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"Elliptical" redirects here. For the exercise machine, see Elliptical trainer.
For other uses, see Ellipse (disambiguation).
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The ellipse and some of its mathematical properties.
An ellipse obtained as the intersection of a cone with a plane.

In mathematics an ellipse (Greek ἔλλειψις (elleipsis), a 'falling short') is a conic section, the locus of points in a plane such that the sum of the distances to two fixed points is equal to a given constant. The two fixed points are then called foci (singular- focus).

Another way is to define it as the path traced out by a point whose distance from a focus maintains a constant ratio less than one with its distance from a straight line not passing through the focus, called the directrix. A fourth way is to define it as the locus of all points satisfying an equation of type

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

where x and y are coordinates of the points defined relative to some rectangular coordinate system.

From this last definition it is seen that an ellipse is obtained from the unit circle

x2 + y2 = 1

by scaling the x and y coordinates with the factors a and b.


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.

Contents

Eccentricity

The eccentricity of an ellipse is the ratio of the distance between the foci to the length of the major axis; this is necessarily between 0 and 1. If the ellipse has the Cartesian equation

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \qquad 0 < b \leq a,

its eccentricity is

e = \sqrt{1-\left(\frac ba\right)^2 }.

The eccentricity is zero if and only if b = a in which case the ellipse is a circle.

The coordinates of the foci are (ae,0) and ( − ae,0). The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse. The distance between the foci is 2ae .

Eliminating b from the above equations gives the alternative equation for the ellipse

y^2= (a^2-x^2)(1-e^2)\,.

The distance from a point (x,y) on the ellipse to the left focal point is

r_1 =\sqrt{(x+a e)^2 + y^2} = \sqrt{x^2 + 2 x a e + a^2 e^2 + (a^2-x^2)(1-e^2)}=\sqrt{(a+e x)^2}=a+e x

The distance from the same point to the right focal point is in the same way

r_2 = a-e x\,

Adding these equations one gets

r_1 + r_2= 2 a\,.

This property of the ellipse, that the sum of the distances to two given points is taking a given value, is often used as the definition of an ellipse. The method to draw an ellipse described below is an application of this.

True anomaly

The polar angle θ of a point on an ellipse relative the focal point F is called the true anomaly of the point. Relative the "canonical coordinate system" with origin at the mid-point between the focii F' and F in which the equation of the ellipse is

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1

one has that

x=ae+ r \cos \theta \,.

As

r = a-e x = a - e (ae+ r \cos \theta)\,

one has that

r = \frac{a(1-e^2)}{1+e\cos \theta}

This is the standard representation of an ellipse in polar coordinates.

Eccentric anomaly

For the point (x,y) on an ellipse with the equation

\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

the eccentric anomaly E \, is defined through

\cos E = \frac{x}{a} \,
\sin E = \frac{y}{b} \,

The direct relation between the eccentric anomaly and the true anomaly is:

\cos E = \frac{x}{a} = e + \frac{ (1 - e^2) \cos \theta }{1 + e \cos \theta }= \frac{ e + \cos \theta }{1 + e \cos \theta } \,
\sin E = \frac{y}{b} = \frac{ \sqrt{1 - e^2} \, \sin \theta }{1 + e \cos \theta }\,

With this relation the eccentric anomaly can be computed from the true anomaly. To compute the true anomaly from the eccentric anomaly a more convenient relation can be derived using using the trigonometric identity

\cos x = \frac{1-\tan^2 \frac{x}{2} }{1+\tan^2 \frac{x}{2}} \,

One gets that

\frac{ e + \cos \theta }{1 + e \cos \theta } = \frac{ 1 - \frac{ 1-e }{1+e} \tan^2 \frac{\theta }{2}} { 1 + \frac{ 1-e }{1+e} \tan^2 \frac{\theta }{2} }\,

and as

\cos E = \frac{1-\tan^2 \frac{E}{2} }{1+\tan^2 \frac{E}{2}} \,

it follows that

\tan^2 \frac{E}{2} = \frac{1-e}{1+e} \tan^2 \frac{\theta }{2}\,

As \sin E \, and \sin \theta \, always have the same sign it follows that \frac{E}{2}\, and \frac{\theta}{2}\, are in the same quadrant.

