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Focus (Geometry) from Wikipedia, the Free Encyclopedia Contents Focus (geometry) From Wikipedia, the free encyclopedia Contents 1 Circle 1 1.1 Terminology .............................................. 1 1.2 History ................................................. 2 1.3 Analytic results ............................................ 2 1.3.1 Length of circumference ................................... 2 1.3.2 Area enclosed ......................................... 2 1.3.3 Equations ........................................... 4 1.3.4 Tangent lines ......................................... 8 1.4 Properties ............................................... 9 1.4.1 Chord ............................................. 9 1.4.2 Sagitta ............................................. 10 1.4.3 Tangent ............................................ 10 1.4.4 Theorems ........................................... 11 1.4.5 Inscribed angles ........................................ 12 1.5 Circle of Apollonius .......................................... 12 1.5.1 Cross-ratios .......................................... 13 1.5.2 Generalised circles ...................................... 13 1.6 Circles inscribed in or circumscribed about other figures ....................... 14 1.7 Circle as limiting case of other figures ................................. 14 1.8 Squaring the circle ........................................... 14 1.9 See also ................................................ 14 1.10 References ............................................... 14 1.11 Further reading ............................................ 15 1.12 External links ............................................. 15 2 Conic section 16 2.1 History ................................................. 17 2.1.1 Menaechmus and early works ................................. 17 2.1.2 Apollonius of Perga ...................................... 17 2.1.3 Al-Kuhi ............................................ 17 2.1.4 Omar Khayyám ........................................ 17 2.1.5 Europe ............................................ 17 2.2 Features ................................................ 18 i ii CONTENTS 2.3 Construction ............................................. 18 2.4 Properties ............................................... 18 2.4.1 Intersection at infinity .................................... 18 2.4.2 Degenerate cases ....................................... 19 2.4.3 Eccentricity, focus and directrix ............................... 19 2.4.4 Generalizations ........................................ 19 2.4.5 In other areas of mathematics ................................ 20 2.5 Cartesian coordinates ......................................... 21 2.5.1 Discriminant classification .................................. 21 2.5.2 Matrix notation ........................................ 21 2.5.3 As slice of quadratic form .................................. 22 2.5.4 Eccentricity in terms of parameters of the quadratic form .................. 22 2.5.5 Standard form ........................................ 22 2.5.6 Invariants of conics ...................................... 23 2.5.7 Modified form ........................................ 23 2.6 Homogeneous coordinates ....................................... 24 2.7 Polar coordinates ........................................... 24 2.8 Pencil of conics ............................................ 25 2.9 Intersecting two conics ........................................ 25 2.10 Applications .............................................. 26 2.11 See also ................................................ 26 2.12 Notes ................................................. 26 2.13 References ............................................... 27 2.14 External links ............................................. 27 3 Focus (geometry) 39 3.1 Conic sections ............................................. 40 3.1.1 Defining conics in terms of two foci ............................. 40 3.1.2 Defining conics in terms of a focus and a directrix ...................... 40 3.1.3 Defining conics in terms of a focus and a directrix circle ................... 40 3.1.4 Astronomical significance ................................... 40 3.2 Cartesian and Cassini ovals ...................................... 41 3.3 Generalization ............................................. 41 3.4 Confocal curves ............................................ 41 3.5 References ............................................... 41 4 Locus (mathematics) 43 4.1 Commonly studied loci ........................................ 43 4.2 Proof of a locus ............................................ 44 4.3 Examples ............................................... 44 4.3.1 First example ......................................... 44 4.3.2 Second example ........................................ 45 CONTENTS iii 4.3.3 Third example ......................................... 46 4.3.4 Fourth example ........................................ 46 4.4 See also ................................................ 46 4.5 References ............................................... 46 5 Point (geometry) 48 5.1 Points in Euclidean geometry ..................................... 48 5.2 Dimension of a point ......................................... 48 5.2.1 Vector space dimension ................................... 49 5.2.2 Topological dimension .................................... 49 5.2.3 Hausdorff dimension ..................................... 50 5.3 Geometry without points ....................................... 50 5.4 Point masses and the Dirac delta function .............................. 50 5.5 See also ................................................ 50 5.6 References ............................................... 51 5.7 External links ............................................. 51 5.8 Text and image sources, contributors, and licenses .......................... 52 5.8.1 Text .............................................. 52 5.8.2 Images ............................................ 54 5.8.3 Content license ........................................ 55 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 from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point 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 everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk. A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations. A circle is a plane figure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. — 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 length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of 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 a segment, 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 chord’s 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 diameter’s endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-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 History The word “circle” derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (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 machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus. Early science, particularly geometry and astrology and astronomy, was connected
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