HyperbolaFrom Wikipedia, the free encyclopediaContents1 Circle 11.1 Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Analytic results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Length of circumference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Area enclosed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.3 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.4 Tangent lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.1 Chord. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4.2 Sagitta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.3 Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4.4 Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.5 Inscribed angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5 Circle of Apollonius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.1 Cross-ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.2 Generalised circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.10References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.11Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.12External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 Conic section 162.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Menaechmus and early works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.2 Apollonius of Perga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.3 Al-Kuhi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.4 Omar Khayym. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.5 Europe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18iii CONTENTS2.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.1 Intersection at innity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4.2 Degenerate cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.3 Eccentricity, focus and directrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.4 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.5 In other areas of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.5 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.1 Discriminant classication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Matrix notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.3 As slice of quadratic form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.4 Eccentricity in terms of parameters of the quadratic form. . . . . . . . . . . . . . . . . . 222.5.5 Standard form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.6 Invariants of conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.5.7 Modied form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Homogeneous coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.7 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Pencil of conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.9 Intersecting two conics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.10Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.11See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.12Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.14External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273 Ellipse 393.1 Elements of an ellipse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.2 Drawing ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.1 Pins-and-string method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.2.2 Trammel method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Parallelogram method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.3 Mathematical denitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.1 In Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.3.2 Projective geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.3.3 In analytic geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.3.4 In trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.3.5 Degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.1 Ellipses in physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.4.2 Ellipses in statistics and nance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.3 Ellipses in computer graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.4.4 Ellipses in optimization theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58CONTENTS iii3.5 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604 Hyperbola 614.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.2 Nomenclature and features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3 Mathematical denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.1 Conic section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.2 Dierence of distances to foci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.3 Directrix and focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.4 Reciprocation of a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3.5 Quadratic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4 True anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Geometrical constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.6 Reections and tangent lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.7 Hyperbolic functions and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.8 Relation to other conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684.9 Conic section analysis of the hyperbolic appearance of circles . . . . . . . . . . . . . . . . . . . . 694.10Derived curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.11Coordinate systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.11.1 Cartesian coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.11.2 Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.11.3 Parametric equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.11.4 Elliptic coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.12Rectangular hyperbola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 724.13Other properties of hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 734.14Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.1 Sundials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.2 Multilateration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.3 Path followed by a particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.4 Korteweg-de Vries equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.5 Angle trisection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.14.6 Ecient portfolio frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 744.15Extensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.16See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.16.1 Other conic sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.16.2 Other related topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.17Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.18References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.19External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76iv CONTENTS5 Parabola 915.1 Introductory images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 915.1.1 Description of nal image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 925.3 Equation in Cartesian coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 Conic section and quadratic form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.4.1 Focal length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.4.2 Position of the focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Other geometric denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.6 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.6.1 Cartesian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.6.2 Latus rectum, semilatus rectum, and polar coordinates . . . . . . . . . . . . . . . . . . . . 995.7 Dimensions of parabolas with axes of symmetry parallel to the y-axis . . . . . . . . . . . . . . . . 