One therefore has that

\tan \frac{E}{2} = \sqrt{\frac{1-e}{1+e}} \tan \frac{\theta }{2}\,

The relation written in this form has singularities for \cos \frac{\theta }{2}\ = 0\, and \cos \frac{E}{2}\ = 0 \,.

But it can also be written in the non-singular form

E = 2 \, \operatorname{arg}\left(\sqrt{1+e} \, \cos \frac{\theta }{2} , \sqrt{1-e}\sin \frac{\theta }{2}\right)\,
\theta = 2 \, \operatorname{arg}\left( \sqrt{1-e} \, \cos \frac{E}{2} , \sqrt{1+e}\sin \frac{E}{2}\right)\,

where

\operatorname{arg}(x \,, y\,),

is the polar argument of the vector (x \, ,\, y\,).

For the numerical computation of \operatorname{arg}(x \,, y\,), the standard function ATAN2(y,x) (or in double precision DATAN2(y,x)) available in for example the programming language FORTRAN can be used.

Reduction to canonical form

It can be proven that any second order polynomial

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

can be reduced to the canonical form

A' x^2 + C' y^2+ F' \,

with x,y defined relative another rectangular coordinate system obtained from the original one by translation and rotation. It can also be proven that if B2 < 4AC relative one coordinate system it is true in all coordinate systems. As in the canonical coordinate system one has that B' = 0 it follows that 0 < A'C'. This means that A' and C' have the same sign and the equation

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

which in the canonical coordinate system takes the form

A' x^2 + C' y^2+ F' = 0 \,

has a solution if and only if F' has the opposite sign to A' and C' In this case the equation can be written

\frac{x^2}{a^2}+\frac{y^2}{b^2}=1


with

a=\sqrt{-\frac{F'}{A'}}
b=\sqrt{-\frac{F'}{C'}}

Drawing

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.

Consider the center of the rectangle to be the origin, halfway between the two pins, and the lengths of its sides to be 2a and 2b, with a being larger than b, and with the long sides parallel to the line containing the two pins. 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 can be seen in the particular case of the point of the ellipse lying on the major axis. Here the length of the string is the distance from the first focus to the center, c, plus the distance from the center to the point on the ellipse, a, plus the distance from that point back to the second focus, a-c, plus the distance between the two foci, 2c, adding to 2a+2c. Subtracting the part of the string between the two foci, we see that the distance from one focus, to the pencil, to the other focus, is 2a.

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

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

where t may be restricted to the interval -\pi\leq t\leq\pi.

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\,\sin\phi\,;\!
The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, R=2r.

The formula for the directrices is

x=h\pm\frac{a^2}{c}=h\pm a\;\csc(o\!\varepsilon)=h\pm\frac{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^2\sin^2\theta+b^2\cos^2\theta}}=\frac{b}{\sqrt{1-\varepsilon^2\cos^2\theta}};\,\!

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

r=\frac{a\cdot(1-\varepsilon^{2})}{1+\varepsilon\cdot\cos\theta};\,\!

A Gauss-mapped form:

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

has normal (cosβ,sinβ).

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted \mathit{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);\,\!

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

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 one-half of the ellipse's major and minor axes respectively.

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

Or:

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

A good approximation is Ramanujan's:

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

or better approximation:

C\approx\pi\left(a+b\right)\left(1+\frac{3\left(\frac{a-b}{a+b}\right)^2}{10+\sqrt{4-3\left(\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.

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 the other 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.

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 {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 = ;
 
  // Angle is given by Degree Value
  var beta = -angle * (Math.PI / 180); //(Math.PI/180) converts Degree Value into Radians
  var sinbeta = Math.sin(beta);
  var cosbeta = Math.cos(beta);
 
  for (var i = 0; i < 360; i += 360 / steps) 
  {
    var alpha = i * (Math.PI / 180) ;
    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

References

External links

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