Generalised circleFrom Wikipedia, the free encyclopedia

Contents

1 Circle 11.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Analytic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3.1 Length of circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Area enclosed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 Tangent lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Chord . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Sagitta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 Inscribed angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5 Circle of Apollonius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Cross-ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.2 Generalised circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.6 Circles inscribed in or circumscribed about other gures . . . . . . . . . . . . . . . . . . . . . . . 141.7 Circle as limiting case of other gures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.8 Squaring the circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.11 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Generalised circle 162.1 Equation in the extended complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.2 The transformation w = 1/z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Representation by Hermitian matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.5.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

i

ii CONTENTS

2.5.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Chapter 1

Circle

This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation).

A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance froma given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a givenpoint is constant. The distance between any of the points and the centre is called the radius.A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everydayuse, the term circle may be used interchangeably to refer to either the boundary of the gure, or to the whole gureincluding its interior; in strict technical usage, the circle is the former and the latter is called a disk.A circle may also be dened as a special ellipse in which the two foci are coincident and the eccentricity is 0, or thetwo-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

A circle is a plane gure bounded by one line, and such that all right lines drawn from a certain pointwithin it to the bounding line, are equal. The bounding line is called its circumference and the point, itscentre.

Euclid. Elements Book I. [1]

1.1 Terminology Arc: any connected part of the circle. Centre: the point equidistant from the points on the circle. Chord: a line segment whose endpoints lie on the circle. Circumference: the length of one circuit along the circle, or the distance around the circle. Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the lengthof such a line segment, which is the largest distance between any two points on the circle. It is a special caseof a chord, namely the longest chord, and it is twice the radius.

Passant: a coplanar straight line that does not touch the circle. Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such asegment, which is half a diameter.

Sector: a region bounded by two radii and an arc lying between the radii. Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chords end-points.

Secant: an extended chord, a coplanar straight line cutting the circle at two points.

1

2 CHAPTER 1. CIRCLE

Semicircle: an arc that extends from one of a diameters endpoints to the other. In non-technical commonusage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called ahalf-disk. A half-disk is a special case of a segment, namely the largest one.

Tangent: a coplanar straight line that touches the circle at a single point.

1.2 HistoryThe word circle derives from the Greek / (kirkos/kuklos), itself a metathesis of the Homeric Greek (krikos), meaning hoop or ring.[2] The origins of the words "circus" and "circuit" are closely related.The circle has been known since before the beginning of recorded history. Natural circles would have been observed,such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand.The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinerypossible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, andcalculus.Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medievalscholars, and many believed that there was something intrinsically divine or perfect that could be found incircles.[3][4]

Some highlights in the history of the circle are:

1700 BCE The Rhind papyrus gives a method to nd the area of a circular eld. The result corresponds to25681 (3.16049...) as an approximate value of .[5]

300 BCE Book 3 of Euclids Elements deals with the properties of circles. In Plato's Seventh Letter there is a detailed denition and explanation of the circle. Plato explains the perfectcircle, and how it is dierent from any drawing, words, denition or explanation.

1880 CE Lindemann proves that is transcendental, eectively settling the millennia-old problem of squaringthe circle.[6]

1.3 Analytic results

1.3.1 Length of circumference

Further information: Circumference

The ratio of a circles circumference to its diameter is (pi), an irrational constant approximately equal to 3.141592654.Thus the length of the circumference C is related to the radius r and diameter d by:

C = 2r = d:

1.3.2 Area enclosedMain article: Area of a disk

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length ofthe circles circumference and whose height equals the circles radius,[7] which comes to multiplied by the radiussquared:

1.3. ANALYTIC RESULTS 3

The compass in this 13th-century manuscript is a symbol of Gods act of Creation. Notice also the circular shape of the halo

Area = r2:

Equivalently, denoting diameter by d,

4 CHAPTER 1. CIRCLE

Circular piece of silk with Mongol images

Area = d2

4 0:7854d2;

that is, approximately 79 percent 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 problemin the calculus of variations, namely the isoperimetric inequality.