995.7.1 Coordinates of the vertex. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7.2 Coordinates of the focus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.7.3 Axis of symmetry, focal length, latus rectum, and directrix . . . . . . . . . . . . . . . . . 1015.8 Proof of the reective property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.8.1 Other consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.8.2 Alternative proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.9 Tangent properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.9.1 Two tangent properties related to the latus rectum. . . . . . . . . . . . . . . . . . . . . . 1045.9.2 Orthoptic property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.9.3 Lamberts theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.9.4 Properties proved elsewhere in this article . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.10Facts related to chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.10.1 Focal length calculated from parameters of a chord . . . . . . . . . . . . . . . . . . . . . 1065.10.2 Area enclosed between a parabola and a chord. . . . . . . . . . . . . . . . . . . . . . . . 1065.11Length of an arc of a parabola . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.12Focal length and radius of curvature at the vertex . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.13Mathematical generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.14Parabolas in the physical world . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.14.1 Gallery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.15See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.16Footnotes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1115.17Citations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.18Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.19External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.20Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.20.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.20.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.20.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119Chapter 1CircleThis 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 TerminologyArc: 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.12 CHAPTER 1. CIRCLESemicircle: 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 problemof squaringthe circle.[6]1.3 Analytic results1.3.1 Length of circumferenceFurther information: CircumferenceThe 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 diskAs 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 3The compass in this 13th-century manuscript is a symbol of Gods act of Creation. Notice also the circular shape of the haloArea = r2.Equivalently, denoting diameter by d,4 CHAPTER 1. CIRCLECircular piece of silk with Mongol imagesArea =d24 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 EquationsCartesian coordinatesIn 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 5Circles 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 tox2+y2= r2.The equation can be written in parametric form using the trigonometric functions sine and cosine asx = 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 +r2t1 +t2.6 CHAPTER 1. CIRCLETughrul Tower from insidey= b +r1 t21 +t2In 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 formx2+y22axz 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 coordinatesIn polar coordinates the equation of a circle is:r22rr0 cos( ) +r20= a2where 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 becomes1.3. ANALYTIC RESULTS 7Area =Circle Area = r2r2Area enclosed by a circle = area of the shaded squarer = 2a cos( ).In the general case, the equation can be solved for r, givingr = r0 cos( ) a2r20 sin2( ),Note that without the sign, the equation would in some cases describe only half a circle.Complex planeIn the complex plane, a circle with a centre at c and radius (r) has the equation |z c| = 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= r2|c|2, since |z c|2= zz cz cz +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)0.5-0.5-1-1.50.5 1 1.5 2rCircle of radius r = 1, centre (a, b) = (1.2, 0.5)1.3.4 Tangent linesMain article: Tangent lines to circlesThe 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 isdydx= x1 ay1 b.1.4. PROPERTIES 9This can also be found using implicit dierentiation.When the centre of the circle is at the origin then the equation of the tangent line becomesx1x +y1y= r2,and its slope isdydx= x1y1.1.4 PropertiesThe 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 ChordChords 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. CIRCLEIf 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+ d2equals 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 8r2 4p2(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.711.4.2 SagittayxThe 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 =y28x+x2.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 TangentA 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 111.4.4 TheoremsEBACDSecant-secant theoremSee also: Power of a pointThe 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. CIRCLE1.4.5 Inscribed anglesSee also: Inscribed angle theoremAn inscribed angle (examples are the blue and green angles in the gure) is exactly half the corresponding central2 Inscribed angle theoremangle (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 13D C Bd2Ad1PApollonius denition of a circle: d1 / d2 constantAPBP=ACBC.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:|[A, B; C, P]| = 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 circleIf C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition|AP||BP|=|AC||BC|14 CHAPTER 1. CIRCLEis 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 also1.10 References[1] OL7227282M1.11. 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 readingPedoe, Dan (1988). Geometry: a comprehensive course. Dover.Circle in The MacTutor History of Mathematics archive1.12 External linksHazewinkel, Michiel, ed. (2001), Circle, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Circle (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 actionMunching on Circles at cut-the-knotArea 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 2Conic sectionTypes of conic sections:1. Parabola2. Circle and ellipse3. HyperbolaIn mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, aright circular conical surface) with a plane. In analytic geometry, a conic may be dened as a plane algebraic curveof degree 2, and as a quadric of dimension 1. There are a number of other geometric denitions possible. One of themost useful, in that it involves only the plane, is that a non-circular conic[1] consists of those points whose distancesto some point, called a focus, and some line, called a directrix, are in a xed ratio, called the eccentricity.Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a specialcase of the ellipse, and is of sucient interest in its own right that it is sometimes called the fourth type of conicsection. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, thosewith eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus-directrix denition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degeneratecases, such as the union of two lines, are included as conics as well.162.1. HISTORY 17The conic sections have been named and studied at least since 200 BC, when Apollonius of Perga undertook asystematic study of their properties.2.1 History2.1.1 Menaechmus and early worksIt is believed that the rst denition of a conic section is due to Menaechmus (died 320 BC). His work did not surviveand is only known through secondary accounts. The denition used at that time diers from the one commonly usedtoday in that it requires the plane cutting the cone to be perpendicular to one of the lines, (a generatrix), that generatesthe cone as a surface of revolution. Thus the shape of the conic is determined by the angle formed at the vertex ofthe cone (between two opposite generatrices): If the angle is acute then the conic is an ellipse; if the angle is rightthen the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola. Note that the circle cannot bedened this way and was not considered a conic at this time.Euclid ( . 