Circle

A circle is the set of points equidistant from a given point $O$. The distance $r$ from the Center is called the Radius, and the point $O$ is called the Center. Twice the Radius is known as the Diameter $d=2r$. The Perimeter $C$ of a circle is called the Circumference, and is given by

$$C=\pi d=2\pi r.$$ (1)

The circle is a Conic Section obtained by the intersection of a Cone with a Plane Perpendicular to the Cone's symmetry axis. A circle is the degenerate case of an Ellipse with equal semimajor and semiminor axes (i.e., with Eccentricity 0). The interior of a circle is called a Disk. The generalization of a circle to 3-D is called a Sphere, and to $n$-D for $n\geq 4$ a Hypersphere.

The region of intersection of two circles is called a Lens. The region of intersection of three symmetrically placed circles (as in a Venn Diagram), in the special case of the center of each being located at the intersection of the other two, is called a Reuleaux Triangle.

The parametric equations for a circle of Radius $a$ are

$$x = a\cos t$$ (2)

$$y = a\sin t.$$ (3)

For a body moving uniformly around the circle,

$$x' = -a\sin t$$ (4)

$$y' = a\cos t,$$ (5)

and

$$x'' = -a\cos t$$ (6)

$$y'' = -a\sin t.$$ (7)

When normalized, the former gives the equation for the unit Tangent Vector of the circle, $(-\sin t, \cos t)$. The circle can also be parameterized by the rational functions

$$x = {1-t^2\over 1+t^2}$$ (8)

$$y = {2t\over 1+t^2},$$ (9)

but an Elliptic Curve cannot. The following plots show a sequence of Normal and Tangent Vectors for the circle.

The Arc Length $s$, Curvature $\kappa$, and Tangential Angle $\phi$ of the circle are

$$s(t) = \int ds=\int \sqrt{x'^2+y'^2}\,dt =at$$ (10)

$$\kappa(t) = {x'y''-y'x''\over(x'^2+y'^2)^{3/2}} = {1\over a}$$ (11)

$$\phi(t) = \int \kappa(t)\,dt ={t\over a}.$$ (12)

The Cesàro Equation is

$$\kappa={1\over a}.$$ (13)

In Polar Coordinates, the equation of the circle has a particularly simple form.

$$r = a$$ (14)

is a circle of Radius $a$ centered at Origin,

$$r = 2a\cos\theta$$ (15)

is circle of Radius $a$ centered at $(a,0)$, and

$$r = 2a\sin\theta$$ (16)

is a circle of Radius $a$ centered on $(0,a)$. In Cartesian Coordinates, the equation of a circle of Radius $a$ centered on $(x_0, y_0)$ is

$$(x-x_0)^2+(y-y_0)^2=a^2.$$ (17)

In Pedal Coordinates with the Pedal Point at the center, the equation is

$$pa=r^2.$$ (18)

The circle having $P_1P_2$ as a diameter is given by

$$(x-x_1)(x-x_2)+(y-y_1)(y-y_2)=0.$$ (19)

The equation of a circle passing through the three points $(x_i, y_i)$ for $i=1$, 2, 3 (the Circumcircle of the Triangle determined by the points) is

$$\left\vert\matrix{ x^2+y^2 & x & y & 1\cr {x_1}^2+{y_1}^2 & ... ...& y_2 & 1\cr {x_3}^2+{y_3}^2 & x_3 & y_3 & 1\cr}\right\vert=0.$$ (20)

The Center and Radius of this circle can be identified by assigning coefficients of a Quadratic Curve

$$ax^2+cy^2+dx+ey+f=0,$$ (21)

where $a=c$ and $b=0$ (since there is no $xy$ cross term). Completing the Square gives

$$a\left({x+{d\over 2a}}\right)^2+a\left({y+{e\over 2a}}\right)^2+f-{d^2+e^2\over 4a}=0.$$ (22)

