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Encyclopedia > Ellipse
The ellipse and some of its mathematical properties.
The ellipse and some of its mathematical properties.
An ellipse obtained as the intersection of a cone with a plane.
An ellipse obtained as the intersection of a cone with a plane.

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. Elliptical trainer in use An elliptical trainer (also cross trainer) is a stationary exercise equipment used to simulate walking or running without causing pressure to the joints and hence decreases the risk of impact injuries. ... Ellipse may refer to: Ellipse, a mathematical term The Ellipse, a 1 km elliptical street in Presidents Park, just south of the White House Ellipse Programmé, a French company that produces childrens cartoons Mini Ellipse, a remote-control glider MacAdam ellipse, a chromaticity diagramen Superellipse, a geometric figure... Image File history File links This is a lossless scalable vector image. ... Image File history File links Elipse. ... Image File history File links Elipse. ... This article is about the geometric object, for other uses see Cone. ... This article is about the mathematical construct. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a locus (Latin for place, plural loci) is a collection of points which share a common property. ... Distance is a numerical description of how far apart objects are at any given moment in time. ... In geometry, the focus (pl. ... In mathematics, a conic section (or just conic) is a curved locus of points, formed by intersecting a cone with a plane. ...

Contents

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. Wikibooks has more on the topic of Conic section Types of conic sections Table of conics, Cyclopaedia, 1728 In mathematics, a conic section (or just conic) is a curve that can be formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ... In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space curve — the directrix — that does not contain the apex. ... Dandelin Spheres—graphics by Hop David In geometry, a nondegenerate conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus: Each Dandelin sphere touches, but does not cross, both the plane and the cone. ...


Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra. ... In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. ... Fig. ...

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

such that B2 < 4AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists.


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. The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... Fig. ... In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ...


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. This article is about the shape and mathematical concept of circle. ... (This page refers to eccentricity in mathematics. ...


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 = PDPT, 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. In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ... Illustration of a unit circle. ... In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. ... In linear algebra, a scalar λ is called an eigenvalue (in some older texts, a characteristic value) of a linear mapping A if there exists a nonzero vector x such that Ax=λx. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... In linear algebra, the eigenvectors (from the German eigen meaning inherent, characteristic) of a linear operator are non-zero vectors which, when operated on by the operator, result in a scalar multiple of themselves. ...


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 (This page refers to eccentricity in mathematics. ... A negative number is a number that is less than zero, such as −2. ... This article is about the shape and mathematical concept of circle. ... In the study of ellipses and related geometry, various parameters in the distortion of a circle into an ellipse are identified and employed: Aspect ratio, flattening and eccentricity. ...

varepsilon=sin(o!varepsilon)!!:;,o!varepsilon=arccosleft(frac{b}{a}right);,!

The greater the eccentricity is, the larger the ratio of a to b, and therefore the more elongated the ellipse. This article is about the mathematical concept. ...


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

varepsilon = frac{c}{a}!!:;,c=asin(o!varepsilon)=sqrt{a^2-b^2};,!

The distance c is known as the linear eccentricity of the ellipse. The distance between the foci is 2c or 2.


Drawing

Two pins, a loop and a pen method
Two pins, a loop and a pen method

An ellipse can be inscribed within a rectangle using 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 loop. The pencil is placed on the paper inside the loop and the string made taut. The string will form a triangle. If the pencil is moved around with the string kept taut, the sum of the distances from the pencil to the pins will remain constant, thus satisfying the definition of an ellipse. A 5 by 4 rectangle In geometry, a rectangle is defined as a quadrilateral where all four of its angles are right angles. ... For other uses, see Triangle (disambiguation). ...


Consider the center of the rectangle to be the origin and the lengths of its sides to be 2a and 2b, with a being larger than b. The major axis then passes through the origin and is parallel to the longer side. The two pins are placed the distance c away from the origin in each direction along the major axis. The required length of the string used to form the loop is 2a + 2c.


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{(x-h)^{2}}{a^{2}}+frac{(y-k)^{2}}{b^{2}}=1;,!

This ellipse can be expressed parametrically as Graph of a butterfly curve, a parametric equation discovered by Temple H. Fay In mathematics, a parametric equation explicitly relates two or more variables in terms of one or more independent parameters. ...

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

where t may be restricted to the interval -pileq tleqpi.


Parametric form of an ellipse rotated counterclockwise 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,sinphi;,!
The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, R=2r.

The formula for the directrices is A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. ...

x=hpmfrac{a^2}{c}=hpm a;csc(o!varepsilon)=hpmfrac{a}{sin(o!varepsilon)};,!