300 BC ) is said to have written four books on conics but these were lost as well.[2] Archimedes (diedc. 212 BC) is known to have studied conics, having determined the area bounded by a parabola and an ellipse. Theonly part of this work to survive is a book on the solids of revolution of conics.2.1.2 Apollonius of PergaThe greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga (died c. 190 BC),whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge. Apolloniussmajor innovation was to characterize a conic using properties within the plane and intrinsic to the curve; this greatlysimplied analysis. With this tool, it was now possible to show that any plane cutting the cone, regardless of its angle,will produce a conic according to the earlier denition, leading to the denition commonly used today.Pappus of Alexandria (died c. 350 CE) is credited with discovering the importance of the concept of a conics focus,and with the discovery of the related concept of a directrix.2.1.3 Al-KuhiAn instrument for drawing conic sections was rst described in 1000 CE by the Islamic mathematician Al-Kuhi.[3][4]2.1.4 Omar KhayymApolloniuss work was translated into Arabic (the technical language of the time) and much of his work only survivesthrough the Arabic version.Persians found applications to the theory; the most notable of these was the Persian[5]mathematician and poet Omar Khayym who used conic sections to solve algebraic equations.2.1.5 EuropeJohannes Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept oflimits.Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and thishelped to provide impetus for the study of this new eld. In particular, Pascal discovered a theorem known as thehexagrammum mysticum from which many other properties of conics can be deduced.Meanwhile, Ren Descartes applied his newly discovered Analytic geometry to the study of conics. This had theeect of reducing the geometrical problems of conics to problems in algebra.18 CHAPTER 2. CONIC SECTION2.2 FeaturesThe three types of conics are the ellipse, parabola, and hyperbola. The circle can be considered as a fourth type (asit was by Apollonius) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and planeis a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of thecone for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to thesymmetry axis of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conicis unbounded and is called a parabola.In the remaining case, the gure is a hyperbola.In this case, the plane willintersect both halves (nappes) of the cone, producing two separate unbounded curves.Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the tablegives the case of a>b, for which the major axis is horizontal; for the reverse case, interchange the symbols a and b.For the hyperbola the east-west opening case is given. In all cases, a and b are positive.)The non-circular conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negativenumber e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L thedirectrix, and e the eccentricity.The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci).The latus rectum (2) is the chord parallel to the directrix and passing through the focus (or one of the two foci).The semi-latus rectum () is half the latus rectum.The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix.The following relations hold: pe = ae = c.2.3 ConstructionThere are many methods to construct a conic. One of them, that is useful in engineering applications, being parallelogrammethod, where a conic is constructed point by point by means of connecting certain equally spaced points on hori-zontal line and vertical line.2.4 PropertiesJust as two (distinct) points determine a line, ve points determine a conic. Formally, given any ve points in theplane in general linear position, meaning no three collinear, there is a unique conic passing through them, which willbe non-degenerate; this is true over both the ane plane and projective plane. Indeed, given any ve points there is aconic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, becauseit contains a line), and may not be unique; see further discussion.Four points in the plane in general linear position determine a unique conic passing through the rst three points andhaving the fourth point as its center. Thus knowing the center is equivalent to knowing two points on the conic forthe purpose of determining the curve.[6]:p. 203Furthermore, a conic is determined by any combination of k points in general position that it passes through and 5klines that are tangent to it, for 0k5.[7]Irreducible conic sections are always smooth. This is important for many applications, such as aerodynamics, wherea smooth surface is required to ensure laminar ow and to prevent turbulence.2.4.1 Intersection at innityAn algebro-geometrically intrinsic formof this classication is by the intersection of the conic with the line at innity,which gives further insight into their geometry:2.4. PROPERTIES 19ellipses intersect the line at innity in 0 pointsrather, in 0 real points, but in 2 complex points, which areconjugate;parabolas intersect the line at innity in 1 double point, corresponding to the axisthey are tangent to the lineat innity, and close at innity, as distended ellipses;hyperbolas intersect the line at innity in 2 points, corresponding to the asymptoteshyperbolas pass throughinnity, with a twist. Going to innity along one branch passes through the point at innity correspondingto the asymptote, then re-emerges on the other branch at the other side but with the inside of the hyperbola(the direction of curvature) on the other side left vs. right (corresponding to the non-orientability of the realprojective plane)and then passing through the other point at innity returns to the rst branch. Hyperbolascan thus be seen as ellipses that have been pulled through innity and re-emerged on the other side, ipped.2.4.2 Degenerate casesFor more details on this topic, see Degenerate conic.There are ve degenerate cases: three in which the plane passes through apex of the cone, and three that arise whenthe cone itself degenerates to a cylinder (a doubled line can occur in both cases).When the plane passes through the apex, the resulting conic is always degenerate, and is either: a point (when theangle between the plane and the axis of the cone is larger than tangential); a straight line (when the plane is tangentialto the surface of the cone); or a pair of intersecting lines (when the angle is smaller than the tangential). Thesecorrespond respectively to degeneration of an ellipse, parabola, and a hyperbola, which are characterized in the sameway by angle. The straight line is more precisely a double line (a line with multiplicity 2) because the plane is tangentto the cone, and thus the intersection should be counted twice.Where the cone is a cylinder, i.e. with the vertex at innity, cylindric sections are obtained;[8] this corresponds tothe apex being at innity. Cylindrical sections are ellipses (or circles), unless the plane is vertical (which correspondsto passing through the apex at innity), in which case three degenerate cases occur: two parallel