The Center can then be identified as

$$x_0 = -{d\over 2a}$$ (23)

$$y_0 = -{e\over 2a}$$ (24)

and the Radius as

$$r=\sqrt{{d^2+e^2\over 4a^2}-{f\over a}}\,,$$ (25)

where

$$a = \left\vert\begin{array}{ccc}x_1 & y_1 & 1\\ x_2 & y_2 & 1\\ x_3 & y_3 & 1\end{array}\right\vert$$ (26)

$$d = -\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2 & y_1 & 1\\ {x_2}^2+{y_2}^2 & y_2 & 1\\ {x_3}^2+{y_3}^2 & y_3 & 1\end{array}\right\vert$$ (27)

$$e = \left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2 & x_1 & 1\\ {x_2}^2+{y_2}^2 & x_2 & 1\\ {x_3}^2+{y_3}^2 & x_3 & 1\end{array}\right\vert$$ (28)

$$f = -\left\vert\begin{array}{ccc}{x_1}^2+{y_1}^2 & x_1 & y_1\\ {x_2}^2+{y_2}^2 & x_2 & y_2\\ {x_3}^2+{y_3}^2 & x_3 & y_3\end{array}\right\vert.$$ (29)

Four or more points which lie on a circle are said to be Concyclic. Three points are trivially concyclic since three noncollinear points determine a circle.

The Circumference-to-Diameter ratio $C/d$ for a circle is constant as the size of the circle is changed (as it must be since scaling a plane figure by a factor $s$ increases its Perimeter by $s$), and $d$ also scales by $s$. This ratio is denoted $\pi$ (Pi), and has been proved Transcendental. With $d$ the Diameter and $r$ the Radius,

$$C=\pi d=2\pi r.$$ (30)

Knowing $C/d$, we can then compute the Area of the circle either geometrically or using Calculus. From Calculus,

$$A=\int_0^{2\pi} d\theta \int_0^r r\,dr = (2\pi)({\textstyle{1\over 2}}r^2) = \pi r^2.$$ (31)

Now for a few geometrical derivations. Using concentric strips, we have

As the number of strips increases to infinity, we are left with a Triangle on the right, so

$$A = {\textstyle{1\over 2}}(2\pi r)r =\pi r^2.$$ (32)

This derivation was first recorded by Archimedes in Measurement of a Circle (ca. 225 BC ). If we cut the circle instead into wedges,

As the number of wedges increases to infinity, we are left with a Rectangle, so

$$A = (\pi r)r = \pi r^2.$$ (33)

See also Arc, Blaschke's Theorem, Brahmagupta's Formula, Brocard Circle, Casey's Theorem, Chord, Circumcircle, Circumference, Clifford's Circle Theorem, Closed Disk, Concentric Circles, Cosine Circle, Cotes Circle Property, Diameter, Disk, Droz-Farny Circles, Euler Triangle Formula, Excircle, Feuerbach's Theorem, Five Disks Problem, Flower of Life, Ford Circle, Fuhrmann Circle, Gersgorin Circle Theorem, Hopf Circle, Incircle, Inversive Distance, Johnson Circle, Kinney's Set, Lemoine Circle, Lens, Magic Circles, Malfatti Circles, McCay Circle, Midcircle, Monge's Theorem, Moser's Circle Problem, Neuberg Circles, Nine-Point Circle, Open Disk, P-Circle, Parry Circle, Pi, Polar Circle, Power (Circle), Prime Circle, Ptolemy's Theorem, Purser's Theorem, Radical Axis, Radius, Reuleaux Triangle, Seed of Life, Seifert Circle, Semicircle, Soddy Circles, Sphere, Taylor Circle, Triangle Inscribing in a Circle, Triplicate-Ratio Circle, Tucker Circles, Unit Circle, Venn Diagram, Villarceau Circles, Yin-Yang

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 125 and 197, 1987.

Casey, J. ``The Circle.'' Ch. 3 in A Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl. Dublin: Hodges, Figgis, & Co., pp. 96-150, 1893.

Courant, R. and Robbins, H. What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 74-75, 1996.

Dunham, W. ``Archimedes' Determination of Circular Area.'' Ch. 4 in Journey Through Genius: The Great Theorems of Mathematics. New York: Wiley, pp. 84-112, 1990.

Eppstein, D. ``Circles and Spheres.'' http://www.ics.uci.edu/~eppstein/junkyard/sphere.html.

Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 65-66, 1972.

MacTutor History of Mathematics Archive. ``Circle.'' http://www-groups.dcs.st-and.ac.uk/~history/Curves/Circle.html.

Pappas, T. ``Infinity & the Circle'' and ``Japanese Calculus.'' The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 68 and 139, 1989.

Pedoe, D. Circles: A Mathematical View, rev. ed. Washington, DC: Math. Assoc. Amer., 1995.

Yates, R. C. ``The Circle.'' A Handbook on Curves and Their Properties. Ann Arbor, MI: J. W. Edwards, pp. 21-25, 1952.