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| {{about|the geometric figure}}
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| [[File:Conicas1.PNG|right|thumb|An ellipse obtained as the intersection of a [[cone (geometry)|cone]] with an inclined plane.]]
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| [[File:Saturn - Lord of the Rings.jpg|right|thumb|The rings of [[Saturn]] are circular, but when seen partially edge on, as in this image, they appear to be ellipses. In addition, the planet itself is an [[ellipsoid]], flatter at the poles than the equator. Picture by [[European Southern Observatory|ESO]]]]
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| In [[mathematics]], an '''ellipse''' is a [[plane curve|curve on a plane]] surrounding two [[focus (geometry)|focal points]] such that a straight line drawn from one of the focal points to any point on the curve and then back to the other focal point has the same length for every point on the curve. As such, it is a generalization of a circle which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how 'elongated' it is) is represented by its [[eccentricity (mathematics)|eccentricity]] which for an ellipse can be any number from 0 (the [[limiting case]] of a [[circle]]) to arbitrarily close to but less than 1.
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| Ellipses are the [[closed curve|closed]] type of [[conic section]]: a plane curve that results from the intersection of a [[Cone (geometry)|cone]] by a [[plane (mathematics)|plane]]. (See figure to the right.) Ellipses have many similarities with the other two forms of conic sections: the [[parabola]]s and the [[hyperbola]]s, both of which are [[open curve|open]] and [[unbounded set|unbounded]].
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| [[Analytical geometry|Analytically]], an ellipse can also be defined as the set of points such that the ratio of the distance of each point on the curve from a given point (called a focus or focal point) to the distance from that same point on the curve to a given line (called the [[directrix (conic section)|directrix]]) is a constant, called the eccentricity of the ellipse.
| | Pozycjonować paginę internetową wolno na wiele sposób - można sumować ją do rejestrów stron, wymieniać się linkami, wierzyć linki, czy tymiż po prostu skoncentrować się na jej rozkręcaniu i zaufać na żywiołowy postęp wartości merytorycznej stronicy, który sam przetłumaczy się na rozwój kwoty użytkowników.<br><br>Czy obstawo więc pozycjonować paginę a jeżeli tak, to w jaki sposób? Pozycjonować stronicę zawsze czato. Niezależnie od owego, czy jedziemy blog o pielęgnacji ogródka, czy tymże stronicę z informatorami dla majsterkowiczów, zaś skończywszy na portalu ze ogarnięciami - solidne funkcje w wyszukiwarką mieszają się na wzrost kwoty odbiorców. W jaki sposób pozycjonować? Tutaj nie ma jednej, błyskotliwej metody. Najlepiej przekazać się na przejście zawodowców, jacy za malusieńką należnością poczęstują się pozycjonowania polskiej witryny zaś pozwolą na siebie wszystkie potencjalne, pejoratywne konsekwencje pozycjonowania.<br><br>Opieko ale pamiętać, że pozycjonowanie nie jest biegiem jednostkowym. Jeśliby tylko zakończeniem hołubić o pozę naszej strony internetowej, to możliwie w cugu kilku najbliższych tygodniach wypadnie ona w efektach wyszukiwań. I niezwyczajne wahania pozy zdołają sprowadzić jej skrupulatne wykluczenie z efektów jako karę za bawienie nieporządnych przedsięwzięć w stosunku do rywalizacji natomiast samemu wyszukiwarki internetowej.<br><br>Here is my web blog; [http://katalog.gazeta.pl/internet,i,komputery/pozycjonowanie,bialystok,s,48672/ pozycjonowanie - Białystok] |
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| Ellipses are common in physics, astronomy and engineering. For example, the [[orbits]] of planets are ellipses with the Sun at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shape of planets and stars are often well described by [[ellipsoids]]. Ellipses also arise as images of a circle under [[parallel projection]] and the bounded cases of [[perspective projection]], which are simply intersections of the projective cone with the plane of projection. It is also the simplest [[Lissajous figure]], formed when the horizontal and vertical motions are [[Sine wave|sinusoid]]s with the same frequency. A similar effect leads to [[elliptical polarization]] of light in [[optics]].
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| The name ἔλλειψις was given by [[Apollonius of Perga]] in his ''Conics'', emphasizing the connection of the curve with "application of areas".
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| ==Elements of an ellipse==
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| {{see also|Conic section#Features|l1=features of conic sections}}
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| [[File:Ellipse Properties of Directrix and String Construction.svg|thumb|right|The ellipse and some of its mathematical properties.|278x278px]]
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| Ellipses have two mutually perpendicular axes about which the ellipse is [[symmetric]]. These axes intersect at the center of the ellipse due to this symmetry. The larger of these two axes, which corresponds to the largest distance between [[antipodal point|antipodal]] points on the ellipse, is called the '''major axis''' or '''transverse diameter'''. (On the figure to the right it is represented by the line between the point labeled '''''−a''''' and the point labeled '''''a'''''.) The smaller of these two axes, and the smallest distance across the ellipse, is called the '''minor axis''' or '''conjugate diameter'''.<ref>
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| {{cite book |authorlink=Charles Haynes Haswell | last = Haswell | first = Charles Haynes | url = http://books.google.com/books?id=Uk4wAAAAMAAJ&pg=RA1-PA381&zoom=3| title = Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas | publisher = Harper & Brothers | year = 1920 }}
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| </ref>
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| (On the figure to the right it is represented by the line between the point labeled '''''−b''''' to the point labeled '''''b'''''.)
