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Ellipse

An ellipse is a curve that is thelocusof all points in theplanethesumof whose distancesandfrom two fixed pointsand(thefoci) separated by a distance ofis a givenpositiveconstant(Hilbert and Cohn-Vossen 1999, p.2). This results in the two-centerbipolar coordinateequation(1)

whereis thesemimajor axisand theoriginof the coordinate system is at one of thefoci. The corresponding parameteris known as thesemiminor axis.The ellipse is aconic sectionand aLissajous curve.An ellipse can be specified inMathematicausingCircle[x,y,a,b].If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse. This is known as the trammel construction of an ellipse (Eves 1965, p.177).

It is possible to construct elliptical gears that rotate smoothly against one another (Brown 1871, pp.14-15; Reuleaux and Kennedy 1876, p.70; Clark and Downward 1930; KMODDL).The ellipse was first studied by Menaechmus, investigated by Euclid, and named by Apollonius. Thefocusandconic section directrixof an ellipse were considered by Pappus. In 1602, Kepler believed that the orbit of Mars wasoval; he later discovered that it was an ellipse with the Sun at onefocus. In fact, Kepler introduced the word "focus" and published his discovery in 1609. In 1705 Halley showed that the comet now named after him moved in an elliptical orbit around the Sun (MacTutor Archive). An ellipse rotated about its minor axis gives anoblate spheroid, while an ellipse rotated about its major axis gives aprolate spheroid.A ray of light passing through afocuswill pass through the other focus after a single bounce (Hilbert and Cohn-Vossen 1999, p.3). Reflections not passing through afocuswill be tangent to a confocalhyperbolaor ellipse, depending on whether the ray passes between thefocior not.Let an ellipse lie along thex-axisand find the equation of the figure (1) whereandare atand. InCartesian coordinates,(2)

Bring the second term to the right side and square both sides,(3)

Now solve for thesquare rootterm and simplify(4)

(5)

(6)

Square one final time to clear the remainingsquare root,(7)

Grouping theterms then gives(8)

which can be written in the simple form(9)

Defining a new constant(10)

puts the equation in the particularly simple form(11)

The parameteris called thesemiminor axisby analogy with the parameter, which is called thesemimajor axis(assuming). The fact thatas defined above is actually thesemiminor axisis easily shown by lettingandbe equal. Then tworight trianglesare produced, each withhypotenuse, base, and height. Since the largest distance along the minor axis will be achieved at this point,is indeed thesemiminor axis.If, instead of being centered at (0, 0), thecenterof the ellipse is at (,), equation () becomes(12)

As can be seen from theCartesian equationfor the ellipse, the curve can also be given by a simple parametric form analogous to that of acircle, but with theandcoordinates having different scalings,(13)

(14)

The generalquadratic curve(15)

is an ellipse when, after defining(16)

(17)

(18)

,, and. Also assume the ellipse is nondegenerate (i.e., it is not acircle, so, and we have already established is not a point, since). In that case, the center of the ellipseis given by(19)

(20)

the semi-axes lengths are(21)

(22)

and the counterclockwise angle of rotation from the-axis to the major axis of the ellipse is(23)

The ellipse can also be defined as thelocusof points whose distance from thefocusis proportional to the horizontal distance from a vertical line known as theconic section directrix, where the ratio is. Lettingbe the ratio andthe distance from the center at which the directrix lies, then in order for this to be true, it must hold at the extremes of the major and minor axes, so(24)

Solving gives(25)

(26)

Thefocal parameterof the ellipse is(27)

(28)

(29)

whereis a characteristic of the ellipse known as theeccentricity, to be defined shortly.

An ellipse whose axes are parallel to the coordinate axes is uniquely determined by any four non-concyclic points on it, and the ellipse passing through the four points,,, andhas equation(30)

Let four points on an ellipse with axes parallel to the coordinate axes have angular coordinatesfor, 2, 3, and 4. Such points areconcyclicwhen(31)

where the intermediate variablehas been defined (Bergeret al.1984; Trott 2006, pp.39-40). Rather surprisingly, this same relationship results after simplification of the above whereis now interpreted as. An equivalent, but more complicated, condition is given by(32)

Likehyperbolas, noncircular ellipses havetwodistinctfociand two associateddirectrices, eachconic section directrixbeingperpendicularto the line joining the two foci (Eves 1965, p.275).Define a new constantcalled theeccentricity(whereis the case of acircle) to replace(33)

from which it follows that(34)

(35)

(36)

(37)

(38)

(39)

Theeccentricitycan therefore be interpreted as the position of thefocusas a fraction of thesemimajor axis.

