is of the form

$\ ax^2+ay^2+2b_1xz+2b_2yz+cz^2 = 0.$

It can be proven that a conic section is a circle if and only if the point I(1: i: 0) and J(1: −i: 0) lie on the conic section. These points are called the circular points at infinity.

In polar coordinates the equation of a circle is

$r^2 - 2 r r_0 \cos(\theta - \varphi) + r_0^2 = a^2.\,$

In the complex plane, a circle with a center at c and radius (r) has the equation $|z-c|^2 = r^2$ . Since $|z-c|^2 = z\overline{z}-\overline{c}z-c\overline{z}+c\overline{c}$ , the slightly generalized equation $pz\overline{z} + gz + \overline{gz} = q$ for real p, q and complex g is sometimes called a generalised circle. Not all generalised circles are actually circles: a generalized circle is either a (true) circle or a line.

Tangent lines

Main article: Tangent lines to circles

The tangent line through a point P on a circle is perpendicular to the diameter passing through P. The equation of the tangent line to a circle of radius r centered at the origin at the point (x₁, y₁) is

$xx_1+yy_1=r^2 \!\$

Hence, the slope of a circle at (x₁, y₁) is given by:

$\frac{dy}{dx} = - \frac{x_1}{y_1}.$

More generally, the slope at a point (x, y) on the circle $(x-a)^2 +(y-b)^2 = r^2$ , i.e., the circle centered at (a, b) with radius r units, is given by

$\frac{dy}{dx} = \frac{a-x}{y-b},$

provided that $y \neq b$ .

Pi (π)

For more details on this topic, see Pi.

Pi or π is the ratio of a circle's circumference to its diameter. It is a constant that takes the same numeric value for all circles.

In modern English, it is pronounced /ˈpaɪ/ (the same way as the English word pie).

Area enclosed

Area of the circle = π × area of the shaded square

Main article: Area of a disk

The area enclosed by a circle is $\pi$ multiplied by the radius squared:

$Area = \pi r^2\,$

Equivalently, denoting diameter by d,

$Area = \frac{\pi d^2}{4} \approx 0{.}7854 \cdot d^2,$

that is, approximately 79% of the circumscribing square (whose side is of length d).

The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

Properties

The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)
The circle is a highly symmetric shape: every line through the center forms a line of reflection symmetry and it has rotational symmetry around the center for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.
All circles are similar.
- A circle's circumference and radius are proportional.
- The area enclosed and the square of its radius are proportional.
  - The constants of proportionality are 2π and π, respectively.
The circle centered at the origin with radius 1 is called the unit circle.
- Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the center of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

Chord properties

Chords are equidistant from the center of a circle if and only if they are equal in length.
The perpendicular bisector of a chord passes through the center of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector:
- A perpendicular line from the center of a circle bisects the chord.
- The line segment (circular segment) through the center bisecting a chord is perpendicular to the chord.
If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.
If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.
If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplemental.
- For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.
An inscribed angle subtended by a diameter is a right angle.
The diameter is the longest chord of the circle.

Sagitta properties

The sagitta (also know as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the circumference of the circle.
Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will fit around the two lines:

$r=\frac{y^2}{8x}+ \frac{x}{2}.$

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y/2)². Solving for r, we find the required result.

Tangent properties

The line drawn perpendicular a radius through the end point of the radius is a tangent to the circle.
A line drawn perpendicular to a tangent through the point of contact with a circle passes through the center of the circle.
Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length.

Theorems

Secant-secant theorem

See also: Power of a point

The chord theorem states that if two chords, CD and EB, intersect at A, then CA×DA = EA×BA.
If a tangent from an external point D meets the circle at C and a secant from the external point D meets the circle at G and E respectively, then DC² = DG×DE. (Tangent-secant theorem.)
If two secants, DG and DE, also cut the circle at H and F respectively, then DH×DG = DF×DE. (Corollary of the tangent-secant theorem.)
The angle between a tangent and chord is equal to the subtended angle on the opposite side of the chord. (Tangent chord property.)
If the angle subtended by the chord at the center is 90 degrees then l = √2 × r, where l is the length of the chord and r is the radius of the circle.
If two secants are inscribed in the circle as shown at right, then the measurement of angle A is equal to one half the difference of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

Inscribed angles

Inscribed angle theorem

An inscribed angle (examples are the blue and green angles in the figure) is exactly half the corresponding central angle (red). Hence, all inscribed angles that subtend the same arc (pink) are equal. Angles inscribed on the arc (brown) are supplementary. In particular, every inscribed angle that subtends a diameter is a right angle (since the central angle is 180 degrees).

Apollonius circle

Apollonius' definition of a circle: d₁/d₂ = constant.

Apollonius of Perga showed that a circle may also be defined as the set of points in plane having a constant ratio of distances to two fixed foci, A and B. That circle is sometimes said to be drawn about two points^[3].

The proof is as follows. A line segment PC bisects the interior angle APB, since the segments are similar:

$\frac{AP}{BP} = \frac{AC}{BC}.$

Analogously, a line segment PD bisects the corresponding exterior angle. Since the interior and exterior angles sum to $180^{\circ}$ , the angle CPD is exactly $90^{\circ}$ , i.e., a right angle. The set of points P that form a right angle with a given line segment CD form a circle, of which CD is the diameter.

Cross-ratios

A closely related property of circles involves the geometry of the cross-ratio of points in the complex plane. If A, B, and C are as above, then the Apollonius circle for these three points is the collection of points P for which the absolute value of the cross-ratio is equal to one:

$|[A,B;C,P]| = 1.$

Stated another way, P is a point on the Apollonius circle if and only if the cross-ratio [A,B;C,P] is on the unit circle in the complex plane.

Generalized circles

See also: Generalized circle

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

$\frac{|AP|}{|BP|} = \frac{|AC|}{|BC|}$ (1)

is not a circle, but rather a line.

Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying (1) is called a generalized circle. It may either be a true circle or a line. In this sense a line is generalized circle of infinite radius.

Notes

↑ Chronology for 30000 BC to 500 BC
↑ Squaring the circle
↑ Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co.. pp. 30. http://dlxs2.library.cornell.edu/cgi/t/text/text-idx?c=math;idno=01680002.

References

^ Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.
"Circle" in The MacTutor History of Mathematics archive

External links

Circle formulas at Geometry Atlas.
Interactive Java applets for the properties of and elementary constructions involving circles.
Interactive Standard Form Equation of Circle Click and drag points to see standard form equation in action
Clifford's Circle Chain Theorems. Step by step presentation of the first theorem. Clifford discovered, in the ordinary Euclidean plane, a "sequence or chain of theorems" of increasing complexity, each building on the last in a natural progression by Antonio Gutierrez from "Geometry Step by Step from the Land of the Incas"
Munching on Circles at cut-the-knot
Ron Blond homepage - interactive applets
calculate circumference and area with your own values
MathAce » Circles MathAce's article about circles - has a good in-depth explanation of unit circles and transforming circular equations.