1.3.3 Equations

Cartesian coordinates

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

(x a)2 + (y b)2 = r2:

1.3. ANALYTIC RESULTS 5

Circles in an old Arabic astronomical drawing.

This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point onthe circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose othersides are of length |x a| and |y b|. If the circle is centred at the origin (0, 0), then the equation simplies to

x2 + y2 = r2:

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 twhere t is a parametric variable in the range 0 to 2, interpreted geometrically as the angle that the ray from (a, b) to(x, y) makes with the positive x-axis.An alternative parametrisation of the circle is:

x = a+ r2t

1 + t2:

6 CHAPTER 1. CIRCLE

Tughrul Tower from inside

y = b+ r1 t21 + t2

In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the linepassing through the centre parallel to the x-axis (see Tangent half-angle substitution). However, this parametrisationworks only if t is made to range not only through all reals but also to a point at innity; otherwise, the bottom-mostpoint of the circle would be omitted.In homogeneous coordinates each conic section with the equation of a circle has the form

x2 + y2 2axz 2byz + cz2 = 0:

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projectiveplane) the points I(1: i: 0) and J(1: i: 0). These points are called the circular points at innity.

Polar coordinates

In polar coordinates the equation of a circle is:

r2 2rr0 cos( ) + r20 = a2

where a is the radius of the circle, (r; ) is the polar coordinate of a generic point on the circle, and (r0; ) is thepolar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). Fora circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on thecircle, the equation becomes

1.3. ANALYTIC RESULTS 7

Area =

Circle Area = r2

r2

Area enclosed by a circle = area of the shaded square

r = 2a cos( ):

In the general case, the equation can be solved for r, giving

r = r0 cos( )qa2 r20 sin2( );

Note that without the sign, the equation would in some cases describe only half a circle.

Complex plane

In the complex plane, a circle with a centre at c and radius (r) has the equation jz cj = r . In parametric form thiscan be written z = reit + c .The slightly generalised equation pzz+gz+gz = q for real p, q and complex g is sometimes called a generalised circle.This becomes the above equation for a circle with p = 1; g = c; q = r2jcj2 , since jzcj2 = zzczcz+cc. Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line.

8 CHAPTER 1. CIRCLE

(a,b)

(x,y)

0.5

-0.5

-1

-1.5

0.5 1 1.5 2r

Circle of radius r = 1, centre (a, b) = (1.2, 0.5)

1.3.4 Tangent linesMain article: Tangent lines to circles

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1)and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1),so it has the form (x1 a)x + (y1 b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that theequation of the tangent is

(x1 a)x+ (y1 b)y = (x1 a)x1 + (y1 b)y1or

(x1 a)(x a) + (y1 b)(y b) = r2:

If y1 b then the slope of this line is

dy

dx= x1 a

y1 b :

1.4. PROPERTIES 9

This can also be found using implicit dierentiation.When the centre of the circle is at the origin then the equation of the tangent line becomes

x1x+ y1y = r2;

and its slope is

dy

dx= x1

y1:

1.4 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 centre forms a line of reection symmetry and ithas rotational symmetry around the centre 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 circles circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2 and , respectively.

The circle which is centred 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 ispossible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of thecoordinates of the three given points. See circumcircle.

1.4.1 Chord Chords are equidistant from the centre of a circle if and only if they are equal in length. The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemmingfrom the uniqueness of the perpendicular bisector are:

A perpendicular line from the centre of a circle bisects the chord. The line segment through the centre 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 ofthe 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 supplementary.

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 (see Thales theorem). The diameter is the longest chord of the circle. If the intersection of any two chords divides one chord into lengths a and b and divides the other chord intolengths c and d, then ab = cd.