lines, known as aribbon (corresponding to an ellipse with one axis innite and the other axis real and non-zero, the distance betweenthe lines), a double line (an ellipse with one innite axis and one axis zero), and no intersection (an ellipse with oneinnite axis and the other axis imaginary).2.4.3 Eccentricity, focus and directrixThe four dening conditions above can be combined into one condition that depends on a xed point F (the focus), aline L (the directrix) not containing F and a nonnegative real number e(the eccentricity). The corresponding conicsection consists of the locus of all points whose distance to F equals etimes their distance to L . For 0 < e < 1 weobtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse orhyperbola. The distance from the center to the directrix is a/e , where ais the semi-major axis of the ellipse, or thedistance from the center to the tops of the hyperbola. The distance from the center to a focus is a e.The circle is a limiting case and is not dened by a focus and directrix in the plane. However, if one were to considerthe directrix to be innitely far from the center (the line at innity), then by taking the eccentricity to be e=0 acircle will have the focus-directrix property, but is still not dened by that property.[9] One must be careful in thissituation to correctly use the denition of eccentricity as the ratio of the distance of a point on the circle to the focus(length of a radius) to the distance of that point to the directrix (this distance is innite) which gives the limiting valueof zero.The eccentricity of a conic section is thus a measure of how far it deviates from being circular.For a given a, the closer eis to 1, the smaller is the semi-minor axis.2.4.4 GeneralizationsConics may be dened over other elds, and may also be classied in the projective plane rather than in the aneplane.20 CHAPTER 2. CONIC SECTIONOver the complex numbers ellipses and hyperbolas are not distinct, since 1 is a square; precisely, the ellipse x2+y2=1 becomes a hyperbola under the substitution y=iw, geometrically a complex rotation, yielding x2 w2=1 ahyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2-way classication: ellipse/hyperbola andparabola. Geometrically, this corresponds to intersecting the line at innity in either 2 distinct points (correspondingto two asymptotes) or in 1 double point (corresponding to the axis of a parabola), and thus the real hyperbola is amore suggestive image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at innity.In projective space, over any division ring, but in particular over either the real or complex numbers, all non-degenerateconics are equivalent, and thus in projective geometry one simply speaks of a conic without specifying a type, astype is not meaningful. Geometrically, the line at innity is no longer special (distinguished), so while some conicsintersect the line at innity dierently, this can be changed by a projective transformation pulling an ellipse out toinnity or pushing a parabola o innity to an ellipse or a hyperbola.A generalization of a non degenerate conic in a projective plane is an oval. An oval is a point set that has the followingproperties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of theoval there exists a unique tangent line.2.4.5 In other areas of mathematicsThe classication into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a eld intosharply distinct subelds.The classication mostly arises due to the presence of a quadratic form (in two variablesthis corresponds to the associated discriminant), but can also correspond to eccentricity.Quadratic form classications:quadratic formsQuadratic forms over the reals are classied by Sylvesters law of inertia, namely by their positiveindex, zero index, and negative index:a quadratic form in n variables can be converted to a diagonal form,as x21 + x22 + + x2k x2k+1 x2k+l, where the number of +1 coecients, k, is the positive index,the number of 1 coecients, l, is the negative index, and the remaining variables are the zero index m, sok +l +m = n. In two variables the non-zero quadratic forms are classied as: x2+y2 positive-denite (the negative is also included), corresponding to ellipses, x2 degenerate, corresponding to parabolas, and x2y2 indenite, corresponding to hyperbolas.In two variables quadratic forms are classied by discriminant, analogously to conics, but in higher dimensions themore useful classication is as denite, (all positive or all negative), degenerate, (some zeros), or indenite (mixof positive and negative but no zeros). This classication underlies many that follow.curvatureThe Gaussian curvature of a surface describes the innitesimal geometry, and may at each point be eitherpositive elliptic geometry, zero Euclidean geometry (at, parabola), or negative hyperbolic geometry;innitesimally, to second order the surface looks like the graph of x2+ y2, x2(or 0), or x2 y2. Indeed,by the uniformization theorem every surface can be taken to be globally (at every point) positively curved,at, or negatively curved. In higher dimensions the Riemann curvature tensor is a more complicated object,but manifolds with constant sectional curvature are interesting objects of study, and have strikingly dierentproperties, as discussed at sectional curvature.Second order PDEsPartial dierential equations (PDEs) of second order are classied at each point as elliptic,parabolic, or hyperbolic, accordingly as their second order terms correspond to an elliptic, parabolic, or hy-perbolic quadratic form.The behavior and theory of these dierent types of PDEs are strikingly dierent representative examples is that the Poisson equation is elliptic, the heat equation is parabolic, and the waveequation is hyperbolic.Eccentricity classications include:Mbius transformationsReal Mbius transformations (elements of PSL2(R) or its 2-fold cover, SL2(R)) areclassied as elliptic, parabolic, or hyperbolic accordingly as their half-trace is 0 | tr |/2 < 1, | tr |/2 = 1, or| tr |/2 > 1, mirroring the classication by eccentricity.2.5. CARTESIAN COORDINATES 21Variance-to-mean ratioThe variance-to-mean ratio classies several important families of discrete probability dis-tributions: the constant distribution as circular (eccentricity 0), binomial distributions as elliptical, Poissondistributions as parabolic, and negative binomial distributions as hyperbolic.This is elaborated at cumulantsof some discrete probability distributions.2.5 Cartesian coordinatesIn the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (thoughit may be degenerate), and all conic sections arise in this way. The equation will be of the formAx2+Bxy +Cy2+Dx +Ey +F= 0 with A, B, Czero. all not[10]As scaling all six constants yields the same locus of zeros, one can consider conics as points in the ve-dimensionalprojective space P5.2.5.1 Discriminant classicationThe conic sections described by this equation can be classied with the discriminant[11]B24AC.If the conic is non-degenerate, then:if B24AC< 0 , the equation represents an ellipse;if A = C and B= 0 , the equation represents a circle, which is a special case of an ellipse;if B24AC= 0 , the equation represents a parabola;if B24AC> 0 , the equation represents a hyperbola;[12]if we also have A+C= 0 , the equation represents a rectangular hyperbola.To distinguish the degenerate cases from the non-degenerate cases, let be the determinant of the 33 matrix [A,B/2, D/2 ; B/2, C, E/2 ; D/2, E/2, F ]: that is, = (AC - B2/4)F + BED/4 - CD2/4 - AE2/4. Then the conic section isnon-degenerate if and only if 0. If =0 we have a point ellipse, two parallel lines (possibly coinciding with eachother) in the case of a parabola, or two intersecting lines in the case of a hyperbola.