Ellipse

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An ellipse obtained as the intersection of a cone with a plane.

In mathematics, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is the finite or bounded case of a conic section, the geometric shape that results from cutting a circular conical or cylindrical surface with an oblique plane (the other two cases being the parabola and the hyperbola). It is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

Ellipses also arise as images of a circle or a sphere under parallel projection, and some cases of perspective projection. Indeed, circles are special cases of ellipses. An ellipse is also the closed and bounded case of an implicit curve of degree 2, and of a rational curve of degree 2. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency.

Elements of an ellipse

The ellipse and some of its mathematical properties.

An ellipse is a smooth closed curve which is symmetric about its center. The distance between antipodal points on the ellipse, or pairs of points whose midpoint is at the center of the ellipse, is maximum and minimum along two perpendicular directions, the major axis or transverse diameter, and the minor axis or conjugate diameter.[1]

The semimajor axis (denoted by a in the figure) and the semiminor axis (denoted by b in the figure) are one half of the major and minor diameters, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,[2][3] the major and minor semiaxes,[4][5] or major radius and minor radius.[6][7][8][9] When a and b are equal, the foci coincide with the center, and the ellipse becomes a circle with radius =b.

There are two special points F1 and F2 on the ellipse's major axis, on either side of the center, such that the sum of the distances from any point of the ellipse to those two points is constant and equal to the major diameter (2 a). Each of these two points is called a focus of the ellipse.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the foci to the length of the major axis. The eccentricity is necessarily between 0 and 1; it is zero if and only if =b, in which case the ellipse is a circle. As the eccentricity tends to 1, the ellipse becomes more elongated, and the ratio /b tends to infinity. The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse.

Drawing ellipses

The pins-and-string method

Two pins, a loop and a pen method

An ellipse can be drawn using two pins, a length of string, and a pencil:

Push the pins into the paper at two points, which will become the ellipse's foci. Tie the string into a loose loop around the two pins. Pull the loop taut with the pencil's tip, so as to form a triangle. Move the pencil around, while keeping the string taut, and its tip will trace out an ellipse.

To draw an ellipse inscribed within a specified rectangle, tangent to its four sides at their midpoints, one must first determine the position of the foci and the length of the loop:

Let A,B,C,D be the corners of the rectangle, in clockwise order, with A-B being one of the long sides. Draw a circle centered on A, having radius the short side A-D. From corner B draw a tangent to the circle. The length L of this tangent is the distance between the foci. Draw two perpendicular lines through the center of the rectangle and parallel to its sides; these will be the major and minor axes of the ellipse. Place the foci on the major axis, symmetrically, at distance L/2 from the center.
To adjust the length of the string loop, insert a pin at one focus, and another pin at the opposite end of the major diameter. Loop the string around the two pins and tie it taut.

Other methods

An ellipse can also be drawn using a ruler, a set square, and a pencil:

Draw two perpendicular lines M,N on the paper; these will be the major and minor axes of the ellipse. Mark three points A, B, C on the ruler. With one hand, move the ruler onto the paper, turning and sliding it so as to keep point A always on line M, and B on line N. With the other hand, keep the pencil's tip on the paper, following point C of the ruler. The tip will trace out an ellipse.

This method can be implemented with a router to cut ellipses from board material. The ellipsograph is a mechanical device that implements this principle. The ruler is replaced by a rod with a pencil holder (point C) at one end, and two adjustable side pins (points A and B) that slide into two perpendicular slots cut into a metal plate. The nothing grinder is a novelty ellipsograph.

Ellipses in physics

Elliptical reflectors and acoustics

If the water's surface is disturbed at one focus of an elliptical water tank, the circular waves created by that disturbance, after being reflected by the walls, will 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 will hold 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 document scanners.

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 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 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.

Planetary orbits

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.

Keplerian elliptical orbits are the result of any radially-directed attraction force whose strength is inversely proportional to 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 effects which become significant when the particles are moving at high speed.)

Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, for instance, of a long pendulum that is free to move in two dimensions, or of a mass attached to a fixed point by a perfectly elastic spring. 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 display is an ellipse, rather than a straight line, the two signals are out of phase.

Elliptical gears

Two gears with the same elliptical outline, each pivoting around one focus and positioned at the proper angle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chain or timing belt. Such elliptical gears may be used in mechanical equipment to vary the torque or angular speed during each turn of the driving axle.

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.)

Mathematical definitions and properties

In Euclidean geometry

Definition

In Euclidean geometry, an ellipse is usually defined as the bounded case of a conic section, or as the locus of the points such that the sum of the distances to two fixed points is constant. The equivalence of these two definitions can be proved using the Dandelin spheres.

Eccentricity

The eccentricity of the ellipse is

$e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2}$

The distance from the center to either focus is ae, or simply $\sqrt{a^2-b^2}$

Directrix

Each focus F of the ellipse is associated to a line D perpendicular to the major axis (the directrix) such that the distance from any point on the ellipse to F is a constant fraction of its distance from D. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse. The ratio between the two distances is the eccentricity e of the ellipse; so the distance from the center to the directrix is a/e.

Ellipse as hypotrochoid

The ellipse is a special case of the hypotrochoid when R=2r.

An ellipse (in red) as a special case of the hypotrochoid with R=2r.

Area

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.

Circumference

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 the better approximation

$C \approx \frac{a}{2} \sqrt{93 + \frac{1}{2} \sqrt{3}}$

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.

