# Dictionary Definition

ellipse n : a closed plane curve resulting from
the intersection of a circular cone and a plane cutting completely
through it; "the sums of the distances from the foci to any point
on an ellipse is constant" [syn: oval]

# User Contributed Dictionary

### Noun

#### Synonyms

- oval (in non-technical use)

#### Translations

### Verb

- In the context of "grammar": To remove from a phrase a word which is grammatically needed, but
which is clearly understood without having to be stated.
- In the exchange:- (A.Would you like to go out?, B.I'd love to), the ellipsed words are go out.

#### Translations

curve

### Related terms

### See also

### Pronunciation

/e.lips/# Extensive Definition

In mathematics, an ellipse
(from the Greek
ἔλλειψις, literally absence) is a locus of
points in a plane such that the sum of the distances to two fixed points
is a constant. The two fixed points are called foci (singular-
focus).
An alternate definition would be that an ellipse is the path traced
out by a point whose distance from a fixed point, called the focus,
maintains a constant ratio less than one with its distance from a
straight line not passing through the focus, called the directrix.

## Overview

An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. For a short elementary proof of this, see Dandelin spheres.Algebraically,
an ellipse is a curve in
the
Cartesian plane defined by an equation of the form

- A x^2 + B xy + C y^2 + D x + E y + F = 0 \,

An ellipse can be drawn with two pins, a loop of
string, and a pencil. The pins are placed at the foci and the pins
and pencil are enclosed inside the string. The pencil is placed on
the paper inside the string, so the string is taut. The string will
form a triangle. If the
pencil is moved around so that the string stays taut, the sum of
the distances from the pencil to the pins will remain constant,
satisfying the definition of an ellipse.

The line segment
AB, that passes through the foci and terminates on the ellipse, is
called the major axis. The major axis is the longest segment that
can be obtained by joining two points on the ellipse. The line
segment CD, which passes through the center (halfway between the
foci), perpendicular to the major
axis, and terminates on the ellipse, is called the minor axis. The
semimajor
axis (denoted by a in the figure) is one half the major axis:
the line segment from the center, through a focus, and to the edge
of the ellipse. Likewise, the semiminor
axis (denoted by b in the figure) is one half the minor
axis.

If the two foci coincide, then the ellipse is a
circle; in other words, a
circle is a special case of an ellipse, one where the eccentricity
is zero.

An ellipse centered at the origin
can be viewed as the image of the unit circle
under a linear map associated with a symmetric
matrix A = PDP^T, D being a diagonal
matrix with the eigenvalues of A, both of
which are real positive, along the main diagonal, and P being a
real unitary
matrix having as columns the eigenvectors of A. Then the
axes of the ellipse will lie along the eigenvectors of A, and the
(square root of the) eigenvalues are the lengths of the semimajor
and semiminor axes.

An ellipse can be produced by multiplying the x
coordinates of all points on a circle by a constant, without
changing the y coordinates. This is equivalent to stretching the
circle out in the x-direction.

## Eccentricity

The shape of an ellipse can be expressed by a number called the eccentricity of the ellipse, conventionally denoted \, \varepsilon. The eccentricity is a non-negative number less than 1 and greater than or equal to 0. It is the value of the constant ratio of the distance of a point on an ellipse from a focus to that from the corresponding directrix. An eccentricity of 0 implies that the two foci occupy the same point and that the ellipse is a circle. It can also be expressed as the sine of the angular eccentricity, o\!\varepsilon\,\!. For an ellipse with semimajor axis a and semiminor axis b, the eccentricity is- \varepsilon=\sin(o\!\varepsilon)\!\!:\;\,o\!\varepsilon=\arccos\left(\frac\right);\,\!

The greater the eccentricity is, the larger the
ratio of a to b, and
therefore the more elongated the ellipse.

If c equals the distance from the center to
either focus, then

- \varepsilon = \frac\!\!:\;\,c=a\sin(o\!\varepsilon)=\sqrt;\,\!