[13]:p.63Moreover, in the case of a non-degenerate ellipse (with B2 4AC 0. An example is x2+y2+ 10 = 0 , which has no real-valued solutions.Note that A and B are polynomial coecients, not the lengths of semi-major/minor axis as dened in some sources.2.5.2 Matrix notationMain article: Matrix representation of conic sectionsThe above equation can be written in matrix notation as[x y].[A B/2B/2 C].[xy]+Dx +Ey +F= 0.The type of conic section is solely determined by the determinant of middle matrix: if it is positive, zero, or negativethen the conic is an ellipse, parabola, or hyperbola respectively (see geometric meaning of a quadratic form). If both22 CHAPTER 2. CONIC SECTIONthe eigenvalues of the middle matrix are non-zero (i.e. it is an ellipse or a hyperbola), we can do a transformation ofvariables to obtain(x ay c)T (AB2B2C)(x ay c) = Gwhere a,c, and G satisfy D + 2aA+Bc = 0,E + 2Cc +Ba = 0, and G = Aa2+Cc2+Bac F .The quadratic can also be written as[x y 1].A B/2 D/2B/2 C E/2D/2 E/2 F.xy1 = 0.If the determinant of this 33 matrix is non-zero, the conic section is not degenerate. If the determinant equalszero, the conic is a degenerate parabola (two parallel or coinciding lines), a degenerate ellipse (a point ellipse), or adegenerate hyperbola (two intersecting lines).Note that in the centered equation with constant term G, G equals minus one times the ratio of the 33 determinantto the 22 determinant.2.5.3 As slice of quadratic formThe equationAx2+Bxy +Cy2+Dx +Ey +F= 0can be rearranged by taking the ane linear part to the other side, yieldingAx2+Bxy +Cy2= (Dx +Ey +F).In this form, a conic section is realized exactly as the intersection of the graph of the quadratic form z=Ax2+Bxy + Cy2and the plane z= (Dx + Ey + F). Parabolas and hyperbolas can be realized by a horizontal plane(D=E=0 ), while ellipses require that the plane be slanted. Degenerate conics correspond to degenerateintersections, such as taking slices such as z= 1 of a positive-denite form.2.5.4 Eccentricity in terms of parameters of the quadratic formWhen the conic section is written algebraically asAx2+Bxy +Cy2+Dx +Ey +F= 0,the eccentricity can be written as a function of the parameters of the quadratic equation.[14] If 4AC = B2the conic isa parabola and its eccentricity equals 1 (if it is non-degenerate). Otherwise, assuming the equation represents eithera non-degenerate hyperbola or a non-degenerate, non-imaginary ellipse, the eccentricity is given bye =2(AC)2+B2(A+C) +(AC)2+B2,where = 1 if the determinant of the 33 matrix is negative and = 1 if that determinant is positive.2.5.5 Standard formThrough a change of coordinates (a rotation of axes and a translation of axes) these equations can be put into standardforms:[15]2.5. CARTESIAN COORDINATES 23Circle: x2+ y2= a2Ellipse: x2/a2+ y2/b2= 1Parabola: y2= 4ax, x2= 4ayHyperbola: x2/a2 y2/b2= 1, x2/b2 y2/a2= 1[16]Rectangular hyperbola: xy = c2The rst four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and hyperbola),or about either but not both (for the parabola). The rectangular hyperbola, however, is instead symmetric about thelines y = x and y = x.These standard forms can be written as parametric equations,Circle: (a cos , a sin ),Ellipse: (a cos , b sin ),Parabola: (at2, 2at),Hyperbola: (a sec , b tan ) or (a cosh u, b sinh u),Rectangular hyperbola: (ct , c/t).2.5.6 Invariants of conicsThe trace and determinant of[A B/2B/2 C] are both invariant with respect to both rotation of axes and translationof the plane (movement of the origin).[12][17]The constant term F is invariant under rotation only.2.5.7 Modied formFor some practical applications, it is important to re-arrange the standard form so that the focal-point can be placed atthe origin. The mathematical formulation for a general conic section, with the other focus if any placed at a positivevalue (for an ellipse) or a negative value (for a hyperbola) on the horizontal axis, is then given in the polar form byr =l1 e cos and in the Cartesian form byx2+y2= (l +ex)(x le1e2l1e2)2+(1 e2)y2l2= 1From the above equation, the linear eccentricity (c) is given by c =(le1e2).From the general equations given above, dierent conic sections can be represented as shown below:Circle:x2+y2= r2Ellipse:(xa2b2)2a2+y2b2= 1Parabola:y2= 4a (x +a)Hyperbola:(x+a2+b2)2a2y2b2= 124 CHAPTER 2. CONIC SECTION2.6 Homogeneous coordinatesIn homogeneous coordinates a conic section can be represented as:A1x2+A2y2+A3z2+ 2B1xy + 2B2xz + 2B3yz= 0.Or in matrix notation[x y z].A1B1B2B1A2B3B2B3A3.xyz = 0.The matrix M=A1B1B2B1A2B3B2B3A3is called the matrix of the conic section.= det(M)= detA1B1B2B1A2B3B2B3A3 is called the determinant of the conic section. If = 0 then the conicsection is said to be degenerate; this means that the conic section is either a union of two straight lines, a repeated line,a point or the empty set.For example, the conic section[x y z].1 0 00 1 00 0 0.xyz = 0 reduces to the union of two lines:{x2y2= 0} = {(x +y)(x y) = 0} = {x +y= 0} {x y= 0}.Similarly, a conic section sometimes reduces to a (single) repeated line:{x2+ 2xy +y2= 0} = {(x +y)2= 0} = {x +y= 0} {x +y= 0} = {x +y= 0}.= det([A1B1B1A2])is called the discriminant of the conic section. If = 0 then the conic section is a parabola, if < 0, it is an hyperbola and if > 0, it is an ellipse. A conic section is a circle if > 0 and A1 = A2 and B1 = 0, it isan rectangular hyperbola if < 0 and A1 = A2. It can be proven that in the complex projective plane CP2two conicsections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersectionpoints and there is always one intersection point (possibilities: four distinct intersection points, two singular intersectionpoints and one double intersection points, two double intersection points, one singular intersection point and 1 withmultiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity> 1, then the two conic sections are said to be tangent. If there is only one intersection point, which has multiplicity4, the two conic sections are said to be osculating.[18]Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is saidto be tangent and it is called the tangent line. Because every straight line intersects a conic section twice, each conicsection has two points at innity (the intersection points with the line at innity). If these points are real, the conicsection must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic sectionhas one double point at innity it is a parabola. If the points at innity are (1,i,0) and (1,-i,0), the conic section is acircle (see circular points at innity). If a conic section has one real and one imaginary point at innity or it has twoimaginary points that are not conjugated then it not a real conic section (its coecients are complex).2.7 Polar coordinatesIn polar coordinates, a conic section with one focus at the origin and, if any, the other at a negative value (for anellipse) or a positive value (for a hyperbola) on the x-axis, is given by the equation2.8. PENCIL OF CONICS 25r =l1 +e cos ,where e is the eccentricity and l is the semi-latus rectum (see above). As above, for e = 0, we have a circle, for 0 < e< 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.2.8 Pencil of conicsA (non-degenerate) conic is completely determined by ve points in general position (no three collinear) in a planeand the system of conics which pass through a xed set of four points (again in a plane and no three collinear) iscalled a pencil of conics.[19] The four common points are called the base points of the pencil. Through any pointother than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles.In a projective plane dened over an algebraically closed eld any two conics meet in four points (counted withmultiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base pointsdetermine three line pairs (degenerate conics through the base points, each line of the pair containing exactly twobase points) and so each pencil of conics will contain at most three degenerate conics.