10 CHAPTER 1. CIRCLE

If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the otherchord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.[8]

The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same asthat of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 4p 2 (where ris the circles radius and p is the distance from the center point to the point of intersection).[9]

The distance from a point on the circle to a given chord times the diameter of the circle equals the product ofthe distances from the point to the ends of the chord.[10]:p.71

1.4.2 Sagitta

y

x

The sagitta is the vertical segment.

The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpointof that chord and the arc of the circle.

Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculatethe radius of the unique circle which will t around the two lines:

r =y2

8x+

x

2:

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

1.4.3 Tangent A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to thecircle.

A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre ofthe circle.

Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal inlength.

If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the anglesBOA and BPA are supplementary.

If AD is tangent to the circle at A and if AQ is a chord of the circle, then DAQ = 12arc(AQ).

1.4. PROPERTIES 11

1.4.4 Theorems

E

B

AC

D

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 thecircle 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. (Corollaryof the tangent-secant theorem.)

The angle between a tangent and chord is equal to one half the subtended angle on the opposite side of thechord (Tangent Chord Angle).

If the angle subtended by the chord at the centre is 90 degrees then l = r 2, where l is the length of the chordand 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 onehalf the dierence of the measurements of the enclosed arcs (DE and BC). This is the secant-secant theorem.

12 CHAPTER 1. CIRCLE

1.4.5 Inscribed anglesSee also: Inscribed angle theoremAn inscribed angle (examples are the blue and green angles in the gure) is exactly half the corresponding central

2

Inscribed angle theorem

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 thecentral angle is 180 degrees).

1.5 Circle of ApolloniusApollonius of Perga showed that a circle may also be dened as the set of points in a plane having a constant ratio(other than 1) of distances to two xed foci, A and B.[11][12] (The set of points where the distances are equal is theperpendicular bisector of A and B, a line.) That circle is sometimes said to be drawn about two points.The proof is in two parts. First, one must prove that, given two foci A and B and a ratio of distances, any point Psatisfying the ratio of distances must fall on a particular circle. Let C be another point, also satisfying the ratio andlying on segment AB. By the angle bisector theorem the line segment PC will bisect the interior angle APB, since thesegments are similar:

1.5. CIRCLE OF APOLLONIUS 13

DC B

d2

A

d1

P

Apollonius denition of a circle: d1 / d2 constant

AP

BP=

AC

BC:

Analogously, a line segment PD through some point D on AB extended bisects the corresponding exterior angle BPQwhere Q is on AP extended. Since the interior and exterior angles sum to 180 degrees, the angle CPD is exactly 90degrees, i.e., a right angle. The set of points P such that angle CPD is a right angle forms a circle, of which CD is adiameter.Second, see[13]:p.15 for a proof that every point on the indicated circle satises the given ratio.

1.5.1 Cross-ratiosA 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 circle of Apollonius for these three points is the collection of points P for which theabsolute value of the cross-ratio is equal to one:

j[A;B;C;P ]j = 1:Stated another way, P is a point on the circle of Apollonius if and only if the cross-ratio [A,B;C,P] is on the unit circlein the complex plane.

1.5.2 Generalised circlesSee also: Generalised circle

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

jAP jjBP j =

jACjjBCj

14 CHAPTER 1. CIRCLE

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 the above equationis called a generalised circle. It may either be a true circle or a line. In this sense a line is a generalised circle ofinnite radius.

1.6 Circles inscribed in or circumscribed about other guresIn every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sidesof the triangle.[14]

About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each ofthe triangles three vertices.[15]

A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribedthat is tangent to each side of the polygon.[16]

A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. Awell-studied example is the cyclic quadrilateral.A hypocycloid is a curve that is inscribed in a given circle by tracing a xed point on a smaller circle that rolls withinand tangent to the given circle.

1.7 Circle as limiting case of other guresThe circle can be viewed as a limiting case of each of various other gures:

A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two xedpoints (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with aneccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle isalso a dierent special case of a Cartesian oval in which one of the weights is zero.