## Equations

An ellipse with a semimajor axis a and semiminor axis b, centered at the point (h,k) and having its major axis parallel to the x-axis may be specified by the equation- \frac+\frac=1;\,\!

This ellipse can be expressed parametrically
as

- x=h+a\,\cos t;\,\!
- y=k+b\,\sin t;\,\!

Parametric form of an ellipse rotated by an angle
\phi\,\!:

- x=h+a\,\cos t\,\cos \phi - b\,\sin t\,\sin \phi;\,\!
- y=k+b\,\sin t\,\cos \phi+a\,\cos t\,\sin\phi;\,\!

The formula for the directrices is

- x=h\pm\frac=h\pm a\;\csc(o\!\varepsilon)=h\pm\frac;\,\!

If h = 0 and k = 0 (i.e., if the center is the
origin (0,0)), then we can express this ellipse in polar
coordinates by the equation

- r=\frac=\frac;\,\!

With one focus at the origin, the ellipse's polar
equation is

- r=\frac;\,\!

A Gauss-mapped
form:

- \left(\frac,\frac\right);

## Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted l\,\! (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. It is related to a\,\! and b\,\! (the ellipse's semi-axes) by the formula al=b^2\,\! or, if using the eccentricity, l=a\cos(o\!\varepsilon)^2=a\cdot(1-\varepsilon^2);\,\!In polar
coordinates, an ellipse with one focus at the origin and the
other on the negative x-axis is given by the equation

- l=r\cdot(1+\sin(o\!\varepsilon)\cos\theta)=r\cdot(1+\varepsilon\cdot\cos\theta);\,\!

An ellipse can also be thought of as a projection
of a circle: a circle on a plane at angle φ to the horizontal
projected vertically onto a horizontal plane gives an ellipse of
eccentricity sin φ, provided φ is not 90°.

## Area and circumference

The area enclosed by an ellipse is πab, where (as before) a and b are the ellipse's semimajor and semiminor axes.The circumference C of an
ellipse is 4 a E(\varepsilon), where the function E is the complete
elliptic
integral of the
second kind.

The exact infinite
series is:

- C = 2\pi a \left[\right];\!\,

Or:

- C = 2\pi a \sum_^\infty ;\,\!

A good approximation is Ramanujan's:

- C \approx \pi \left[3(a+b) - \sqrt\right]\!\,

or better approximation:

- C\approx\pi\left(a+b\right)\left(1+\frac\right);\!\,

For the special case where the minor axis is half
the major axis, we can use:

- C \approx \frac;\!\,

Or:

C \approx \frac \sqrt;\!\, (better
approximation).

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. The inverse
function, the angle subtended as a function of the arc length,
is given by the elliptic
functions.

## Stretching and projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Similarly, any oblique projection onto a plane results in a conic section. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.## Reflection property

Assume an elliptic mirror with a light source at one of the foci. Then all rays are reflected to a single point — the second focus. Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are orthogonal to the radius.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 another focus remarkably well. Such a room is
called a whisper chamber. Examples are the
National Statuary Hall Collection at the U.S. Capitol
(where John
Quincy Adams is said to have used this property for
eavesdropping on political matters), at an exhibit on sound at the
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.

## Ellipses in physics

In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses in his first law of planetary motion. Later, Isaac Newton explained this as a corollary of his law of universal gravitation.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 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.

The general solution for a harmonic
oscillator in two or more dimensions is also an ellipse,
but this time with the origin of the force located at the center of
the ellipse.

In optics, an index
ellipsoid describes the refractive
index of a material as a function of the direction through that
material. This only applies to materials that are optically
anisotropic. Also
see birefringence.

## Ellipses in computer graphics

Drawing an ellipse as a graphics
primitive is common in standard display libraries, such as the
Macintosh QuickDraw API,
the Windows Graphics
Device Interface (GDI) and the
Windows Presentation Foundation (WPF). Often such libraries are
limited and can only draw an ellipse with either the major axis or
the minor axis horizontal. 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. An
efficient generalization to draw ellipses was invented in 1984 by
Jerry Van
Aken (IEEE CG&A, Sept. 1984).

