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1. From Wikipedia, the free encyclopedia2. Lexicographical order

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  • 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 i

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