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{ab}{sqrt{a^2sin^2theta+b^2cos^2theta}}=frac{b}{sqrt{1-varepsilon^2cos^2theta}};,!

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

r=frac{acdot(1-varepsilon^{2})}{1+varepsiloncdotcostheta};,!

A Gauss-mapped form: In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere . ...

left(frac{acosbeta}{sqrt{a^2cos^2beta+b^2sin^2beta}},frac{bsinbeta}{sqrt{a^2cos^2beta+b^2sin^2beta}}right);

has normal (cosβ,sinβ).


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=acos(o!varepsilon)^2=acdot(1-varepsilon^2);,! In mathematics, the latus rectum of a conic section is the chord parallel to the directrix and passing through the single focus, or one of the two foci. ... Minuscule, or lower case, is the smaller form (case) of letters (in the Roman alphabet: a, b, c, ...). Originally alphabets were written entirely in majuscule (capital) letters which were spaced between well-defined upper and lower bounds. ... For other uses of L, see L (disambiguation). ... Fig. ...

Ellipse, showing semi-latus rectum

In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation Image File history File links Elps-slr. ... A polar grid with several angles labeled in degrees In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by an angle and a distance. ...

l=rcdot(1+sin(o!varepsilon)costheta)=rcdot(1+varepsiloncdotcostheta);,!

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 area of a circle with radius r is . ...


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 circumference is the distance around a closed curve. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ...


The exact infinite series is: In mathematics, a series is a sum of a sequence of terms. ...

C = 2pi a left[{1 - left({1over 2}right)^2varepsilon^2 - left({1cdot 3over 2cdot 4}right)^2{varepsilon^4over 3} - left({1cdot 3cdot 5over 2cdot 4cdot 6}right)^2{varepsilon^6over5} - dots}right];!,

Or:

C = 2pi a sum_{n=0}^infty {leftlbrace - left[prod_{m=1}^n left({ 2m-1 over 2m}right)right]^2 {varepsilon^{2n}over 2n - 1}rightrbrace};,!

A good approximation is Ramanujan's: It has been suggested that this article or section be merged with estimation. ... Ramanujan redirects here. ...

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

or better approximation: It has been suggested that this article or section be merged with estimation. ...

Capproxpileft(a+bright)left(1+frac{3left(frac{a-b}{a+b}right)^2}{10+sqrt{4-3left(frac{a-b}{a+b}right)^2}}right);!,

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

C approx frac{pi a (9 - sqrt{35})}{2};!,

Or:


C approx frac{a}{2} sqrt{93 + frac{1}{2} sqrt{3}};!, (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. Determining the length of an irregular arc segment—also called rectification of a curve—was historically difficult. ... In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Giulio Fagnano and Leonhard Euler. ... A function ƒ and its inverse ƒ–1. ... In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions. ...


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. This article needs to be cleaned up to conform to a higher standard of quality. ...


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. This article is about wave reflectors (mainly, specular reflection of visible light). ... The reflection of a bridge in Indianapolis, Indianas Central Canal. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...


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 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. National Statuary Hall The National Statuary Hall is an area in the United States Capitol devoted to statues of people and symbols important in American history. ... United States Capitol The United States Capitol is the building which serves as home for the legislative branch of the United States government. ... John Quincy Adams (July 11, 1767 – February 23, 1848) was a diplomat, politician, and the sixth President of the United States (March 4, 1825 – March 4, 1829). ... A view from the lagoon behind the Museum of Science and Industry, the only in-place surviving building from the 1893 World Columbian Exposition and a National Historic Landmark. ... For other uses, see Chicago (disambiguation). ... A Corner of Main Quad The University of Illinois at Urbana-Champaign (UIUC, U of I, or simply Illinois), is the oldest, largest, and most prestigious campus in the University of Illinois system. ... The Alhambra (Arabic: الحمراء = Al-Ħamrā; literally the red fortress) is a palace and fortress complex of the Moorish monarchs of Granada in southern Spain (known as Al-Andalus when the fortress was constructed), occupying a hilly terrace on the southeastern border of the city of Granada. ...


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. Kepler redirects here. ... Two bodies with a slight difference in mass orbiting around a common barycenter. ... Illustration of Keplers three laws with two planetary orbits. ... Sir Isaac Newton FRS (4 January 1643 – 31 March 1727) [ OS: 25 December 1642 – 20 March 1727][1] was an English physicist, mathematician, astronomer, natural philosopher, and alchemist. ... Isaac Newtons theory of universal gravitation (part of classical mechanics) states the following: Every single point mass attracts every other point mass by a force pointing along the line combining the two. ...


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. In classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other. ... Several equivalence relations in mathematics are called similarity. ... In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it was concentrated. ...