[20]A pencil of conics can represented algebraically in the following way. Let C1 and C2 be two distinct conics in aprojective plane dened over an algebraically closed eld K. For every pair , of elements of K, not both zero, theexpression:C1 +C2represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete witha slight abuse of notation (using the same notation to denote the object as well as the equation dening the object.)Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the conic C1". Another concreterealization would be obtained by thinking of C1 as the 33 symmetric matrix which represents it. If C1 and C2 havesuch concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneouscoordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic ifthey dier by a non-zero multiplicative constant.2.9 Intersecting two conicsThe solutions to a system of two second degree equations in two variables may be viewed as the coordinates ofthe points of intersection of two generic conic sections. In particular two conics may possess none, two or fourpossibly coincident intersection points. An ecient method of locating these solutions exploits the homogeneousmatrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters.The procedure to locate the intersection points follows these steps, where the conics are represented by matrices:given the two conics C1 and C2 , consider the pencil of conics given by their linear combination C1 +C2.identify the homogeneous parameters (, ) which correspond to the degenerate conic of the pencil. This canbe done by imposing the condition that det(C1 + C2)=0 and solving for and . These turn out to bethe solutions of a third degree equation.given the degenerate conic C0 , identify the two, possibly coincident, lines constituting it.intersect each identied line with either one of the two original conics; this step can be done eciently usingthe dual conic representation of C0the points of intersection will represent the solutions to the initial equation system.26 CHAPTER 2. CONIC SECTION2.10 ApplicationsConic sections are important in astronomy: the orbits of two massive objects that interact according to Newtons lawof universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are boundtogether, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas.See two-body problem.In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective trans-formations.For specic applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola.For certain fossils in paleontology, understanding conic sections can help understand the three-dimensional shape ofcertain organisms.2.11 See alsoCircumconic and inconicConic Sections RebellionDandelin spheresDirector circleElliptic coordinate systemFocus (geometry), an overview of properties of conic sections related to the fociLambert conformal conic projectionMatrix representation of conic sectionsNine-point conicParabolic coordinatesProjective conicsQuadratic functionQuadrics, the higher-dimensional analogs of conics2.12 Notes[1] Eves 1963, p. 319[2] Heath, T.L., The Thirteen Books of Euclids Elements, Vol. I, Dover, 1956, pg.16[3] Stillwell, John (2010). Mathematics and its history (3rd ed.). New York: Springer. p. 30. ISBN 1-4419-6052-X.[4] Apollonius of Perga Conics Books One to Seven (PDF). Retrieved 10 June 2011.[5] Turner, Howard R. (1997). Science in medieval Islam: an illustrated introduction. University of Texas Press. p. 53. ISBN0-292-78149-0., Chapter , p. 53[6] Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimen-sions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+coordinates&t=books[7] Paris Pamlos, A gallery of conics by ve elements, Forum Geometricorum 14, 2014, 295-348. http://forumgeom.fau.edu/FG2014volume14/FG201431.pdf[8] MathWorld: Cylindric section.2.13. REFERENCES 27[9] Eves 1963, p. 320[10] Protter & Morrey (1970, p. 316)[11] Fanchi, John R. (2006), Math refresher for scientists and engineers, John Wiley and Sons, pp. 4445, ISBN0-471-75715-2,Section 3.2, page 45[12] Protter & Morrey (1970, p. 326)[13] Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.[14] Ayoub, Ayoub B., The eccentricity of a conic section, The College Mathematics Journal 34(2), March 2003, 116121.[15] Protter & Morrey (1970, pp. 314328,585589)[16] Protter & Morrey (1970, pp. 290314)[17] Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, pp. 101-111.[18] Wilczynski, E. J. (1916), Some remarks on the historical development and the future prospects of the dierential geometryof plane curves, Bull. Amer. Math. Soc. 22: 317329, doi:10.1090/s0002-9904-1916-02785-6.[19] Faulkner 1952, pg. 64[20] Samuel 1988, pg. 502.13 ReferencesAkopyan, A.V. and Zaslavsky, A.A. (2007). Geometry of Conics.American Mathematical Society.p.134.ISBN 0-8218-4323-0.Eves, Howard (1963), A Survey of Geometry (Volume One), Boston: Allyn and BaconFaulkner, T. E. (1952), Projective Geometry (2nd ed.), Edinburgh: Oliver and BoydProtter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading:Addison-Wesley, LCCN 76087042Samuel, Pierre (1988), Projective Geometry, Undergraduate Texts in Mathematics (Readings in Mathematics),New York: Springer-Verlag, ISBN 0-387-96752-42.14 External linksDerivations of Conic Sections at ConvergenceConic sections at Special plane curves.Weisstein, Eric W., Conic Section, MathWorld.Determinants and Conic Section CurvesOccurrence of the conics. Conics in nature and elsewhere.Conics. An essay on conics and how they are generated.See Conic Sections at cut-the-knot for a sharp proof that any nite conic section is an ellipse and Xah Lee fora similar treatment of other conics.Cone-plane intersection MATLAB codeEight Point Conic at Dynamic Geometry SketchesAn interactive Java conics grapher; uses a general second-order implicit equation.28 CHAPTER 2. CONIC SECTIONTable of conics, Cyclopaedia, 17282.14. EXTERNAL LINKS 29Diagram from Apollonius Conics, in a 9th century Arabic translation30 CHAPTER 2. CONIC SECTIONCircleEllipseParabolaHyperbolaConics are of three types: parabolas, ellipses, including circles, and hyperbolas.2.14. EXTERNAL LINKS 31Major axisMinor axisLatus rectumfocal parameterlinear eccentricitydirectricesfociConic parameters in the case of an ellipse32 CHAPTER 2. CONIC SECTIONe=0.5e=1e=2e=FMM'Ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with xed focus F and directrix (e=).2.14. EXTERNAL LINKS 33Standard forms of an ellipse34 CHAPTER 2. CONIC SECTIONStandard forms of a parabola2.14. EXTERNAL LINKS 35Standard forms of a hyperbola36 CHAPTER 2. CONIC SECTION(a) (b)(c)Three dierent types of conic sections. Focal-points corresponding to all conic sections are placed at the origin.2.14. EXTERNAL LINKS 37Development of the conic section as the eccentricity e increases38 CHAPTER 2. CONIC SECTIONThe paraboloid shape of Archeocyathids produces conic sections on rock facesChapter 3EllipseThis article is about the geometric gure. For other uses, see Ellipse (disambiguation).Elliptical redirects here. For the exercise machine, see Elliptical trainer.In mathematics, an ellipse is a curve on a plane surrounding two focal points such that the sum of the distances toAn ellipse obtained as the intersection of a cone with an inclined plane.3940 CHAPTER 3. ELLIPSEthe two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is aspecial type of an ellipse that has both focal points at the same location. The shape of an ellipse (how 'elongated' itis) is represented by its eccentricity, which for an ellipse can be any number from 0 (the limiting case of a circle) toarbitrarily close to but less than 1.Ellipses are the closed type of conic section: a plane curve that results from the intersection of a cone by a plane.