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

In mathematics, 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 orthogonal to the cone's axis. An ellipse is also the locus of all points of the plane whose distances to two fixed points add to the same constant.

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 arise from the intersection of a right circular cylinder with a plane that is not parallel to the cylinder's main axis of symmetry. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of 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[link]

The ellipse and some of its mathematical properties.

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

The semi-major axis (denoted by a in the figure) and the semi-minor axis (denoted by b in the figure) are one half of the major and minor axes, 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]

The foci of the ellipse are two special points F₁ and F₂ on the ellipse's major axis and are equidistant from the center point. The sum of the distances from any point P on the ellipse to those two foci is constant and equal to the major axis ( PF₁ + PF₂ = 2a ). Each of these two points is called a focus of the ellipse.

Refer to the lower Directrix section of this article for a second equivalent construction of an ellipse.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to the length of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0<e<1). When the eccentricity is 0 the foci coincide with the center point and the figure is a circle. As the eccentricity tends toward 1, the ellipse gets a more elongated shape. It tends towards a line segment (see below) if the two foci remain a finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away.
The distance ae from a focal point to the centre is called the linear eccentricity of the ellipse (f = ae).

Drawing ellipses[link]

Pins-and-string method[link]

The characterization of an ellipse as the locus of points so that sum of the distances to the foci is constant leads to a method of drawing one using two drawing pins, a length of string, and a pencil.^[10] In this method, pins are pushed into the paper at two points which will become the ellipse's foci. A string tied at each end to the two pins and the tip of a pen is used to pull the loop taut so as to form a triangle. The tip of the pen will then trace an ellipse if it is moved while keeping the string taut. Using two pegs and a rope, this procedure is traditionally used by gardeners to outline an elliptical flower bed; thus it is called the gardener's ellipse.^[11]

Other methods[link]

Trammel of Archimedes (ellipsograph) 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. A->C being the length of the major axis and B->C the length of the minor axis. With one hand, move the ruler on the paper, turning and sliding it so as to keep point A always on line N, and B on line M. 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.^[12] 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[link]

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. 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.

Mathematical definitions and properties[link]

In Euclidean geometry[link]

Definition[link]

In Euclidean geometry, an ellipse is usually defined as the bounded case of a conic section, or as the set of 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.

Equations[link]

The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is Failed to parse (Missing texvc executable; please see math/README to configure.): \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1.

Focus[link]

The distance from the center C to either focus is f = ae, which can be expressed in terms of the major and minor radii:

Failed to parse (Missing texvc executable; please see math/README to configure.): f = \sqrt{a^2-b^2}.

Eccentricity[link]

The eccentricity of the ellipse (commonly denoted as either e or Failed to parse (Missing texvc executable; please see math/README to configure.): \epsilon ) is

Failed to parse (Missing texvc executable; please see math/README to configure.): e=\varepsilon=\sqrt{\frac{a^2-b^2}{a^2}} =\sqrt{1-\left(\frac{b}{a}\right)^2} =f/a

(where again a and b are one-half of the ellipse's major and minor axes respectively, and f is the focal distance) or, as expressed in terms using the flattening factor Failed to parse (Missing texvc executable; please see math/README to configure.): g=1-\frac {b}{a}=1-\sqrt{1-e^2},

Failed to parse (Missing texvc executable; please see math/README to configure.): e=\sqrt{g(2-g)}.

When the ellipse is expressed in general quadratic form, its eccentricity is given by this expression. Other expressions for the eccentricity are given here.

Directrix[link]

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to the illustration on the right. The distance from any point P on the ellipse to the focus F is a constant fraction of that point's perpendicular distance to the directrix resulting in the equality, e=PF/PD. The ratio of these two distances is the eccentricity of the ellipse. This property (which can be proved using the Dandelin spheres) can be taken as another definition of the ellipse.
Besides the well–known ratio e=f/a, it is also true that e=a/d.

Circular directrix[link]

The ellipse can also be defined as the set of points that are equidistant from one focus and a particular circle, the directrix circle, that is centered on the other focus. The radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle, as is the entire ellipse.

Ellipse as hypotrochoid[link]

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[link]

The area enclosed by an ellipse is πab, where a and b are one-half of the ellipse's major and minor axes respectively.

If the ellipse is given by the implicit equation Failed to parse (Missing texvc executable; please see math/README to configure.): A x^2+ B x y + C y^2 = 1 , then the area is Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{2\pi}{\sqrt{ 4 A C - B^2 }} .

Circumference[link]

The circumference Failed to parse (Missing texvc executable; please see math/README to configure.): C

of an ellipse is:

Failed to parse (Missing texvc executable; please see math/README to configure.): C = 4 a E(e)

where again e is the eccentricity and where the function Failed to parse (Missing texvc executable; please see math/README to configure.): E

is the complete elliptic integral of the second kind.

The exact infinite series is:

Failed to parse (Missing texvc executable; please see math/README to configure.): C = 2\pi a \left[{1 - \left({1\over 2}\right)^2e^2 - \left({1\cdot 3\over 2\cdot 4}\right)^2{e^4\over 3} - \left({1\cdot 3\cdot 5\over 2\cdot 4\cdot 6}\right)^2{e^6\over5} - \cdots}\right]\,\!

or

Failed to parse (Missing texvc executable; please see math/README to configure.): C = 2\pi a \left[1 - \sum_{n=1}^\infty {e^{2n}\over 2n - 1} \prod_{m=1}^n \left({ 2m-1 \over 2m}\right)^2\right] \,\!