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| The '''[[semi-major axis]]''' (denoted by ''a'' in the figure) and the '''[[semi-minor axis]]''' (denoted by ''b'' in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the '''major''' and '''minor semi-axes''',<ref>
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| {{cite book |authorlink=John Herschel |first=Sir John Frederick William |last=Herschel |title=A treatise on astronomy |url=http://books.google.com/books?id=hh0uNybw1ZUC&pg=PA256 |year=1842 |publisher=Lea & Blanchard |page=256}}
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| </ref><ref> | |
| {{cite book |first=John |last=Lankford |title=History of Astronomy: An Encyclopedia |url=http://books.google.com/books?id=berWESi5c5QC&pg=PA194 |year=1997 |publisher=Taylor & Francis |isbn=978-0-8153-0322-0 |pages=194}}
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| </ref> the '''major''' and '''minor semiaxes''',<ref>
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| {{cite book |first1=Viktor Vasilʹevich |last1=Prasolov |first2=Vladimir Mikhaĭlovich |last2=Tikhomirov |title=Geometry |url=http://books.google.com/books?id=t7kbhDDUFSkC&pg=PA80 |year=2001 |publisher=American Mathematical Society |isbn=978-0-8218-2038-4|page=80}}
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| </ref><ref>
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| {{cite book |first=Donald |last=Fenna |title=Cartographic Science: A Compendium of Map Projections, With Derivations |url=http://books.google.com/books?id=8LZeu8RxOIsC&pg=PA24 |year=2007 |publisher=CRC Press |isbn=978-0-8493-8169-0 |page=24}}
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| </ref> or '''major [[radius]]''' and '''minor radius'''.<ref>
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| {{cite book |title=AutoCAD release 13 command reference |url=http://books.google.com/books?id=q4hRAAAAMAAJ |year=1994 |publisher=Autodesk, Inc. |page=216}}
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| </ref><ref>
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| {{cite book |first=David |last=Salomon |title=Curves And Surfaces for Computer Graphics |url=http://books.google.com/books?id=m0Je92uycVAC&pg=PA365 |year=2006 |publisher=Birkhäuser |isbn=978-0-387-24196-8 |pages=365}}
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| </ref><ref>
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| {{cite book |first1=Frank |last1=Kreith |first2=D. Yogi |last2=Goswami |title=The CRC Handbook Of Mechanical Engineering |url=http://books.google.com/books?id=_wlZ5LHTyBIC&pg=SA11-PA8 |year=2005 |publisher=CRC Press |isbn=978-0-8493-0866-6 |pages=11–8 |quote=Circles and Ellipses (11.3.2)}}
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| </ref><ref>
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| [[The Mathematical Association of America]] (1976), [http://books.google.com.br/books?id=Xpk0AAAAIAAJ&q=ellipse+%22major+radius%22&dq=ellipse+%22major+radius%22&lr=&num=20&client=firefox-a&hl=en The American Mathematical Monthly, vol. 83, page 207]
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| </ref>
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| The four points where these axes cross the ellipse are the '''[[vertex (curve)|vertices]]''' and are marked as '''''a''''', '''''−a''''', '''''b''''', and '''''−b'''''. In addition to being at the largest and smallest distance from the center, these points are where the curvature of the ellipse is maximum and minimum.<ref>
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| {{citation|title=Elementary Geometry of Differentiable Curves: An Undergraduate Introduction|first=C. G.|last=Gibson|publisher=Cambridge University Press|year=2001|isbn=9780521011075|page=127}}.
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| </ref>
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| The two '''[[focus (geometry)|foci]] ''' (plural of '''focus''' and the term '''focal points''' is also used) of an ellipse are two special points ''F<sub>1</sub>'' and ''F<sub>2</sub>'' on the ellipse's major axis that are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis (''PF<sub>1</sub>'' + ''PF<sub>2</sub>'' = 2''a''). (On the figure to the right this corresponds to the sum of the two green lines equaling the length of the major axis that goes from '''''−a''''' to '''''a'''''.)
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| The distance to the focal point from the center of the ellipse is sometimes called the '''linear eccentricity''', ''f'', of the ellipse. Here it is denoted by ''f'', but it is often denoted by ''c''. Due to the [[Pythagorean theorem]] and the definition of the ellipse explained in the previous paragraph: ''f''<sup>2</sup> = ''a''<sup>2</sup> −''b''<sup>2</sup>.
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| A second equivalent method of constructing an ellipse using a directrix is shown on the plot as the three blue lines. (See the [[#Directrix|Directrix section]] of this article for more information about this method). The dashed blue line is the directrix of the ellipse shown.
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| The '''[[eccentricity (mathematics)|eccentricity]]''' of an ellipse, usually denoted by ''ε'' or ''e'', is the ratio of the distance between the two foci, to the length of the major axis or ''e'' = 2''f''/2''a'' = ''f''/''a''. For an ellipse the eccentricity is between 0 and 1 (0 < ''e'' < 1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity [[limit of a function|tends toward]] 1, the ellipse gets a more elongated shape. It tends towards a line segment ([[#Line segment as a type of degenerate ellipse|see below]]) if the two foci remain a finite distance apart and a [[parabola]] if one focus is kept fixed as the other is allowed to move arbitrarily far away. The eccentricity is also equal to the ratio of the distance (such as the (blue) line ''PF<sub>2</sub>'') from any particular point on an ellipse to one of the foci to the perpendicular distance to the directrix from the same point (line ''PD''), ''e'' = ''PF<sub>2</sub>''/''PD''.
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| ==Drawing ellipses==
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| ===Pins-and-string method===
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| [[File:Drawing an ellipse via two tacks a loop and a pen.jpg|thumb|right|Drawing an ellipse with two pins, a loop, and a pen.|300x300px]]
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| The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two [[drawing pin]]s, a length of string, and a pencil.<ref>{{harvnb|Besant|1907|p=57}}</ref> In this method, pins are pushed into the paper at two points which will become the ellipse's foci. A string tied at each end to the two pins and the tip 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 is moved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the ''gardener's ellipse''.<ref>{{cite book
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| |title=The Practical Draughtsman's Book of Industrial Design
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| |first=Aîné|last=Armengaud |publisher=Longman, Brown, Green, and Longmans|year=1853|page=16|chapter=Ovals, Ellipses, Parabolas, Volutes, etc. §53
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| |url=http://books.google.com/books?id=6skOAAAAQAAJ&pg=PA15#v=onepage&f=false}}</ref>
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| ===Trammel method===
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| An ellipse can also be drawn using a [[ruler]], a [[set square]], and a pencil:
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| :Draw two perpendicular lines ''M'',''N'' on the paper; these will be the major (''M'') and minor (''N'') axes of the ellipse. Mark three points ''A'', ''B'', ''C'' on the ruler. ''A->C'' being the length of the semi-major axis and ''B->C'' the length of the semi-minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point ''A'' always on line ''N'', and ''B'' on line ''M''. With the other hand, keep the pencil's tip on the paper, following point ''C'' of the ruler. The tip will trace out an ellipse.