(See gure to the right.)Ellipses have many similarities with the other two forms of conic sections: the parabolasand the hyperbolas, both of which are open and unbounded. The cross section of a cylinder is an ellipse, unless thesection is parallel to the axis of the cylinder.Analytically, an ellipse can also be dened as the set of points such that the ratio of the distance of each point on thecurve from a given point (called a focus or focal point) to the distance from that same point on the curve to a givenline (called the directrix) is a constant, called the eccentricity of the ellipse.Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the solar systemis an ellipse with the barycenter of the planet-Sun pair at one of the focal points. The same is true for moons orbitingplanets and all other systems having two astronomical bodies. The shape of planets and stars are often well describedby ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspectiveprojection, which are simply intersections of the projective cone with the plane of projection. It is also the simplestLissajous gure, formed when the horizontal and vertical motions are sinusoids with the same frequency. A similareect leads to elliptical polarization of light in optics.The name, (lleipsis, omission), was given by Apollonius of Perga in his Conics, emphasizing the connec-tion of the curve with application of areas.3.1 Elements of an ellipseSee also: Features of conic sectionsEllipses have two mutually perpendicular axes about which the ellipse is symmetric. These axes intersect at theC f aF2b f aF1b e = f a 0 < e < 1 e = PF2PDdPD PF1+PF2 = 2a P The ellipse and some of its mathematical properties.center of the ellipse due to this symmetry.The larger of these two axes, which corresponds to the largest distancebetween antipodal points on the ellipse, is called the major axis. (On the gure to the right it is represented by theline segment between the point labeled a and the point labeled a.) The smaller of these two axes, and the smallestdistance across the ellipse, is called the minor axis.[1] (On the gure to the right it is represented by the line segmentbetween the point labeled b to the point labeled b.)The semi-major axis (denoted by a in the gure) and the semi-minor axis (denoted by b in the gure) are one half3.2. DRAWING ELLIPSES 41The distance traveled from one focus to another, via some point on the ellipse, is the same regardless of the point selected.of the major and minor axes, respectively. These are sometimes called (especially in technical elds) the major andminor semi-axes,[2][3] the major and minor semiaxes,[4][5] or major radius and minor radius.[6][7][8][9]The four points where these axes cross the ellipse are the vertices and are marked as a, a, b, and b. In additionto being at the largest and smallest distance from the center, these points are where the curvature of the ellipse ismaximum and minimum.[10]The two foci (plural of focus and the term focal points is also used) of an ellipse are two special points F1 and F2on the ellipses major axis that are equidistant from the center point. The sum of the distances from any point P onthe ellipse to those two foci is constant and equal to the major axis (PF1 + PF2 = 2a). (On the gure to the right thiscorresponds to the sum of the two green lines equaling the length of the major axis that goes from a to a.)The distance to the focal point from the center of the ellipse is sometimes called the linear eccentricity, f, of theellipse. Here it is denoted by f, but it is often denoted by c. Due to the Pythagorean theorem and the denition ofthe ellipse explained in the previous paragraph: f2= a2b2.A second equivalent method of constructing an ellipse using a directrix is shown on the plot as the three blue lines.(See the Directrix section of this article for more information about this method). The dashed blue line is the directrixof the ellipse shown.The eccentricity of an ellipse, usually denoted by or e, is the ratio of the distance between the two foci, to thelength of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e < 1). When theeccentricity is 0 the foci coincide with the center point and the gure is a circle. As the eccentricity tends toward 1,the ellipse gets a more elongated shape.It tends towards a line segment (see below) if the two foci remain a nitedistance apart and a parabola if one focus is kept xed as the other is allowed to move arbitrarily far away. Theeccentricity is also equal to the ratio of the distance (such as the (blue) line PF2) from any particular point on anellipse to one of the foci to the perpendicular distance to the directrix from the same point (line PD), e = PF2/PD.3.2 Drawing ellipses3.2.1 Pins-and-string methodThe characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to amethod of drawing one using two drawing pins, a length of string, and a pencil.[11] In this method, pins are pushed42 CHAPTER 3. ELLIPSEDrawing an ellipse with two pins, a loop, and a peninto the paper at two points which will become the ellipses foci.A string tied at each end to the two pins and thetip of a pen is used to pull the loop taut so as to form a triangle. The tip of the pen will then trace an ellipse if it ismoved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners tooutline an elliptical ower bed; thus it is called the gardeners ellipse.[12]3.2.2 Trammel methodAn ellipse can also be drawn using a ruler, a set square, and a pencil:Draw two perpendicular lines M,N on the paper; these will be the major (M) and minor (N) axes of theellipse. Mark three points A, B, C on the ruler. A->C being the length of the semi-major axis and B->Cthe length of the semi-minor axis. With one hand, move the ruler on the paper, turning and sliding it soas to keep point A always on line N, and B on line M. With the other hand, keep the pencils tip on thepaper, following point C of the ruler. The tip will trace out an ellipse.The trammel of Archimedes, or ellipsograph, is a mechanical device that implements this principle. The ruler isreplaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slideinto two perpendicular slots cut into a metal plate.[13] The mechanism can be used with a router to cut ellipses fromboard material. The mechanism is also used in a toy called the nothing grinder.3.2.3 Parallelogram methodIn the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontallines and equally spaced points on two vertical lines. It is based on Steiners theorem on the generation of conicsections. Similar methods exist for the parabola and hyperbola.3.3. MATHEMATICAL DEFINITIONS AND PROPERTIES 43Trammel of Archimedes (ellipsograph) animation3.3 Mathematical denitions and properties3.3.1 In Euclidean geometryDenitionIn Euclidean geometry, the ellipse is usually dened as the bounded case of a conic section, or as the set of pointssuch that the sum of the distances to two xed points (the foci) is constant. The ellipse can also be dened as the setof points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positivefraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called thedirectrix). Yet another equivalent denition of the ellipse is that it is the set of points that are equidistant from onepoint in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).The equivalence of these denitions can be proved using the Dandelin spheres.44 CHAPTER 3. ELLIPSEEllipse construction applying the parallelogram methodEquationsThe equation of an ellipse whose major and minor axes coincide with the Cartesian axes is(xa)2+(yb)2=1 .This can be explained as follows:If we letx = a cos .Andy= b