Ellipse

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

For other uses, see Ellipse (disambiguation).

In geometry, an ellipse (from Greek ἔλλειψις elleipsis, a "falling short") is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is perpendicular to the axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

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

The rings of Saturn are circular, but when seen partially edge on, as in this photograph, they appear to be ellipses. Photo by ESO

Ellipses are closed curves and are the bounded case of the conic sections, the curves that result from the intersection of a circular cone and a plane that does not pass through its apex; the other two (open and unbounded) cases are parabolas and hyperbolas. Ellipses also arise as images of a circle under parallel projection and some cases of perspective projection. 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.

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, the major and minor semiaxes, or major radius and minor radius. When a and b are equal, the foci coincide with the center, and the ellipse becomes a circle with radius a=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 (2a). 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 a=b, in which case the ellipse is a circle. As the eccentricity tends to 1, the ellipse gets a more elongated shape and tends either towards a line segment (see below) or a parabola, and the ratio a/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

Drawing an ellipse with two pins, a loop and a pen.

An ellipse can be drawn using two drawing 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.

If the ellipse is to be 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 string 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, whose radius is 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. Then draw the ellipse as above; it should fit snugly in the original rectangle.

Other methods

Trammel of Archimedes (elpsograph) animation

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.

The trammel of Archimedes or 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 mechanism can be used with a router to cut ellipses from board material. The mechanism is also used in a toy called the "nothing grinder".

Approximations to ellipses

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. With the exception of Mercury, all the planets have an orbit whose minor axis differs from the major axis by less than half of one percent. To draw the orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to the eccentricity multiplied by the radius.

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 United States 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

Further information: Elliptic orbit

The idea of planets moving in an elliptic orbit was proposed in the 5th century by the Indian astronomer Aryabhata and later in the 11th century by the Islamic astronomers Biruni and Arzachel, though in a geocentric context. In the 15th century, the Kerala astronomer Nilakantha Somayaji proposed elliptic orbits in a geoheliocentric context (where the planets orbit the Sun, which in turn orbits the Earth). In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun are ellipses with the Sun at one focus, 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 the square of the 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; of a mass attached to a fixed point by a perfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportional to its distance from a fixed attractor. 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 produce variable angular speed or torque from a constant rotation of the driving axle. An example application would be a device that winds thread onto a conical bobbin on a spinning machine. The bobbin would need to wind faster when the thread is near the apex than when it is near the base.

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 raniel moralde.

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

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

The ellipse is a special case of the hypotrochoid when 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.

If the ellipse is given by the implicit equation $Ax^{2}+Bxy+Cy^{2}+1=0$, then the area is ${\frac {2\pi }{\sqrt {B^{2}-4AC}}}$.

Circumference

The circumference $C$ of an ellipse is $4aE(\varepsilon ^{2})$, where the function $E$ is the complete elliptic integral of the second kind. The exact infinite series is:

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

oder

$C=-2\pi a\sum _{n=0}^{\infty }{\varepsilon ^{2n} \over 2n-1}\prod _{m=1}^{n}\left({2m-1 \over 2m}\right)^{2}\,\!$

For computational purposes a much faster series where the denominators vanish at a rate ${\tfrac {27}{1024}}\left({\tfrac {a-b}{a+b}}\right)^{8}$ is given by:

$C={\frac {8\pi }{Q^{5/4}}}\sum _{n=0}^{\infty }{\frac {({\tfrac {1}{12}})_{n}({\tfrac {5}{12}})_{n}(v_{1}+nv_{2})r^{n}}{(n!)^{2}}}$

$r={\tfrac {432(a^{2}-b^{2})^{2}(a-b)^{6}ba}{Q^{3}}}$

$Q=b^{4}+60ab^{3}+134a^{2}b^{2}+60a^{3}b+a^{4}$

$v_{1}=ba(15b^{4}+68ab^{3}+90a^{2}b^{2}+68a^{3}b+15a^{4})$

$v_{2}=-a^{6}-b^{6}+126ab^{5}+1041a^{2}b^{4}+1764a^{3}b^{3}+1041a^{4}b^{2}+126a^{5}b$

A good approximation is Ramanujan's:

$C\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]=\pi (3(a+b)-{\sqrt {10ab+3(a^{2}+b^{2})}})$

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 hyperbolae and parabolae. 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 of any of those geometric objects from a point O onto a plane P, provided that the plane Q that goes 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

$~AX^{2}+BXY+CY^{2}+DX+EY+F=0$

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

$~AX^{2}+BXY+CY^{2}+DX+EY=1$

with $~B^{2}-4AC<0$

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 $(-ea,0)$ and $(+ea,0)$.

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 $\mathbb {R} ^{2}$, 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 $a+eX$ and $a-eX$, 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

$X(t)=X_{c}+a\,\cos t\,\cos \phi -b\,\sin t\,\sin \phi$

$Y(t)=Y_{c}+a\,\cos t\,\sin \phi +b\,\sin t\,\cos \phi$

as the parameter t varies from 0 to 2π. 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

Parametric equation for the ellipse (red) in canonical position. The eccentric anomaly t is the angle of the blue line with the X-axis.

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

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate $\theta =0$ measured from the major axis, the ellipse's equation is

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

Polar form relative to focus

Polar coordinates centered at focus.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate $\theta =0$ still measured from the major axis, the ellipse's equation is

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

where the sign in the denominator is negative if the reference direction $\theta =0$ points towards the center (as illustrated on the right), and positive if that direction points away from the center.

In the slightly more general case of an ellipse with one focus at the origin and the other focus at angular coordinate $\phi$, the polar form is

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

The angle $\theta$ in these formulas is called the true anomaly of the point. The numerator $a(1-\varepsilon ^{2})$ of these formulas 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.

Semi-latus rectum.

General polar form

The following equation on the polar coordinates (r,θ) describes a general ellipse with semidiameters a and b, centered at a point (r₀,θ₀), with the a axis rotated by φ relative to the polar axis:

$r(\theta )={\frac {P(\theta )+Q(\theta )}{R(\theta )}}$

where

$P(\theta )=r_{0}\left(\left(b^{2}-a^{2}\right)\cos \left(\theta +\theta _{0}-2\varphi \right)+\left(a^{2}+b^{2}\right)\cos \left(\theta -\theta _{0}\right)\right)$

$Q(\theta )={\sqrt {2}}ab{\sqrt {R(\theta )-2r_{0}^{2}\sin ^{2}\left(\theta -\theta _{0}\right)}}$

$R(\theta )=\left(b^{2}-a^{2}\right)\cos(2\theta -2\varphi )+a^{2}+b^{2}$

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:

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

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

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), defining its position, orientation, shape, and scale. In comparison, circles have only three degrees of freedom (position and scale), while parabolae have four. 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, for example, the coefficients A,B,C,D,E of the implicit equation, or with the coefficients X_c, Y_c, φ, a, b of the general parametric form.

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.

Degenerate ellipse

A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends. Although the eccentricity is 1 this is not a parabola. A radial elliptic trajectory is a non-trivial special case of an elliptic orbit, where the ellipse is a line segment.

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
• Proofs involving the ellipse
• Ellipsis
• Elliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae

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
• Weisstein, Eric W. "Ellipse". MathWorld.

External links

• Apollonius' Derivation of the Ellipse at Convergence
• Ellipse & Hyperbola Construction - Two interactive applets showing how to trace the curves of the ellipse and hyperbola. (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.
• Ellipse as hypotrochoid