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| [[File:Archimedes Trammel.gif|thumb|left|[[Trammel of Archimedes]] (ellipsograph) animation|300x300px]]
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| The [[trammel of Archimedes]] or ellipsograph is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point ''C'') at one end, and two adjustable side pins (points ''A'' and ''B'') that slide into two perpendicular slots cut into a metal plate.<ref>{{cite book |first=Henry T. |last=Brown |title=Five Hundred and Seven Mechanical Movements: Embracing All Those which are Most Important in Dynamics, Hydraulics, Hydrostatics, Pneumatics, Steam Engines, Mill and Other Gearing, Presses, Horology, and Miscellaneous Machinery; and Including Many Movements Never Before Published, and Several which Have Only Recently Come Into Use |url=http://books.google.com/books?id=TFwOAAAAYAAJ&pg=PA41 |year=1881 |publisher=Brown & Brown |pages=40–41 section 152}}</ref> The mechanism can be used with a [[router (woodworking)|router]] to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".
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| ===Parallelogram method===
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| [[File:Ellipse construction - parallelogram method.gif|thumb|right|Ellipse construction applying the parallelogram method|278x278px]]
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| In the parallelogram method, an ellipse is constructed point by point using equally spaced points on two horizontal lines and equally spaced points on two vertical lines. It is based on [[Steiner's theorem (geometry)|Steiner's theorem]] on the generation of [[conic sections]]. Similar methods exist for the [[parabola]] and [[hyperbola]].
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| ===Approximations to ellipses===
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| An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.
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| <br style="clear:both" /> | |
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| ==Mathematical definitions and properties==
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| ===In Euclidean geometry===
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| ====Definition====
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| In [[Euclidean geometry]], the ellipse is usually defined as the bounded case of a conic section, or as the set of points such that the sum of the distances to two fixed points (the [[#Focus|foci]]) is constant. The ellipse can also be defined as the set of points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positive fraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called the [[#Directrix|directrix]]). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from one point in the plane (a focus) and a particular circle, the [[#Circular directrix|directrix circle]] (whose center is the other focus).
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| The equivalence of these definitions can be proved using the [[Dandelin spheres]].
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| ====Equations====
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| The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is
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| <math>\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.</math>
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| This means any noncircular ellipse is a compressed (or stretched) circle. If a circle is treated like an ellipse, then the area of the ellipse would be proportional to the length of either axis (i.e. doubling the length of an axis in a circular ellipse would create an ellipse with double the area of the original circle).
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| ==== Focus ====
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| The distance from the center ''C'' to either focus is ''f'' = ''ae'', which can be expressed in terms of the major and minor radii:
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| :<math>f = \sqrt{a^2-b^2}.</math>
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| ====Eccentricity====
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| The [[Eccentricity (mathematics)|eccentricity]] of the ellipse (commonly denoted as either ''e'' or <math>\varepsilon</math>) is
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| : <math>e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}}
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| =\sqrt{1-\left(\frac{b}{a}\right)^2}
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| =f/a</math>
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| (where again ''a'' and ''b'' are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the [[flattening]] factor <math>g=1-\frac {b}{a}=1-\sqrt{1-e^2},</math>
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| :<math>e=\sqrt{g(2-g)}.</math>
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| [[Eccentricity (mathematics)#Other formulas for the eccentricity of an ellipse|Other formulas for the eccentricity of an ellipse]] are listed in the article on [[Eccentricity (mathematics)|eccentricity of conic sections]].
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| [[Conic section#Eccentricity in terms of parameters of the quadratic form|Formulas for the eccentricity of an ellipse]] that is expressed in the more general [[quadratic form]] are described in the article dedicated to [[conic section]]s.
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| ====Directrix====
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| [[File:Ellipse Properties of Directrix.svg|thumb|300px]]
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| Each focus ''F'' of the ellipse is associated with a line parallel to the minor axis called a [[directrix (conic section)|directrix]]. Refer to the illustration on the right, in which the ellipse is centered at the origin. The distance from any point ''P'' on the ellipse to the focus ''F'' is a constant fraction of that point's perpendicular distance to the directrix, resulting in the equality ''e'' = ''PF''/''PD''. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the [[Dandelin spheres]]) can be taken as another definition of the ellipse.<br>
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| Besides the well-known ratio ''e'' = ''f''/''a'', where ''f'' is the distance from the center to the focus and ''a'' is the distance from the center to a vertex (most sharply curved point of the ellipse), it is also true that ''e'' = ''a''/''d'', where ''d'' is the distance from the center to the directrix.
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| ====Circular directrix====
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| The ellipse can also be defined 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 ellipse's major axis, so the focus and the entire ellipse are inside the directrix circle.
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| ====Ellipse as hypotrochoid====
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| [[File:Ellipse as hypotrochoid.gif|right|300|thumb|An ellipse (in red) as a special case of the [[hypotrochoid]] with ''R'' = 2''r''.]]
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| The ellipse is a special case of the [[hypotrochoid]] when ''R'' = 2''r''.
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| ==== Area ====
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| The [[area]] <math>A_\text{ellipse}</math> enclosed by an ellipse is:
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| :<math>A_\text{ellipse} = \pi a b</math>
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| where ''a'' and ''b'' are one-half of the ellipse's major and minor axes respectively.
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| An ellipse defined implicitly by <math>A x^2+ B x y + C y^2 = 1 </math> has area <math>\frac{2\pi}{\sqrt{ 4 A C - B^2 }}</math>.
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| The area formula ''πab'' is easy to understand: start with a circle of radius ''b''
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| (so its area is π''b''<sup>2</sup>) and stretch it by a factor ''a''/''b'' to make an ellipse.
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| This increases the area by the same factor: π''b''<sup>2</sup>(''a''/''b'') = π''ab''.
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| For the ellipse in standard form, <math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>, and hence <math>y=\pm \sqrt{\frac{a^2b^2-b^2x^2}{a^2}}</math>, with horizontal intercepts at ± ''a'', the area <math>A_\text{ellipse}</math> can be computed as twice the integral of the positive square root:
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| :<math>\begin{align}
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| A_\text{ellipse} &= \int_{-a}^a 2b\sqrt{1-x^2/a^2}\,dx\\
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| &= \frac ba \int_{-a}^a 2\sqrt{a^2-x^2}\,dx.
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| \end{align}
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| </math>
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| The second integral is the area of a circle of radius <math>a</math>, i.e.,
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| <math>\pi a^2</math>; thus we have:
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| :<math>A_\text{ellipse} = \frac{b}{a}A_\text{circle} = \pi ab.</math>
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| The area formula can also be proven in terms of [[polar coordinates]] using the coordinate transformation
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| <math> \bold{T}(r,\theta) = ( ra \cos\theta , rb \sin\theta ). </math>
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| Any point inside the ellipse with ''x''-intercept ''a'' and ''y''-intercept ''b'' can be defined in terms of ''r'' and <math>\theta</math>, where <math> 0 \leqslant r \leqslant 1</math> and <math> 0 \leqslant \theta \leqslant 2\pi </math>.