Ifandare measured from afocusinstead of from the center(as they commonly are in orbital mechanics) then the equations of the ellipse are(40)

(41)

and () becomes(42)

Clearing thedenominatorsgives(43)

Substituting ingives(44)

Plugging in to re-expressandin terms ofand,(45)

Dividing byand simplifying gives(46)

which can be solved forto obtain(47)

The sign can be determined by requiring thatmust bepositive. When, (47) becomes, but sinceis alwayspositive, we must take thenegativesign, so (47) becomes(48)

(49)

(50)

The distance from afocusto a point with horizontal coordinate(where the origin is taken to lie at the center of the ellipse) is found from(51)

Plugging this into (50) yields(52)

(53)

Inpedal coordinateswith thepedal pointat thefocus, the equation of the ellipse is(54)

Thearc lengthof the ellipse is(55)

(56)

(57)

whereis an incompleteelliptic integral of the second kindwithelliptic modulus(the eccentricity).The relationship between the polar angle from the ellipse centerand the parameterfollows from(58)

This function is illustrated above withshown as the solid curve andas the dashed, with. Care must be taken to make sure that the correct branch of theinverse tangentfunction is used. As can be seen,weaves back and forth around, with crossings occurring at multiples of. Thecurvatureandtangential angleof the ellipse are given by(59)

(60)

The entireperimeterof the ellipse is given by setting(corresponding to), which is equivalent to four times the length of one of the ellipse'squadrants,(61)

(62)

(63)

whereis acomplete elliptic integral of the second kindwithelliptic modulus(the eccentricity). Theperimetercan be computed using the rapidly convergingGauss-Kummer seriesas(64)

(65)

(Sloane'sA056981andA056982), whereis abinomial coefficientand(66)

This can also be written analytically as(67)

(68)

whereis ahypergeometric function,is a completeelliptic integral of the first kind.

Approximations to theperimeterinclude(69)

(70)

(71)

where the last two are due to Ramanujan (1913-1914), and (71) has a relative error offor small values of. The error surfaces are illustrated above for these functions.The maximum and minimum distances from thefocusare called theapoapsisandperiapsis, and are given by(72)

(73)

Theareaof an ellipse may be found by directintegration(74)

(75)

(76)

(77)

(78)

(79)

Theareacan also be computed more simply by making the change of coordinatesandfrom the elliptical regionto the new region. Then the equation becomes(80)

or, sois acircleofradius. Since(81)

theJacobianis(82)

(83)

Theareais therefore(84)

(85)

(86)

(87)

as before. Theareaof an arbitrary ellipse given by thequadratic equation(88)

is(89)

Theareaof an ellipse with semiaxesandwith respect to apedal pointis(90)

The unittangent vectorof the ellipse so parameterized is(91)

(92)