The following is example JavaScript code using
the parametric formula for an ellipse to calculate a set of points.
The ellipse can be then approximated by connecting the points with
lines.

/*

- This functions returns an array containing 36 points to draw an
- ellipse.
- @param x X coordinate
- @param y Y coordinate
- @param a Semimajor axis
- @param b Semiminor axis
- @param angle Angle of the ellipse
- /

## See also

- Ellipsoid, a higher dimensional analog of an ellipse
- Spheroid, the ellipsoids obtained by rotating an ellipse about its major or minor axis.
- Superellipse, a generalization of an ellipse that can look more rectangular
- Hyperbola
- Parabola
- Oval
- True, eccentric, and mean anomalies
- Matrix representation of conic sections
- Kepler's Laws of Planetary Motion
- Ellipse/Proofs

## References

- Charles D.Miller, Margaret L.Lial, David I.Schneider: Fundamentals of College Algebra. 3rd Edition Scott Foresman/Little 1990. ISBN 0-673-38638-4. Page 381
- Coxeter, H. S. M.: Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115-119, 1969.
- Ellipse at the Encyclopedia of Mathematics (Springer)
- Ellipse at Planetmath

## External links

- Apollonius' Derivation of the Ellipse at Convergence
- Ellipse & Hyperbola Construction - An interactive sketch showing how to trace the curves of the ellipse and hyperbola. (Requires Java.)
- Ellipse Construction - Another interactive sketch, this time showing a different method of tracing the ellipse. (Requires Java.)
- The Shape and History of The Ellipse in Washington, D.C. by Clark Kimberling
- Collection of animated ellipse demonstrations. Ellipse, axes, semi-axes, area, perimeter, tangent, foci.
- Woodworking videos showing how to work with ellipses in wood.
- Ellipse as hypotrochoid

ellipse in Afrikaans: Ellips

ellipse in Arabic: قطع ناقص

ellipse in Asturian: Elipse

ellipse in Belarusian: Эліпс

ellipse in Bosnian: Elipsa

ellipse in Bulgarian: Елипса

ellipse in Catalan: El·lipse

ellipse in Czech: Elipsa

ellipse in Danish: Ellipse (geometri)

ellipse in German: Ellipse

ellipse in Estonian: Ellips (geomeetria)

ellipse in Modern Greek (1453-): Έλλειψη

ellipse in Spanish: Elipse

ellipse in Esperanto: Elipso (matematiko)

ellipse in Persian: بیضی

ellipse in French: Ellipse (mathématiques)

ellipse in Galician: Elipse (lingua)

ellipse in Korean: 타원

ellipse in Croatian: Elipsa

ellipse in Indonesian: Elips

ellipse in Interlingua (International Auxiliary
Language Association): Ellipse

ellipse in Italian: Ellisse

ellipse in Hebrew: אליפסה

ellipse in Georgian: ელიფსი

ellipse in Latvian: Elipse

ellipse in Lithuanian: Elipsė

ellipse in Hungarian: Ellipszis (görbe)

ellipse in Dutch: Ellips (wiskunde)

ellipse in Japanese: 楕円

ellipse in Norwegian: Ellipse

ellipse in Norwegian Nynorsk: Ellipse

ellipse in Polish: Elipsa (matematyka)

ellipse in Portuguese: Elipse

ellipse in Romanian: Elipsă

ellipse in Quechua: Lump'u

ellipse in Russian: Эллипс

ellipse in Sicilian: Ellissi

ellipse in Simple English: Ellipse

ellipse in Slovak: Elipsa

ellipse in Slovenian: Elipsa

ellipse in Serbian: Елипса

ellipse in Finnish: Ellipsi

ellipse in Swedish: Ellips (matematik)

ellipse in Tamil: நீள்வட்டம்

ellipse in Vietnamese: Elíp

ellipse in Turkish: Elips

ellipse in Ukrainian: Еліпс

ellipse in Urdu: بیضہ

ellipse in Chinese: 椭圆