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| To define the area differential in such coordinates we use the [[Jacobian matrix and determinant|Jacobian matrix]] of the coordinate transformation times <math>dr\,d\theta</math>:
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| :<math>
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| \begin{align}
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| dA_\text{ellipse} &= \det \begin{pmatrix}
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| \frac{\partial \bold{T}}{\partial r} & \frac{\partial \bold{T}}{\partial \theta}\\
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| \end{pmatrix}
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| \,dr\,d\theta \\
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| &= \det \begin{pmatrix}
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| a\cos\theta & -ra\sin\theta \\
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| b\sin\theta & rb\cos\theta
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| \end{pmatrix}
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| \,dr\,d\theta \\
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| &= abr\,dr\,d\theta.
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| \end{align}
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| </math>
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| We now integrate over the ellipse to find the area:
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| :<math>A_\text{ellipse} = \int_\text{ellipse} dA_\text{ellipse} = \iint_\text{ellipse} abr\,dr\,d\theta = ab \int_{0}^{2\pi} \int_{0}^{1}r\,dr\,d\theta = ab\pi.</math>
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| ====Circumference====
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| The [[circumference]] <math>C</math> of an ellipse is:
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| :<math>C = 4 a E(e)</math>
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| where again ''a'' is the length of the semi-major axis and ''e'' is the eccentricity and where the function <math>E</math> is the [[Elliptic integral#Complete elliptic integral of the second kind|complete elliptic integral of the second kind]]. This
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| may be evaluated directly using the
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| [[Carlson symmetric form]].<ref>{{cite doi
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| |10.1007/BF02198293}}</ref> This gives a succinct and rapidly converging method for evaluating the circumference.<ref>
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| {{citation
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| |title = Python code for the circumference of an ellipse in terms of the complete elliptic integral of the second kind
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| |url = http://paulbourke.net/geometry/ellipsecirc/python.code
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| |accessdate = 2013-12-28}}</ref>
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| The exact [[infinite series]] is:
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| :<math>C = 2\pi a \left[{1 - \left({1\over 2}\right)^2e^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{e^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{e^6\over5} - \cdots}\right]</math>
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| or
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| :<math>C = 2\pi a \left[1 - \sum_{n=1}^\infty \left(\frac{(2n - 1)!!}{2^n n!}\right)^2 \frac{e^{2n}}{2n - 1}\right],</math>
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| where <math>n!!</math> is the [[double factorial]].
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| Unfortunately, this series converges rather slowly; however, by expanding in terms of <math> h = (a-b)^2/(a+b)^2 </math>,
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| Ivory<ref>{{cite journal
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| |ref = {{harvid|Ivory|1798}}
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| |last = Ivory
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| |first = J.
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| |title = A new series for the rectification of the ellipsis
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| |authorlink = James Ivory (mathematician)
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| |journal = Transactions of the Royal Society of Edinburgh
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| |year = 1798
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| |volume = 4
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| |pages = 177–190
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| |url = http://books.google.com/books?id=FaUaqZZYYPAC&pg=PA177
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| }}</ref>
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| and
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| Bessel<ref>{{cite journal
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| |ref = {{harvid|Bessel|1825}}
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| |last = Bessel
| |
| |first = F. W.
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| |title = The calculation of longitude and latitude from geodesic measurements (1825)
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| |authorlink = Friedrich Bessel
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| |journal = Astron. Nachr.
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| |year = 2010
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| |volume = 331
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| |number = 8
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| |pages = 852–861
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| |arxiv = 0908.1824
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| |doi = 10.1002/asna.201011352
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| |postscript=. English translation of Astron. Nachr. '''4''', 241–254 (1825).
| |
| }}</ref>
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| derived an expression which converges much more rapidly,
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| :<math>C = \pi (a + b) \left[1 + \sum_{n=1}^\infty \left(\frac{(2n - 1)!!}{2^n n!}\right)^2 \frac{h^n}{(2n - 1)^2}\right].</math>
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| | |
| A good [[approximation]] is [[Srinivasa Ramanujan|Ramanujan]]'s:
| |
| | |
| :<math>C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)}\right]</math>
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| | |
| and a better [[approximation]] is
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| :<math>C\approx\pi\left(a+b\right)\left(1+\frac{3h}{10+\sqrt{4-3h}}\right).\!\,</math>
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| For the special case where the minor axis is half the major axis, these become:
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| :<math>C \approx \frac{\pi a (9 - \sqrt{35})}{2}</math>
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| or, as an estimate of the better approximation,
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| :<math>C \approx \frac{a}{2} \sqrt{93 + \frac{1}{2} \sqrt{3}}</math>
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| | |
| More generally, the [[arc length]] of a portion of the circumference, as a function of the angle subtended, is given by an incomplete [[elliptic integral]].
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| {{See also|Meridian arc#Meridian distance on the ellipsoid}}
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| The [[inverse function]], the angle subtended as a function of the arc length, is given by the [[elliptic functions]].{{Citation needed|date=October 2010}}
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| | |
| ====Chords====
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| | |
| The midpoints of a set of parallel [[Chord (geometry)|chord]]s of an ellipse are collinear.<ref>Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in ''Mathematical Plums'' (R. Honsberger, editor). Washington, DC: Mathematical Association of America, 1979.</ref>{{rp|p.147}}
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| ===== Latus rectum =====
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| The chords of an ellipse which are [[perpendicular]] to the major axis and pass through one of its foci are called the [[latus rectum|latera recta]] of the ellipse. The length of each latus rectum is {{math|{{sfrac|2''b''<sup>2</sup>|''a''}}}}.
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| | |
| ===== Curvature =====
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| The curvature is <math>\frac{1}{a^{2}b^{2}}\left(\frac{x^{2}}{a^{4}}+\frac{y^{2}}{b^{4}}\right)^{-\frac{3}{2}}</math>.
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| | |
| ===In projective geometry===
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| In [[projective geometry]], an ellipse can be defined as the set of all points of intersection between corresponding lines of two [[pencil (mathematics)|pencils of lines]] which are related by a [[projective map]] (see [[Steiner's theorem (geometry)|Steiner's theorem]]). By [[projective duality]], an ellipse can be defined also as the [[envelope (mathematics)|envelope]] of all lines that connect corresponding points of two lines which are related by a projective map.
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| This definition also generates hyperbolae and parabolae. However, in projective geometry every conic section is equivalent to an ellipse. A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.