In projective geometry

In projective geometry, an ellipse can be defined as the set of all points of intersection between corresponding lines of two pencils of lines which are related by a projective map. By projective duality, an ellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which are related by a projective map.

This definition also generates hyperbolas and parabolas. However, in projective geometry every conic section is equivalent to an ellipse. A parabola is an ellipse that is tangent to the line at infinity Ω, and the hyperbola is an ellipse that crosses Ω.

An ellipse is also the result of projecting a circle, sphere, or ellipse in three dimensions onto a plane, by parallel lines. It is also the result of conical (perspective) projection any of those geometric objects from a point O onto a plane P, provided that the plane Q that goes through through O and is parallel to P does not cut the object. The image of an ellipse by any affine map is an ellipse, and so is the image of an ellipse by any projective map M such that the line M-1(Ω) does not touch or cross the ellipse.

In analytic geometry

General ellipse

In analytic geometry, the ellipse is defined as the set of points $(X,Y)$ of the Cartesian plane that satisfy the implicit equation

$~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0$

provided that F is not zero and $F(B^2 - 4 A C)$ is positive; or of the form

$~A X^2 + B X Y + C Y^2 + D X + E Y = 1$

with $~B^2 - 4 A C < 0$

Canonical form

By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

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

Here $(x,y)$ are the point coordinates in the canonical system, whose origin is the center $(X_c,Y_c)$ of the ellipse, whose $x$-axis is the unit vector $(X_a,Y_a)$ parallel to the major axis, and whose $y$-axis is the perpendicular vector $(-Y_a,X_a)$ That is, $x = X_a(X - X_c) + Y_a(Y - Y_c)$ and $y = -Y_a(X - X_c) + X_a(Y - Y_c)$.

In this system, the center is the origin $(0,0)$ and the foci are $(-e a, 0)$ and $(+e a, 0)$.

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters. Moreover, any canonical ellipse can be obtained by scaling the unit circle of $\reals^2$, defined by the equation

$X^2+Y^2=1\,$

by factors a and b along the two axes.

For an ellipse in canonical form, we have

$Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)}$

The distances from a point $(X,Y)$ on the ellipse to the left and right foci are $a + e X$ and $a - e X$, respectively.

In trigonometry

General parametric form

An ellipse in general position can be expressed parametrically as the path of a point $(X(t),Y(t))$, where

$X(t)=X_c + a\,\cos t\,\cos \phi - b\,\sin t\,\sin\phi$
$Y(t)=Y_c + a\,\cos t\,\sin \phi + b\,\sin t\,\cos\phi$

for $t \in [0,2 \pi)$. Here $(X_c,Y_c)$ is the center of the ellipse, and $\phi$ is the angle between the $X$-axis and the major axis of the ellipse.

Parametric form in canonical position

For an ellipse in canonical position (center at origin, major axis along the X-axis), the equation simplifies to

$X(t)=a\,\cos t$
$Y(t)=b\,\sin t$

Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of $(X(t),Y(t))$ with the X-axis.

Polar form relative to center

For an ellipse in canonical position, with the origin at the center, the polar equation is

$r=\frac{ab}{\sqrt{b^2 \cos^2\theta + a^2 \sin^2\theta}}$.

Polar form relative to focus

Polar coordinates for the ellipse, with origin at F1

Using polar coordinates with the origin at a focus and with $\theta = 0,\pi$ along the major axis, the ellipse's polar equation is

$r=\frac{a (1-\varepsilon^{2})}{1 \pm \varepsilon \cos\theta},$

where the sign in the denominator is positive if the origin is at F2 and negative if the origin is at F1. The latter case is illustrated on the right.

The angle $\theta$ is called the true anomaly of the point. The numerator $a (1-\varepsilon^{2})$ of this formula is the semi-latus rectum of the ellipse, usually denoted $l$. It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.

Ellipse, showing semi-latus rectum

Gauss-mapped form

The Gauss-mapped equation of the ellipse gives the coordinates of the point on the ellipse where the normal makes an angle $\beta$ with the X-axis:

$X(\beta) = \frac{a^2\cos\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}}$
$Y(\beta) =\frac{b^2\sin\beta}{\sqrt{a^2\cos^2\beta+b^2\sin^2\beta}}$

Angular eccentricity

The angular eccentricity $o\!\varepsilon$ is the angle whose sine is the eccentricity e; that is,

$o\!\varepsilon=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!$

Degrees of freedom

An ellipse in the plane has five degrees of freedom, the same as a general conic section. Said another way, the set of all ellipses in the plane, with any natural metric (such as the Hausdorff distance) is a five-dimensional manifold. These degrees can be identified with the coefficients $A,B,C,D,E$ of the implicit equation. In comparison, circles have only three degrees of freedom, while parabolas have four.

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 to drawing ellipses with the major axis horizontal or vertical. 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

• Conic section
• 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

References

1. Haswell, Charles Haynes (1920). "Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas". Harper & Brothers. Retrieved 2007-04-09.
2. John Herschel (1842) A Treatise on Astronomy‎, page 256
3. John Lankford (1996), History of Astronomy: An Encyclopedia, page 194
4. V. Prasolov and V. Tikhomirov (2001), Geometry‎, page 80
5. David Salomon (2006), Curves and surfaces for computer graphics‎, page 365
6. CRC Press (2004), The CRC handbook of mechanical engineering, page 11-8
7. The Mathematical Association of America (1976), The American Mathematical Monthly, vol. 83, page 207