Editing Ellipse (section)

== Applications ==
===Physics===
==== Elliptical reflectors and acoustics ====
{{See also|Fresnel zone}}

[[File: "Wave pattern of a little droplet dropped into mercury in one focus of the ellipse " by Weber Bros..jpg|thumb|Wave pattern of a little droplet dropped into mercury in the foci of the ellipse]]

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves of that disturbance, after [[reflection (physics)|reflecting]] off the walls, 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.

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 ellipsoidal mirror (specifically, a [[prolate spheroid]]), this property holds 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.

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]].

==== Planetary orbits ====
{{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 [approximately] 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]].

More generally, in the gravitational [[two-body problem]], if the two bodies are bound to each other (that is, 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. The orbit of either body in the reference frame of the other is also an ellipse, with the other body at the same focus.

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.)

For [[elliptical orbit]]s, useful relations involving the eccentricity <math>e</math> are:
<math display="block">\begin{align}
    e &= \frac{r_a - r_p}{r_a + r_p} = \frac{r_a - r_p}{2a} \\
  r_a &= (1 + e)a \\
  r_p &= (1 - e)a
\end{align}</math>

where
* <math>r_a</math> is the radius at [[apoapsis]], i.e., the farthest distance of the orbit to the [[barycenter]] of the system, which is a [[Focus (geometry)|focus]] of the ellipse
* <math>r_p</math> is the radius at [[periapsis]], the closest distance
* <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>\ell</math> is their [[harmonic mean]]. In other words,
<math display="block">\begin{align}
     a &= \frac{r_a + r_p}{2} \\[2pt]
     b &= \sqrt{r_a r_p} \\[2pt]
  \ell &= \frac{2}{\frac{1}{r_a} + \frac{1}{r_p}} = \frac{2r_ar_p}{r_a + r_p}.
\end{align}</math>

==== 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.

==== 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 [[Lissajous figure]] display is an ellipse, rather than a straight line, the two signals are out of phase.

==== Elliptical gears ====
Two [[non-circular gear]]s with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, 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=https://books.google.com/books?id=fPoOAAAAYAAJ&pg=PA72 |year=1906 |publisher=Philadelphia Gear Works |page=72}}</ref>

==== 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.)
* 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 Encyclopedia of Laser Physics and Technology - lamp-pumped lasers, arc lamps, flash lamps, high-power, Nd:YAG laser<!-- Bot generated title -->]</ref>
* 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>{{cite web |url=http://www.cymer.com/plasma_chamber_detail/ |title=Cymer - EUV Plasma Chamber Detail Category Home Page |access-date=2013-06-20 |url-status=dead |archive-url=https://web.archive.org/web/20130517100847/http://www.cymer.com/plasma_chamber_detail |archive-date=2013-05-17 }}</ref>

===Statistics and finance===
In [[statistics]], a bivariate [[random vector]] <math>(X, Y)</math> 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 ellipsoids. A special case is the [[multivariate normal distribution]]. The elliptical distributions are important in the financial field 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 }}</ref><ref>{{cite journal |author1=Owen, J. |author2=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= 3|pages=745–752 |date=June 1983 |jstor=2328079 |doi=10.1111/j.1540-6261.1983.tb02499.x}}</ref>

=== Computer graphics ===
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 |doi-access=free }}</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 |s2cid=18995215 }}</ref>

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 |doi-access=free }}</ref> These algorithms need only a few multiplications and additions to calculate each vector.

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.

;Drawing with Bézier paths:
[[Composite Bézier curve]]s 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 behave appropriately under such transformations.

=== Optimization theory ===
It is sometimes useful to find the minimum bounding ellipse on a set of points. The [[ellipsoid method]] is quite useful for solving this problem.