Ellipse

From Wikipedia, the free encyclopedia.

"Elliptical" redirects here. For the exercise machine, see Elliptical trainer.

For other uses, see Ellipse (disambiguation).

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 the apparent shape of a circle viewed obliquely from outside it, as distinct from a hyperbola which is the shape seen from inside. It is the finite or bounded case of a conic section as a shape cut in a cone by a plane, the unbounded cases being the parabola, which like the ellipse remains connected, and the hyperbola, which separates into two connected components or branches.

Equivalently an ellipse can be defined as the locus of points, or path traced out, in a plane such that the sum of the distances from the moving point to two fixed points remains constant. The two fixed points are then called foci (singular- focus). When the foci coincide the ellipse becomes a circle and the two distances then coincide as its radius. A variant of this replaces one of the foci with a straight line not passing through the remaining focus, called the directrix; in this case the locus is of a point whose distance from the remaining focus maintains a constant ratio less than one with its distance from the directrix.

Yet another definition of an ellipse, the algebraic or implicit definition, is, up to rotation and translation (geometry), any set of points $(x, y)$ in the Cartesian plane satisfying an equation of the form

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

where a and b are any positive real numbers.

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

x 2 + y 2 = 1,

of radius 1 and hence width and height 2, by scaling the x and y coordinates with the respective factors a and b; thus $\frac{x^2}{49}+\frac{y^2}{9}=1$ determines an ellipse centered on the origin of width 14 and height 6.

The informal "apparent shape" definition is formalized by defining an ellipse to be the result of projecting a circle onto any plane in three dimensions that does not intersect the circle. (Cutting the circle at two points yields a hyperbola, one point a parabola.) More generally an ellipse results when any conic section, whether ellipse, parabola, or hyperbola, is projected onto a plane, provided its connected components lie properly within distinct halves of the space cut by the plane (i.e. not even touching the plane, and with the two branches of a hyperbola placed on opposite sides), and that no axis of symmetry of the conic section be parallel to the plane (to force the diverging arms of parabolas to close up).

In defining an ellipse, the vertical diameter (or "minor axis") passing through its center is known as the conjugate diameter or axis, and the horizontal diameter (or major axis)——perpendicular, or "transverse", to the conjugate——is the transverse diameter or axis passing through its center.^[1]

The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximal where the major axis cuts the ellipse and minimal where the minor axis cuts it. 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 line segment from the center to a nearest point of the ellipse. In the example of the preceding paragraph a = 7 and b = 3.

Common to all of these definitions is that they specify an ellipse in a plane with five real numbers, the degrees of freedom of any general conic section in a plane. In the case of the algebraic definition the translation and rotation needed to put the ellipse in general position accounts for three of the numbers. Circles as a special case of the ellipse have only three degrees of freedom, while parabolas as the limiting case of a projection of a circle where the circle touches the plane of projection have four.

1 Eccentricity
2 True anomaly
3 Eccentric anomaly
4 Reduction to canonical form
5 Drawing
6 Equations
7 Semi-latus rectum and polar coordinates
8 Area and circumference
9 Stretching and projection
10 Reflection property
11 Ellipses in physics
12 Ellipses in computer graphics
13 See also
14 References
15 External links

[edit] 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 $(a e,0)$ and $( - a e,0).$ The distance $a e$ from a focal point to the centre is called the linear eccentricity of the ellipse. The distance between the foci is $2 a e$ .

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.

[edit] True anomaly

The polar angle $θ$ of a point on an ellipse relative the focal point $F 1$ is called the true anomaly of the point. Relative the "canonical coordinate system" with origin, $C$ , at the mid-point between the focii $F 1$ and $F 2$ in which the equation of the ellipse is

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

one has that

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

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

one has that

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

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

[edit] 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 $\left(x, y\right)$ .

For the numerical computation of $\operatorname{arg}(x, y),$ the standard function atan2(y, x) available in many programming languages can be used.

[edit] 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 $B 2 < 4 A C$ 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

$a=\sqrt{-\frac{F'}{A'}}$

$b=\sqrt{-\frac{F'}{C'}}$

[edit] 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 resulting ellipse will have each of the four sides of the rectangle as a tangent parallel to one of its axes, with each long side being parallel to the major axis and each short to the minor.

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.

The foci are placed as follows. First draw a circle centered on a corner of the rectangle, having radius the short side, namely 2b. The two long sides, of length 2a, are then respectively tangent to and centered on this circle. From the far end of the latter long side, draw a tangent to the circle. By Pythagoras's theorem the length of this tangent is $\sqrt{(2a)^2-(2b)^2}$ , which is the distance between the foci of an ellipse of these dimensions. Place the foci symmetrically on the line parallel to and midway between the two long sides, at the separation just determined.

The length of the loop is determined simply by hooking it over one focus and adjusting it so as to just reach the middle of the more remote short side.

The length of the portion of the loop from the center of the rectangle to the hooking focus and back to the center equals the separation of the foci, by symmetry of their placement. The other portion of the loop, from the center to the side and back, is of length 2a, which is therefore also the distance from one focus, to the pencil, thence to the other focus.

[edit] 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 $(cosβ,sinβ)$ .

[edit] 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, $l=a\cos(o\!\varepsilon)^2=a\cdot(1-\varepsilon^2);\,\!$

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°.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] 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.

[edit] See also

Apollonius of Perga, the classical authority
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 or more "pointy"
Hyperbola
Parabola
Oval
True, eccentric, and mean anomaly
Matrix representation of conic sections
Kepler's Laws of Planetary Motion
Ellipse/Proofs
Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae

[edit] References

^ Haswell, Charles Haynes (1920). "Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas". Harper & Brothers. Retrieved on 2007-04-09.

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.

[edit] External links

Wikimedia Commons has media related to: Ellipses

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

This is an extract from Wikipedia, the Free Encyclopedia

From Wikipedia, the free encyclopedia.

Contents

[edit] Eccentricity

[edit] True anomaly

[edit] Eccentric anomaly

[edit] Reduction to canonical form

[edit] Drawing

[edit] Equations

[edit] Semi-latus rectum and polar coordinates

[edit] Area and circumference

[edit] Stretching and projection

[edit] Reflection property

[edit] Ellipses in physics

[edit] Ellipses in computer graphics

[edit] See also

[edit] References

[edit] External links