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. An undamped spring-mass system is a simple harmonic oscillator. ... For other uses, see Dimension (disambiguation). ...


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. The index ellipsoid is a diagram of an ellipsoid that depictes the orientation and relative magnitude of refractive indices in a crystal. ... The refractive index (or index of refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves) is reduced inside the medium. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ... A calcite crystal laid upon a paper with some letters showing the double refraction Birefringence, or double refraction, is the decomposition of a ray of light into two rays (the ordinary ray and the extraordinary ray) when it passes through certain types of material, such as calcite crystals or boron...


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). Two quickdraws. ... The Graphics Device Interface (GDI, sometimes called Graphical Device Interface) is one of the three core components or subsystems, together with the kernel and the Windows API for the user interface (GDI window manager) of Microsoft Windows. ... This subsystem is a part of . ... Jack E. Bresenham is a professor of computer science. ...


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 {double} X coordinate * @param y {double} Y coordinate * @param a {double} Semimajor axis * @param b {double} Semiminor axis * @param angle {double} Angle of the ellipse */ function calculateEllipse(x, y, a, b, angle, steps) { if (steps == null) steps = 36; var points = []; var beta = -angle / 180 * Math.PI; var sinbeta = Math.sin(beta); var cosbeta = Math.cos(beta); for (var i = 0; i < 360; i += 360 / steps) { var alpha = i / 180 * Math.PI; var sinalpha = Math.sin(alpha); var cosalpha = Math.cos(alpha); var X = x + (a * cosalpha * cosbeta - b * sinalpha * sinbeta); var Y = y + (a * cosalpha * sinbeta + b * sinalpha * cosbeta); points.push(new OpenLayers.Geometry.Point(X, Y)); } return points; } 

One beneficial consequence of using the parametric formula is that 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.


See also

3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. ... Squircle, the superellipse for n = 4, a = b = 1, approximates a chamfered square. ... In mathematics, a hyperbola (Greek literally overshooting or excess) is a type of conic section defined as the intersection between a right circular conical surface and a plane which cuts through both halves of the cone. ... A parabola A graph showing the reflective property, the directrix (light blue), and the lines connecting the focus and directrix to the parabola (blue) In mathematics, the parabola (from the Greek: παραβολή) (IPA pronunciation: ) is a conic section generated by the intersection of a right circular conical surface and a plane... This oval, with only one axis of symmetry, resembles a chicken egg. ... In astronomy, the true anomaly (, also written ) is the angle between the direction z-s of periapsis and the current position p of an object on its orbit, measured at the focus s of the ellipse (the point around which the object orbits). ... The eccentric anomaly is the angle between the direction of periapsis and the current position of an object on its orbit, projected onto the ellipses circumscribing circle perpendicularly to the major axis, measured at the centre of the ellipse. ... In the study of orbital dynamics the mean anomaly is a measure of time, specific to the orbiting body p, which is a multiple of 2π radians at and only at periapsis. ... Look up anomaly in Wiktionary, the free dictionary. ... In mathematics, the matrix representation of conic sections is one way of studying a conic section, its axis, vertices, foci, tangents, and the relative position of a given point. ... Illustration of Keplers three laws with two planetary orbits. ... The derivation of the cartesian form for an ellipse is simple and instructive. ...

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
  • Eric W. Weisstein, Ellipse at MathWorld.

Eric W. Weisstein (born March 18, 1969, in Bloomington, Indiana) is an encyclopedist who created and maintains MathWorld and Eric Weissteins World of Science (ScienceWorld). ... MathWorld is an online mathematics reference work, sponsored by Wolfram Research Inc. ...

External links

Wikimedia Commons has media related to:
Ellipses
A hypotrochoid is a roulette traced by a point attached to a circle of radius r rolling around the inside of a fixed circle of radius R, where the point is a distance d from the center of the interior circle. ...

  Results from FactBites:
 
Ellipse (1425 words)
Ellipse is commonly defined as the locus of points P such that the sum of the distances from P to two fixed points F1, F2 (called foci) are constant.
The pedal of a ellipse with respect to a focus is a circle, conversely, the negative pedal of a circle with respect to a point inside the circle is a ellipse.
Ellipse's inversion with respect to a focus is a dimpled limacon of Pascal.
Ellipse - MSN Encarta (485 words)
The ellipse is used in engineering in the arches of some bridges and the design of gears for certain types of machinery such as Wankel engines and punch presses.
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Ellipses are a type of conic section, a class of curves formed by a plane that cuts through a right circular cone.
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COMMENTARY     

DVdm
22nd May 2010
The Gauss-mapped form should have the a and b in the numerators squared. Check by verifying (X/a)^2+(Y/b)^2=1.
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