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| | |
| An ellipse is also the result of [[oblique projection|projecting]] a circle, sphere, or ellipse in three dimensions onto a plane, by [[parallel (geometry)|parallel]] lines. It is also the result of conical (perspective) projection of any of those geometric objects from a point ''O'' onto a plane ''P'', provided that the plane ''Q'' that goes through ''O'' and is parallel to ''P'' does not cut the object. The image of an ellipse by any [[affine map]] is an ellipse, and so is the image of an ellipse by any projective map ''M'' such that the line ''M''<sup>−1</sup>(Ω) does not touch or cross the ellipse.
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| | |
| ===In analytic geometry===
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| ====General ellipse====
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| In [[analytic geometry]], the ellipse is defined as the set of points <math>(X,Y)</math> of the [[Cartesian plane]] that, in non-degenerate cases, satisfy the [[Implicit and explicit functions|implicit]] equation<ref>{{cite book
| |
| |title=Precalculus with Limits
| |
| |first1=Ron
| |
| |last1=Larson
| |
| |first2=Robert P.
| |
| |last2=Hostetler
| |
| |first3=David C.
| |
| |last3=Falvo
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| |publisher=Cengage Learning
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| |year=2006
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| |isbn=0-618-66089-5
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| |page=767
| |
| |url=http://books.google.com/books?id=yMdHnyerji8C |chapterurl=http://books.google.com/books?id=yMdHnyerji8C&pg=PA767 |chapter=Chapter 10}}
| |
| </ref><ref>{{cite book
| |
| |title=Precalculus
| |
| |first1=Cynthia Y.
| |
| |last1=Young
| |
| |publisher=John Wiley and Sons
| |
| |year=2010
| |
| |isbn=0-471-75684-9
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| |page=831
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| |url=http://books.google.com/books?id=9HRLAn326zEC |chapterurl=http://books.google.com/books?id=9HRLAn326zEC&pg=PA831 |chapter=Chapter 9}}
| |
| </ref>
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| | |
| :<math>~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0</math>
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| provided <math>B^2 - 4AC < 0.</math>
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| | |
| To distinguish the degenerate cases from the non-degenerate case, let ''∆'' be the [[determinant]]
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| :<math>\begin{vmatrix} A & B/2 & D/2\\B/2 & C & E/2\\D/2 & E/2 & F \end{vmatrix}</math>
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| that is,
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| :<math>\Delta = \left( AC - \frac{B^2}{4} \right) F + \frac{BED}{4} - \frac{CD^2}{4} - \frac{AE^2}{4}</math>
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| Then the ellipse is a non-degenerate real ellipse if and only if ''C∆'' < 0. If ''C∆'' > 0, we have an imaginary ellipse, and if ''∆'' = 0, we have a point ellipse.<ref>Lawrence, J. Dennis, ''A Catalog of Special Plane Curves'', Dover Publ., 1972.</ref>{{rp|p.63}}
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| ====Canonical form====
| |
| | |
| Let <math>a>b</math>. Through change of coordinates (a [[rotation of axes]] and a translation) the general ellipse can be described by the [[canonical form|canonical implicit equation]]
| |
| | |
| :<math>\frac{x^2}{a^2}+\frac{y^2}{b^2}=1</math>
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| | |
| Here <math>(x,y)</math> are the point coordinates in the canonical system, whose origin is the center <math>(X_c,Y_c)</math> of the ellipse, whose <math>x</math>-axis is the unit vector <math>(X_a,Y_a)</math> coinciding with the major axis, and whose <math>y</math>-axis is the perpendicular vector <math>(-Y_a,X_a)</math> coinciding with the minor axis. That is, <math>x = X_a(X - X_c) + Y_a(Y - Y_c)</math> and <math>y = -Y_a(X - X_c) + X_a(Y - Y_c)</math>.
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| In this system, the center is the origin <math>(0,0)</math> and the foci are <math>(-e a, 0)</math> and <math>(+e a, 0)</math>.
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| | |
| Any ellipse can be obtained by [[rotation (geometry)|rotation]] and [[translation (geometry)|translation]] of a canonical ellipse with the proper semi-diameters. Translation of an ellipse centered at <math>(X_c,Y_c)</math> is expressed as
| |
| :<math>\frac{(x - X_c)^2}{a^2}+\frac{(y - Y_c)^2}{b^2}=1</math>
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| | |
| Moreover, any canonical ellipse can be obtained by scaling the [[unit circle]] of <math>\reals^2</math>, defined by the equation
| |
| | |
| :<math>X^2+Y^2=1\,</math>
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| by factors ''a'' and ''b'' along the two axes.
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| | |
| For an ellipse in canonical form, we have
| |
| | |
| :<math> Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)}</math>
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| | |
| The distances from a point <math>(X,Y)</math> on the ellipse to the left and right foci are <math>a + e X</math> and <math>a - e X</math>, respectively.
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| | |
| ===In trigonometry===
| |
| | |
| ==== General parametric form ====
| |
| An ellipse in general position can be expressed [[parametric equation|parametrically]] as the path of a point <math>(X(t),Y(t))</math>, where
| |
| :<math>X(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi</math>
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| :<math>Y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi</math>
| |
| as the parameter ''t'' varies from 0 to 2''π''. Here <math>(X_c,Y_c)</math> is the center of the ellipse, and <math>\varphi</math> is the angle between the <math>X</Math>-axis and the major axis of the ellipse.
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| | |
| ==== Parametric form in canonical position ====
| |
| [[File:Parametric ellipse.gif|thumb|right|200px|Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly ''t'' is the angle of the blue line with the X-axis.]]
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| For an ellipse in canonical position (center at origin, major axis along the ''X''-axis), the equation simplifies to
| |
| :<math>X(t)=a\,\cos t</math>
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| :<math>Y(t)=b\,\sin t</math>
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| Note that the parameter ''t'' (called the '''[[eccentric anomaly]]''' in astronomy) is ''not'' the angle of <math>(X(t),Y(t))</math> with the ''X''-axis.