sin .Then plotting x and y values for all angles of between 0 and 2 results in an ellipse (e.g. at = 0, x = a, y = 0 andat = /2, y = b, x = 0).Squaring both equations gives:3.3. MATHEMATICAL DEFINITIONS AND PROPERTIES 45x2= a2cos2.Andy2= b2sin2.Dividing these two equations by a2and b2respectively gives:x2a2= cos2.Andy2b2= sin2.Adding these two equations together gives:x2a2+y2b2= cos2 + sin2.Applying the Pythagorean identity to the right hand side gives:x2a2+y2b2= 1.This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then thearea of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circularellipse would create an ellipse with double the area of the original circle).FocusThe distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minorradii:f=a2b2.The sum of the distances from any point P = P(x,y) on the ellipse to those two foci is constant and equal to the majoraxis (proof):PF1 +PF2=(x +f)2+y2+(x f)2+y2= 2aEccentricityThe eccentricity of the ellipse (commonly denoted as either e or ) ise = =a2b2a2=1 (ba)2= f/a46 CHAPTER 3. ELLIPSE(where again a and b are one-half of the ellipses major and minor axes respectively, and f is the focal distance) or,as expressed in terms using the attening factor g= 1 ba= 1 1 e2,e =g(2 g).Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulasfor the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the articlededicated to conic sections.DirectrixC f a dPD 0 < e < 1 e = PFPDEach focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustrationon the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse to the focus Fis a constant fraction of that points perpendicular distance to the directrix, resulting in the equality e = PF/PD. Theratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelinspheres) can be taken as another denition of the ellipse.Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance fromthe center to the farthest vertices (most sharply curved points of the ellipse), it is also true that e = a/d, where d is thedistance from the center to the directrix.Circular directrixThe ellipse can also be dened as the set of points that are equidistant from one focus and a circle, the directrix circle,that is centered on the other focus. The radius of the directrix circle equals the ellipses major axis, so the focus andthe entire ellipse are inside the directrix circle.Ellipse as hypotrochoidThe ellipse is a special case of the hypotrochoid when R = 2r.AreaThe area Aellipse enclosed by an ellipse is:3.3. MATHEMATICAL DEFINITIONS AND PROPERTIES 47An ellipse (in red) as a special case of the hypotrochoid with R = 2r.Aellipse= abwhere a and b are the semi-major and semi-minor axes (12 of the ellipses major and minor axes), respectively.An ellipse dened implicitly by Ax2+Bxy +Cy2= 1 has area24ACB2 .The area formula ab is intuitive: start with a circle of radius b (so its area is b2) and stretch it by a factor a/b tomake an ellipse. This intuitively justies the area by the same factor:b2(a/b) = ab. However, a more rigorousproof requires integration as follows:For the ellipse in standard form,x2a2+y2b2= 1 , and hence y= a2b2b2x2a2, with horizontal intercepts at a, thearea Aellipse can be computed as twice the integral of the positive square root:Aellipse=aa2b1 x2/a2dx=baaa2a2x2dx.48 CHAPTER 3. ELLIPSEThe second integral is the area of a circle of radius a , i.e., a2; thus we have:Aellipse=baAcircle= ab.The area formula can also be proven in terms of polar coordinates using the coordinate transformation T(r, )=(ra cos , rb sin ).Any point inside the ellipse with x-intercept a and y-intercept b can be dened in terms of r and , where 0 r 1and 0 2 .To dene the area dierential in such coordinates we use the Jacobian matrix of the coordinate transformation timesdr d :dAellipse= det(TrT)dr d= det(a cos ra sin b sin rb cos )dr d= abr dr d.We now integrate over the ellipse to nd the area:Aellipse=ellipsedAellipse=ellipseabr dr d = ab2010r dr d = ab.CircumferenceThe circumference C of an ellipse is:C= 4aE(e)where again a is the length of the semi-major axis and e is the eccentricity and where the function E is the completeelliptic integral of the second kind (the arc length of an ellipse, in general, has no closed-formsolution in terms of ele-mentary functions and the elliptic integrals were motivated by this problem). This may be evaluated directly using theCarlson symmetric form.[14] This gives a succinct and rapidly converging method for evaluating the circumference.[15]The exact innite series is:C= 2a[1 (12)2e2(1 32 4)2e43(1 3 52 4 6)2e65 ]orC= 2a[1 n=1((2n 1)!!2nn!)2e2n2n 1],where n!! is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in termsof h = (a b)2/(a +b)2, Ivory[16] and Bessel[17] derived an expression which converges much more rapidly,C= (a +b)[1 +n=1((2n 1)!!2nn!)2hn(2n 1)2].Ramanujan gives two good approximations for the circumference in 16 of;[18] they are3.3. MATHEMATICAL DEFINITIONS AND PROPERTIES 49C [3(a +b) (3a +b)(a + 3b)] = [3(a +b) 10ab + 3(a2+b2)]andC (a +b)(1 +3h10 +4 3h).The errors in these approximations, which were obtained empirically, are of order h3and h5, respectively.More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by anincomplete elliptic integral.See also: Meridian arc Meridian distance on the ellipsoidThe inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions.Some lower and upper bounds on the circumference of the canonical ellipsex2a2+y2b2= 1 with a b are[19]C 2a,(a +b) C 4(a +b),4a2+b2 C 2a2+b2,C 4(a b) + 2.Here the upper bound 2a is the circumference of a circumscribed concentric circle passing through the endpointsof the ellipses major axis, and the lower bound 4a2+b2is the perimeter of an inscribed rhombus with vertices atthe endpoints of the major and minor axes.ChordsThe midpoints of a set of parallel chords of an ellipse are collinear.[20]:p.147Latus rectum The chords of an ellipse which are perpendicular to the major axis and pass through one of its fociare called the latera recta of the ellipse. The length of each latus rectum is 2b2/a.Curvature The curvature is1a2b2(x2a4+y2b4)32. A local normal to the ellipse bisects the angle F1PF2 shownin the gure above. This is evident graphically in the parallelogram method of construction, and can be provenanalytically, for example by using the parametric form in canonical position, as given below.3.3.2 Projective geometryIn a projective geometry dened over a eld, a conic section can be dened as the set of all points of intersectionbetween corresponding lines of two pencils of lines in a plane which are related by a projective, but not perspective,map (see Steiners theorem). By projective duality, a conic section can also be dened as the envelope of all lines thatconnect corresponding points of two lines which are related by a projective, but not perspective, map.In a pappian projective plane (one dened over a eld), all conic sections are equivalent to each other, and the dierenttypes of conic sections are determined by how they intersect the line at innity, denoted by . An ellipse is a conicsection which does not intersect this line. A parabola is a conic section that is tangent to , and a hyperbola is onethat crosses twice.[21] Since an ellipse does not intersect the line at innity, it properly belongs to the ane planedetermined by removing the line at innity and all of its points from the projective plane.50 CHAPTER 3. ELLIPSEAne spaceAn ellipse is also the result of projecting a circle, sphere, or ellipse in a three dimensional ane space onto a plane(at), by parallel lines. This is a special case of conical (perspective) projection of any of those geometric objects inthe ane space from a point O onto a plane P, when the point O lies in the plane at innity of the ane space.