A sequence ofnormalandtangent vectorsare plotted above for the ellipse.Thelocusof the apex of a variableconecontaining an ellipse fixed in three-space is ahyperbolathrough thefociof the ellipse. In addition, thelocusof the apex of aconecontaining thathyperbolais the original ellipse. Furthermore, theeccentricitiesof the ellipse andhyperbolaare reciprocals. Thelocusof centers of aPappus chainofcirclesis an ellipse. Surprisingly, the locus of the end of a garage door mounted on rollers along a vertical track but extending beyond the track is a quadrant of an ellipse (Wells 1991, p.66). (Theenvelopeof the door's positions is anastroid.)SEE ALSO:Circle,Circumellipse,Conic Section,Eccentric Anomaly,Eccentricity,Elliptic Cone,Ellipse Tangent,Elliptic Curve,Elliptic Cylinder,Hyperbola,Inellipse,Lissajous Curve,One-Seventh Ellipse,Oval,Parabola,Paraboloid,Quadratic Curve,Rectellipse,Reflection Property,Rounded Rectangle,Salmon's Theorem,Squircle,Stadium,Steiner Circumellipse,Steiner InellipseREFERENCES:Abbott, P. "On the Perimeter of an Ellipse."MathematicaJ.11, 172-185, 2009.Berger, M.; Pansu, P.; Berry, J.-P.; and Saint-Raymond, X.Problems in Geometry.New York: Springer-Verlag, 1984.Beyer, W.H.CRC Standard Mathematical Tables, 28th ed.Boca Raton, FL: CRC Press, pp.126, 198-199, and 217, 1987.Brown, H.T.Five Hundred and Seven Mechanical Movements. Embracing All Those Which Are Most Important in Dynamics, Hydraulics, Hydrostatics, Pneumatics, Steam Engines, Mill and Other Gearing ... and Including Many Movements Never Before Published, and Several Which Have Only Recently Come Into Use.New York: Brown, Coombs & Co., 1871.Casey, J. "The Ellipse." Ch.6 inA Treatise on the Analytical Geometry of the Point, Line, Circle, and Conic Sections, Containing an Account of Its Most Recent Extensions, with Numerous Examples, 2nd ed., rev. enl.Dublin: Hodges, Figgis, & Co., pp.201-249, 1893.Clark, W.M. and Downward, V.Mechanical Models: A Series of Working Models on the Art and Science of Mechanics.Newark, NJ: Newark Museum, 1930.Courant, R. and Robbins, H.What Is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed.Oxford, England: Oxford University Press, p.75, 1996.Coxeter, H.S.M. "Conics" 8.4 inIntroduction to Geometry, 2nd ed.New York: Wiley, pp.115-119, 1969.Eves, H.A Survey of Geometry, rev. ed.Boston, MA: Allyn & Bacon, 1965.Fukagawa, H. and Pedoe, D. "Ellipses," "Ellipses and One Circle," "Ellipses and Two Circles," "Ellipses and Three Circles," "Ellipses and Many Circles," "Ellipses and Triangles," "Ellipses and Quadrilaterals," "Ellipses, Circles, and Rectangles," and "Ellipses, Circles and Rhombuses." 5.1, 6.1-8.2 inJapanese Temple Geometry Problems.Winnipeg, Manitoba, Canada: Charles Babbage Research Foundation, pp.50-68, 135-160, 1989.Harris, J.W. and Stocker, H. "Ellipse." 3.8.7 inHandbook of Mathematics and Computational Science.New York: Springer-Verlag, p.93, 1998.Hilbert, D. and Cohn-Vossen, S.Geometry and the Imagination.New York: Chelsea, pp.2-3, 1999.Kern, W.F. and Bland, J.R.Solid Mensuration with Proofs, 2nd ed.New York: Wiley, p.4, 1948.KMODDL: Kinetic Models for Design Digital Library. "Model: 067 Elliptical Gears."http://kmoddl.library.cornell.edu/model.php?m=557.Lawrence, J.D.A Catalog of Special Plane Curves.New York: Dover, pp.72-78, 1972.Lockwood, E.H. "The Ellipse." Ch.2 inA Book of Curves.Cambridge, England: Cambridge University Press, pp.13-24, 1967.MacTutor History of Mathematics Archive. "Ellipse."http://www-groups.dcs.st-and.ac.uk/~history/Curves/Ellipse.html.Ramanujan, S. "Modular Equations and Approximations to."Quart. J. Pure. Appl. Math.45, 350-372, 1913-1914.Reuleaux, F. and Kennedy, A.B.W. (Eds.).Kinematics of Machinery: Outlines of a Theory of Machines.London: Macmillan, 1876. Reprinted by New York: Dover, 1976.Sloane, N.J.A. SequencesA056981andA056982in "The On-Line Encyclopedia of Integer Sequences."Trott, M.TheMathematicaGuideBook for Symbolics.New York: Springer-Verlag, 2006.http://www.mathematicaguidebooks.org/.Wells, D.The Penguin Dictionary of Curious and Interesting Geometry.London: Penguin, pp.63-67, 1991.Yates, R.C. "Conics."A Handbook on Curves and Their Properties.Ann Arbor, MI: J.W.Edwards, pp.36-56, 1952.CITE THIS AS:Weisstein, Eric W."Ellipse." FromMathWorld--A Wolfram Web Resource.http://mathworld.wolfram.com/Ellipse.htmlWolfram Web ResourcesMathematicaThe #1 tool for creating Demonstrations and anything technical.Wolfram|AlphaExplore anything with the first computational knowledge engine.Wolfram Demonstrations ProjectExplore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more.

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