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| | |
| For a given point on an ellipse, formulae connecting the [[tangential angle]] <math>\phi</math>, the polar angle from the ellipse center <math>\theta</math>, and the [[Parametric equation|parametric angle]] ''t''<ref>If the ellipse is illustrated as a [[meridional]] one for the earth, the tangential angle is equal to geodetic [[latitude]], the angle <math>\theta</math> is the geocentric latitude, and parametric angle ''t'' is a parametric (or reduced) latitude of auxiliary circle</ref> are:<ref>[http://mathworld.wolfram.com/Ellipse.html ''Ellipse'' at MathWorld, derived from formula (58) and (60)]</ref><ref>[http://math.stackexchange.com/a/410339/41584 clarifies problems with MathWorld formula (60)]</ref><ref>[http://www.math.uoc.gr/~pamfilos/eGallery/problems/Auxiliary.html Auxiliary circle and various ellipse formulas]</ref><ref>{{cite book |author=Meeus, J. |chapter=Ch. 10: The Earth's Globe |title=Astronomical Algorithms |publisher=Willmann-Bell |year=1991 |isbn=0-943396-35-2 |page=78 }}</ref>
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| | |
| :<math>-\cot\phi = \frac{a}{b} \tan t = \frac{\tan \theta}{(1 - g)^2} = \frac{\tan \theta}{1 - e^2},</math>
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| :<math> -\tan t = \frac{b}{a} \cot\phi = \sqrt{(1 - e^2)} \cot\phi = (1 - g) \cot\phi = \frac{-\tan\theta}{\sqrt{(1 - e^2)}} = -\frac{a}{b} \tan\theta.</math>
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| ==== Polar form relative to center ====
| |
| [[File:Ellipse Polar center.svg|thumb|right|200px|Polar coordinates centered at the center.]]
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| In [[polar coordinates]], with the origin at the center of the ellipse and with the angular coordinate <math>\theta</math> measured from the major axis, the ellipse's equation is
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| :<math>r(\theta)=\frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}</math>
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| | |
| ==== Reflexive property ====
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| When a ray of light originating from one focus reflects off the inner surface of an ellipse, it always passes through the other focus.
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| ==== Polar form relative to focus ====
| |
| [[File:Ellipse Polar.svg|thumb|right|200px|Polar coordinates centered at focus.]]
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| If instead we use polar coordinates with the origin at one focus, with the angular coordinate <math>\theta = 0</math> still measured from the major axis, the ellipse's equation is
| |
| :<math>r(\theta)=\frac{a (1-e^{2})}{1 \pm e\cos\theta}</math>
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| where the sign in the denominator is negative if the reference direction <math>\theta = 0</math> points towards the center (as illustrated on the right), and positive if that direction points away from the center.
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| | |
| In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate <math>\phi</math>, the polar form is
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| :<math>r=\frac{a (1-e^{2})}{1 - e\cos(\theta - \phi)}.</math>
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| | |
| The angle <math>\theta</math> in these formulas is called the '''[[true anomaly]]''' of the point. The numerator <math>a (1-e^{2})</math> of these formulas is the '''[[semi-latus rectum]]''' of the ellipse, usually denoted <math>l</math>. It is the distance from a focus of the ellipse to the ellipse itself, measured along a line [[perpendicular]] to the major axis.
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| [[File:Elps-slr.svg|thumb|right|200px|Semi-latus rectum.]]
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| | |
| ==== General polar form ====
| |
| The following equation on the polar coordinates (''r'', ''θ'') describes a general ellipse with semidiameters ''a'' and ''b'', centered at a point (''r''<sub>0</sub>, ''θ''<sub>0</sub>), with the ''a'' axis rotated by ''φ'' relative to the polar axis:{{citation needed|date=October 2012}}
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| | |
| :<math>r(\theta )=\frac{P(\theta )+Q(\theta )}{R(\theta )}</math>
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| | |
| where
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| | |
| :<math>P(\theta )=r_0 \left[\left(b^2-a^2\right) \cos \left(\theta +\theta _0-2 \varphi
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| \right)+\left(a^2+b^2\right) \cos \left(\theta -\theta_0\right)\right]</math>
| |
| | |
| :<math>Q(\theta )=\sqrt{2} a b \sqrt{R(\theta )-2 r_0^2 \sin ^2\left(\theta -\theta_0\right)}</math>
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| | |
| :<math>R(\theta )=\left(b^2-a^2\right) \cos (2 \theta -2 \varphi )+a^2+b^2</math>
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| <!--
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| The following Gauss Map section is probably wrong!
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| Please don't uncomment it unless corrected and verified!
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| ====Gauss-map====
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| The [[Gauss map|Gauss-map]] of the ellipse, mapping every point onto the unit circle through the components of the unit normal vector in the point where the normal makes an angle <math>\beta</math> with the ''X''-axis is given by:{{Citation needed|date=May 2010}}
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| -->
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| <!-- The equations are indeed correct with the non-squared a and b in the numerators. See talk page.-->
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| <!--
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| :<math>X(\beta) = \frac{a\cos\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}}</math>
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| :<math>Y(\beta) =\frac{b\sin\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}}.</math>
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| It is easily verified that
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| :<math>X(\beta)^2 + Y(\beta)^2=1 \, .</math>
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| | |
| '''Note''': the Gauss map is not to be confused with another way to write the equation of the ellipse itself:
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| :<math>x(\beta) = \frac{a^2\cos\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}}</math>
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| :<math>y(\beta) =\frac{b^2\sin\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}},</math>
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| for which it is easily verified that
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| :<math>\left(\frac{x(\beta)}{a}\right)^2 + \left(\frac{y(\beta)}{b}\right)^2=1.</math>
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| -->
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| | |
| ==== Angular eccentricity ====
| |
| | |
| The [[angular eccentricity]] <math>\alpha</math> is the angle whose sine is the eccentricity ''e''; that is,
| |
| | |
| :<math>\alpha=\sin^{-1}(e)=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!</math>
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| | |
| ===Degrees of freedom===
| |
| An ellipse in the plane has five [[Degrees of freedom (physics and chemistry)|degrees of freedom]] (the same as a general conic section), defining its position, orientation, shape, and scale. In comparison, circles have only three degrees of freedom (position and scale), while parabolae have four. Said another way, the set of all ellipses in the plane, with any natural metric (such as the [[Hausdorff metric|Hausdorff distance]]) is a five-dimensional [[manifold (mathematics)|manifold]]. These degrees can be identified with, for example, the coefficients ''A'',''B'',''C'',''D'',''E'' of the implicit equation, or with the coefficients ''X''<sub>c</sub>, ''Y''<sub>c</sub>, ''φ'', ''a'', ''b'' of the general parametric form.
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| ==Ellipses in physics==
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| | |
| === Elliptical reflectors and acoustics ===
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| If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves created by that disturbance, after being [[reflection (physics)|reflected]] by the walls, will converge simultaneously to a single point — the second focus. This is a consequence of the total travel length being the same along any wall-bouncing path between the two foci.