A superellipse has an equation of the form xa n+ yb n= 1 for positive a, b, and n. A supercircle has b = a.A circle is the special case of a supercircle in which n = 2.

A Cassini oval is a set of points such that the product of the distances from any of its points to two xed pointsis a constant. When the two xed points coincide, a circle results.

A curve of constant width is a gure whose width, dened as the perpendicular distance between two distinctparallel lines each intersecting its boundary in a single point, is the same regardless of the direction of thosetwo parallel lines. The circle is the simplest example of this type of gure.

1.8 Squaring the circleSquaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as agiven circle by using only a nite number of steps with compass and straightedge.In 1882, the task was proven to be impossible, as a consequence of the LindemannWeierstrass theorem whichproves that pi () is a transcendental number, rather than an algebraic irrational number; that is, it is not the root ofany polynomial with rational coecients.

1.9 See also

1.10 References[1] OL7227282M

1.11. FURTHER READING 15

[2] krikos, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus

[3] Arthur Koestler, The Sleepwalkers: A History of Mans Changing Vision of the Universe (1959)

[4] Proclus, The Six Books of Proclus, the Platonic Successor, on the Theology of Plato Tr. Thomas Taylor (1816) Vol.2, Ch.2,Of Plato

[5] Chronology for 30000 BC to 500 BC. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

[6] Squaring the circle. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

[7] Measurement of a Circle by Archimedes

[8] Posamentier and Salkind, Challenging Problems in Geometry, Dover, 2nd edition, 1996: pp. 104105, #423.

[9] College Mathematics Journal 29(4), September 1998, p. 331, problem 635.

[10] Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.

[11] Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co. p. 30.

[12] Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 1417.

[13] Altshiller-Court, Nathan, College Geometry, Dover, 2007 (orig. 1952).

[14] Incircle from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

[15] Circumcircle from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

[16] Tangential Polygon from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

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

1.12 External links Hazewinkel, Michiel, ed. (2001), Circle, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 Circle (PlanetMath.org website) Weisstein, Eric W., Circle, MathWorld. 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 Munching on Circles at cut-the-knot Area of a Circle Calculate the basic properties of a circle. MathAces Circle article has a good in-depth explanation of unit circles and transforming circular equations. How to nd the area of a circle. There are many types of problems involving how to nd the area of circle; forexample, nding area of a circle from its radius, diameter or circumference.

Chapter 2

Generalised circle

A generalized circle, also referred to as a cline or circline, is a straight line or a circle. The concept is mainlyused in inversive geometry, because straight lines and circles have very similar properties in that geometry and arebest treated together.Inversive plane geometry is formulated on the plane extended by one point at innity. A straight line is then thoughtof as a circle that passes through the point at innity. The fundamental transformations in inversive geometry, theinversions, have the property that they map generalized circles to generalized circles. Mbius transformations, whichare compositions of inversions, inherit that property. These transformations do not necessarily map lines to lines andcircles to circles: they can mix the two.Inversions come in two kinds: inversions at circles and reections at lines. Since the two have very similar properties,we combine them and talk about inversions at generalized circles.Given any three distinct points in the extended plane, there exists precisely one generalized circle that passes throughthe three points.The extended plane can be identied with the sphere using a stereographic projection. The point at innity thenbecomes an ordinary point on the sphere, and all generalized circles become circles on the sphere.

2.1 Equation in the extended complex planeThe extended plane of inversive geometry can be identied with the extended complex plane, so that equations ofcomplex numbers can be used to describe lines, circles and inversions.A circle is the set of points z in a plane that lie at radius r from a center point .