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| Similarly, if a light source is placed at one focus of an elliptic [[mirror]], all light rays on the plane of the ellipse are reflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternative definition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back to the center.) If the ellipse is rotated along its major axis to produce an [[ellipsoid]]al mirror (specifically, a [[prolate spheroid]]), this property will hold for all rays out of the source. Alternatively, a cylindrical mirror with elliptical cross-section can be used to focus light from a linear [[fluorescent lamp]] along a line of the paper; such mirrors are used in some [[image scanner|document scanner]]s.
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| | |
| Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear a person standing at the other focus remarkably well. The effect is even more evident under a [[cupola|vaulted roof]] shaped as a section of a prolate spheroid. Such a room is called a ''[[whisper chamber]]''. The same effect can be demonstrated with two reflectors shaped like the end caps of such a spheroid, placed facing each other at the proper distance. Examples are the [[National Statuary Hall]] at the [[United States Capitol]] (where [[John Quincy Adams]] is said to have used this property for eavesdropping on political matters); the [[Mormon Tabernacle]] at [[Temple Square]] in [[Salt Lake City]], [[Utah]]; at an exhibit on sound at the [[Museum of Science and Industry (Chicago)|Museum of Science and Industry]] in [[Chicago]]; in front of the [[University of Illinois at Urbana-Champaign]] Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in the [[Alhambra]].
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| | |
| === Planetary orbits ===
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| {{Main|Elliptic orbit}}
| |
| | |
| In the 17th century, [[Johannes Kepler]] discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, in his [[Kepler's laws of planetary motion|first law of planetary motion]]. Later, [[Isaac Newton]] explained this as a corollary of his [[Newton's law of universal gravitation|law of universal gravitation]].
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| More generally, in the gravitational [[two-body problem]], if the two bodies are bound to each other (i.e., the total energy is negative), their orbits are [[Similarity (geometry)|similar]] ellipses with the common [[barycenter]] being one of the foci of each ellipse. The other focus of either ellipse has no known physical significance. Interestingly, the orbit of either body in the reference frame of the other is also an ellipse, with the other body at one focus.
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| | |
| Keplerian elliptical orbits are the result of any radially directed attraction force whose strength is inversely proportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles in empty space would also be an ellipse. (However, this conclusion ignores losses due to [[electromagnetic radiation]] and [[quantum mechanics|quantum effects]] which become significant when the particles are moving at high speed.)
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| | |
| For [[elliptical orbit]]s, useful relations involving the eccentricity <math>e</math> are:
| |
| | |
| :<math>e=\frac{r_{a}-r_{p}}{r_{a}+r_{p}}=\frac{r_{a}-r_{p}}{2a}</math>
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| | |
| :<math>r_{a}=(1+e)a</math>
| |
| | |
| :<math>r_{p}=(1-e)a</math>
| |
| | |
| where
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| *<math>r_{a}</math> is the radius at [[apoapsis]] (the farthest distance)
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| *<math>r_{p}</math> is the radius at [[periapsis]] (the closest distance)
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| *<math>a</math> is the length of the [[semi-major axis]]
| |
| | |
| Also, in terms of <math>r_{a}</math> and <math>r_{p}</math>, the semi-major axis <math>a</math> is their [[arithmetic mean]], the semi-minor axis <math>b</math> is their [[geometric mean]], and the [[conic section#Features|semi-latus rectum]] <math>l</math> is their [[harmonic mean]]. In other words,
| |
| | |
| :<math>a=\frac{r_{a}+r_{p}}{2}</math>
| |
| | |
| :<math>b=\sqrt[2]{r_{a}\cdot r_{p}}</math>
| |
| | |
| :<math>l=\frac{2}{\frac{1}{r_{a}}+\frac{1}{r_{p}}}=\frac{2r_{a}r_{p}}{r_{a}+r_{p}}</math>.
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| | |
| === Harmonic oscillators ===
| |
| | |
| The general solution for a [[harmonic oscillator]] in two or more [[dimension]]s is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by a perfectly elastic [[spring (mechanics)|spring]]; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center of attraction at the geometric center of the ellipse, and have fairly simple equations of motion.
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| | |
| === Phase visualization ===
| |
| | |
| In [[electronics]], the relative phase of two sinusoidal signals can be compared by feeding them to the vertical and horizontal inputs of an [[oscilloscope]]. If the display is an ellipse, rather than a straight line, the two signals are out of phase.
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| | |
| === Elliptical gears ===
| |
| | |
| Two [[non-circular gear]]s with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a [[link chain]] or [[toothed belt|timing belt]], or in the case of a bicycle the main [[chainwheel#ovoid chainwheels|chainring]] may be elliptical, or an [[ovoid]] similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable [[angular speed]] or [[torque]] from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying [[mechanical advantage]].
| |
| | |
| Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.<ref>
| |
| David Drew.
| |
| "Elliptical Gears".
| |
| [http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Drew/Emat6890/Elliptical%20Gears.htm]
| |
| </ref>
| |
| | |
| An example gear application would be a device that winds thread onto a conical [[bobbin]] on a [[Spinning (textiles)|spinning]] machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.<ref>{{cite book |first=George B. |last=Grant |title=A treatise on gear wheels |url=http://books.google.com/books?id=fPoOAAAAYAAJ&pg=PA72 |year=1906 |publisher=Philadelphia Gear Works |page=72}}</ref>
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| | |
| === Optics ===
| |
| * In a material that is optically [[anisotropic]] ([[birefringent]]), the [[refractive index]] depends on the direction of the light. The dependency can be described by an [[index ellipsoid]]. (If the material is optically [[isotropic]], this ellipsoid is a sphere.)