(; r) = fz : the distance between z and is rg

Using the complex plane, we can treat as a complex number and circle as a set of complex numbers.Using the property that a complex number multiplied by its conjugate gives us the square of the modulus of thenumber, and that its modulus is its Euclidean distance from the origin, we can express the equation for as follows:

jz j = r

jz j2 = r2

(z )(z ) = r2

zz z z + = r2

zz z z + r2 = 0:We can multiply this by a real constant A to get an equation of the form

16

2.2. THE TRANSFORMATION W = 1/Z 17

Azz +Bz + Cz +D = 0

where A and D are real, and B and C are complex conjugates. Reversing the steps, we see that in order for this to bea circle, the radius squared must be equal to BC/A^2 - D/A > 0. So the above equation denes a generalized circlewhenever AD < BC. Note that when A is zero, this equation denes a straight line.

2.2 The transformation w = 1/zIt is now easy to see that the transformation w = 1/z maps generalized circles to generalized circles:

Azz +Bz + Cz +D = 0

A1

w

1

w+B

1

w+ C

1

w+D = 0

A+B w + Cw +Dw w = 0

D ww + Cw +B w +A = 0:

We see that straight lines through the origin (A = D = 0) are mapped to straight lines through the origin, straight linesnot containing the origin (A = 0; D 0) to circles containing the origin, circles containing the origin (A 0; D = 0) tostraight lines not containing the origin, and circles not containing the origin (A 0; D 0) to circles not containingthe origin.

2.3 Representation by Hermitian matricesThe data dening the equation of a generalized circle

Azz +Bz + Cz +D = 0

can be usefully put into the form of an invertible hermitian matrix

C =

A BC D

= Cy:

Two such invertible hermitian matrices specify the same generalized circle if and only if they dier by a real multiple.To transform a generalized circle described by C by the Mbius transformation H , you simply do

C 7! (H1)TC(H1):

2.4 References Hans Schwerdtfeger, Geometry of Complex Numbers, Courier Dover Publications, 1979 Michael Henle, Modern Geometry: Non-Euclidean, Projective, and Discrete, 2nd edition, Prentice Hall,2001

18 CHAPTER 2. GENERALISED CIRCLE

2.5 Text and image sources, contributors, and licenses2.5.1 Text

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2.5. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 19

Alexzhang2, Cyberbot II, The Illusive Man, ChrisGualtieri, Loda103, Jiikim, Gareth1893, Dexbot, Codename Lisa, RayMan51207,Mynanisgay, Smeg2, BScMScMD, AleksanderVatov, Lugia2453, Graphium, Savlaraj, Wizdomain, AIRYTAILAWESOMENESS, Lin-graphKaaz, Wenpin~enwiki, Jake randall21, Corndog21, AcheronSS, Tomgammon, Funker2001, Jordanbakerissexy19999, Lambosboy,Brockhole7, Alevorn, DavidLeighEllis, Wamiq, Sheetal.singh001, Babitaarora, Nigellwh, Ugog Nizdast, We.are.young.1, NottNott,FDMS4, Agoshtas, Jianhui67, Jjbernardiscool, Nickad21345, Asdfghjklpoiuytrew, Hungryrefanatic123, Jakerandall1228, Laxlife23,Musamaster, SantiLak, PAPALIAM, Joenick2121, Fridayjunior, WillemienH, Loraof, TrackerAR, Sahil Jain 2002, DiscantX, Launch-ingRibbons, Math Expert X, Anyaykumar, Drstanley3532, Crazyhobbo123, KasparBot, Girlynot, Say yourmind (SYM), N.hLovesCheeseand Anonymous: 1062

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CircleTerminologyHistoryAnalytic resultsLength of circumferenceArea enclosedEquationsTangent lines

PropertiesChordSagittaTangentTheoremsInscribed angles

Circle of ApolloniusCross-ratiosGeneralised circles

Circles inscribed in or circumscribed about other figuresCircle as limiting case of other figuresSquaring the circleSee alsoReferencesFurther readingExternal links

Generalised circleEquation in the extended complex planeThe transformation w = 1/zRepresentation by Hermitian matricesReferences Text and image sources, contributors, and licensesTextImagesContent license