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| * In lamp-[[laser pumping|pumped]] solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from the pump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal axis).<ref>http://www.rp-photonics.com/lamp_pumped_lasers.html</ref>
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| * In laser-plasma produced [[Extreme ultraviolet|EUV]] light sources used in microchip [[Extreme ultraviolet lithography|lithography]], EUV light is generated by plasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the input of the lithography machine.<ref>http://www.cymer.com/plasma_chamber_detail/</ref>
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| | |
| ==Ellipses in statistics and finance==
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| In [[statistics]], a bivariate [[random vector]] (''X'', ''Y'') is [[elliptical distribution|jointly elliptically distributed]] if its iso-density contours — loci of equal values of the density function — are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are [[ellipsoid]]s. A special case is the [[multivariate normal distribution]]. The elliptical distributions are important in [[finance]] because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance — that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.<ref>{{cite journal |author=Chamberlain, G. |title=A characterization of the distributions that imply mean—Variance utility functions |journal=[[Journal of Economic Theory]] |volume=29 |issue=1 |pages=185–201 |date=February 1983 |doi=10.1016/0022-0531(83)90129-1 |url=http://www.sciencedirect.com/science/article/pii/0022053183901291}}</ref><ref>{{cite journal |author=Owen, J.; Rabinovitch, R. |title=On the class of elliptical distributions and their applications to the theory of portfolio choice |journal=[[Journal of Finance]] |volume=38 |issue= |pages=745–752 |date=June 1983 |jstor=2328079}}</ref>
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| == Ellipses in computer graphics ==
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| Drawing an ellipse as a [[graphics primitive]] is common in standard display libraries, such as the MacIntosh [[QuickDraw]] API, and [[Direct2D]] on Windows. [[Jack Bresenham]] at IBM is most famous for the invention of 2D drawing primitives, including line and circle drawing, using only fast integer operations such as addition and branch on carry bit. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.<ref>{{cite journal |author=Pitteway, M.L.V. |title=Algorithm for drawing ellipses or hyperbolae with a digital plotter |journal=The Computer Journal |volume=10 |issue=3 |pages=282–9 |year=1967 |doi=10.1093/comjnl/10.3.282 |url=http://comjnl.oxfordjournals.org/content/10/3/282.abstract}}</ref> Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.<ref>{{cite journal |author=Van Aken, J.R. |title=An Efficient Ellipse-Drawing Algorithm |journal=IEEE Computer Graphics and Applications |volume=4 |issue=9 |pages=24–35 |date=September 1984 |doi=10.1109/MCG.1984.275994 }}</ref>
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| In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.<ref>{{cite journal |author=Smith, L.B. |title=Drawing ellipses, hyperbolae or parabolae with a fixed number of points |journal=The Computer Journal |volume=14 |issue=1 |pages=81–86 |year=1971 |doi=10.1093/comjnl/14.1.81 |url=http://comjnl.oxfordjournals.org/content/14/1/81.short}}</ref> These algorithms need only a few multiplications and additions to calculate each vector.
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| It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.
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| ===Drawing with Bézier spline paths===
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| [[Bézier splines]] may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an [[affine transformation]] of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent [[Bézier curve]]s will behave appropriately under such transformations.
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| ==Line segment as a type of degenerate ellipse==
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| A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends.<ref>{{cite web |author=Seligman, Courtney |title=Orbital Motions Ellipses and Other Conic Sections |date=1993–2010 |work=Online Astronomy eText |url=http://cseligman.com/text/history/ellipses.htm}}</ref> Although the eccentricity is 1 this is not a parabola. A [[radial elliptic trajectory]] is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.
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| == Ellipses in optimization theory ==
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| It is sometimes useful to find the minimum bounding ellipse on a set of points. The [[ellipsoid method]] is quite useful for attacking this problem.
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| == See also ==
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| <div style="-moz-column-count:2; column-count:2;">
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| * [[Apollonius of Perga]], the classical authority
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| * [[Cartesian oval]], a generalization of the ellipse
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| * [[Circumconic and inconic]]
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| * [[Conic section]]
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| * [[Ellipsoid]], a higher dimensional analog of an ellipse
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| * [[Elliptic coordinates]], an orthogonal coordinate system based on families of ellipses and [[hyperbola]]e
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| * [[Elliptical distribution]], in statistics
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| * [[Elliptic partial differential equation]]
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| * [[Great ellipse]]
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| * [[Hyperbola]]
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| * [[Kepler's laws of planetary motion]]
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| * [[Matrix representation of conic sections]]
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| * [[n-ellipse|''n''-ellipse]], a generalization of the ellipse for ''n'' foci
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| * [[Oval]]
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| * [[Parabola]]
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| * [[Proofs involving the ellipse]]
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| * [[Spheroid]], the ellipsoid obtained by rotating an ellipse about its major or minor axis
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| * [[Steiner circumellipse]], the unique ellipse circumscribing a triangle and sharing its centroid
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| * [[Steiner inellipse]], the unique ellipse inscribed in a triangle with tangencies at the sides' midpoints
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| * [[Superellipse]], a generalization of an ellipse that can look more rectangular or more "pointy"
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| * [[true anomaly|True]], [[eccentric anomaly|eccentric]], and [[mean anomaly]]
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| *[[Geodesics on an ellipsoid]]
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| </div>
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| == References ==
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| *{{cite book |first=W.H. |last=Besant | title=Conic Sections |chapter=Chapter III. The Ellipse
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| | publisher=George Bell and Sons| location=London | year=1907
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| |url=http://books.google.com/books?id=TRJLAAAAYAAJ&pg=PA50 |page=50 |ref=harv}}
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| *{{cite book |author=Miller, Charles D.; Lial, Margaret L.; Schneider, David I. |title=Fundamentals of College Algebra |publisher=Scott Foresman/Little |location= |year=1990 |isbn=0-673-38638-4 |page=381 |edition=3rd}}
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| *{{cite book |author=Coxeter, H.S.M. |title=Introduction to Geometry |publisher=Wiley |location=New York |year=1969 |isbn= |pages=115–9 |edition=2nd}}
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| *[http://planetmath.org/encyclopedia/Ellipse2.html Ellipse at Planetmath]
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| *{{MathWorld |id=Ellipse |title=Ellipse}}
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| ==Notes==
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| {{Reflist}}
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| == External links ==
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| {{Commons category|Ellipses}}
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| *[http://www.youtube.com/watch?v=7UD8hOs-vaI Video: How to draw Ellipse]
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| *[http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=203 Apollonius' Derivation of the Ellipse] at [http://mathdl.maa.org/convergence/1/ Convergence]
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| *[http://faculty.evansville.edu/ck6/ellipse.pdf The Shape and History of The Ellipse in Washington, D.C.] by Clark Kimberling
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| *[http://www.mathopenref.com/tocs/ellipsetoc.html Collection of animated ellipse demonstrations.] Ellipse, axes, semi-axes, area, perimeter, tangent, foci.
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| *{{MathWorld |id=Hypotrochoid.html |title=Ellipse as [[hypotrochoid]]}}
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| *{{springer| title=Ellipse | id=Ellipse&oldid=11394 | last=Ivanov | first=A.B. }}
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| [[Category:Conic sections]]
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| [[Category:Curves]]
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| [[Category:Elementary shapes]]
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