For computational purposes a much faster series where the denominators vanish at a rate Failed to parse (Missing texvc executable; please see math/README to configure.): \tfrac{27}{1024} \left (\tfrac{a-b}{a+b} \right )^{8}

is given by:[13]

Failed to parse (Missing texvc executable; please see math/README to configure.): 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}}

Failed to parse (Missing texvc executable; please see math/README to configure.): r = \frac{432(a^{2}-b^{2})^{2}(a-b)^{6}ba}{Q^3}

Failed to parse (Missing texvc executable; please see math/README to configure.): Q = b^{4}+60ab^{3}+134a^{2}b^{2}+60a^{3}b+a^{4}\,

Failed to parse (Missing texvc executable; please see math/README to configure.): v_{1} = ba(15b^{4}+68ab^{3}+90a^{2}b^{2}+68a^{3}b+15a^{4})\,

Failed to parse (Missing texvc executable; please see math/README to configure.): 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): C \approx \pi \left[3(a+b) - \sqrt{(3a+b)(a+3b)}\right]= \pi \left[3(a+b)-\sqrt{10ab+3(a^2+b^2)}\right]

and a better approximation is

Failed to parse (Missing texvc executable; please see math/README to configure.): 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, these become:

Failed to parse (Missing texvc executable; please see math/README to configure.): C \approx \frac{\pi a (9 - \sqrt{35})}{2}

or, as an estimate of the better approximation,

Failed to parse (Missing texvc executable; please see math/README to configure.): 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.^{[citation needed]}

Chords[link]

The midpoints of a set of parallel chords of an ellipse are collinear.^[14]:p.147

In projective geometry[link]

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[link]

General ellipse[link]

In analytic geometry, the ellipse is defined as the set of points Failed to parse (Missing texvc executable; please see math/README to configure.): (X,Y)

of the Cartesian plane that, in non-degenerate cases, satisfy the implicit equation[15][16]

Failed to parse (Missing texvc executable; please see math/README to configure.): ~A X^2 + B X Y + C Y^2 + D X + E Y + F = 0

provided Failed to parse (Missing texvc executable; please see math/README to configure.): B^2 - 4AC < 0.

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant of the 3×3 matrix [A, B/2, D/2 ; B/2, C, E/2 ; D/2, E/2, F ]: that is, ∆ = (AC - B²/4)F + BED/4 - CD²/4 - AE²/4. Then the ellipse is a non-degenerate real ellipse if and only if C∆<0. If C∆>0 we have an imaginary ellipse, and if ∆=0 we have a point ellipse.^[17]:p.63

Canonical form[link]

Let Failed to parse (Missing texvc executable; please see math/README to configure.): a>b . By a proper choice of coordinate system, the ellipse can be described by the canonical implicit equation

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{x^2}{a^2}+\frac{y^2}{b^2}=1

Here Failed to parse (Missing texvc executable; please see math/README to configure.): (x,y)

are the point coordinates in the canonical system, whose origin is the center Failed to parse (Missing texvc executable; please see math/README to configure.): (X_c,Y_c)
of the ellipse, whose Failed to parse (Missing texvc executable; please see math/README to configure.): x

-axis is the unit vector Failed to parse (Missing texvc executable; please see math/README to configure.): (X_a,Y_a)

coinciding with the major axis, and whose Failed to parse (Missing texvc executable; please see math/README to configure.): y

-axis is the perpendicular vector Failed to parse (Missing texvc executable; please see math/README to configure.): (-Y_a,X_a)

coinciding with the minor axis.  That is, Failed to parse (Missing texvc executable; please see math/README to configure.): x = X_a(X - X_c) + Y_a(Y - Y_c)
and Failed to parse (Missing texvc executable; please see math/README to configure.): y = -Y_a(X - X_c) + X_a(Y - Y_c)

.

In this system, the center is the origin Failed to parse (Missing texvc executable; please see math/README to configure.): (0,0)

and the foci are Failed to parse (Missing texvc executable; please see math/README to configure.): (-e a, 0)
and Failed to parse (Missing texvc executable; please see math/README to configure.): (+e a, 0)

.

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters. Translation of an ellipse centered at Failed to parse (Missing texvc executable; please see math/README to configure.): (X_c,Y_c)

is expressed as

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{(x - X_c)^2}{a^2}+\frac{(y - Y_c)^2}{b^2}=1

Moreover, any canonical ellipse can be obtained by scaling the unit circle of Failed to parse (Missing texvc executable; please see math/README to configure.): \reals^2 , defined by the equation

Failed to parse (Missing texvc executable; please see math/README to configure.): X^2+Y^2=1\,

by factors a and b along the two axes.

For an ellipse in canonical form, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): Y = \pm b\sqrt{1 - (X/a)^2} = \pm \sqrt{(a^2-X^2)(1 - e^2)}

The distances from a point Failed to parse (Missing texvc executable; please see math/README to configure.): (X,Y)

on the ellipse to the left and right foci are Failed to parse (Missing texvc executable; please see math/README to configure.): a + e X
and Failed to parse (Missing texvc executable; please see math/README to configure.): a - e X

, respectively.

In trigonometry[link]

General parametric form[link]

An ellipse in general position can be expressed parametrically as the path of a point Failed to parse (Missing texvc executable; please see math/README to configure.): (X(t),Y(t)) , where

Failed to parse (Missing texvc executable; please see math/README to configure.): X(t)=X_c + a\,\cos t\,\cos \varphi - b\,\sin t\,\sin\varphi

Failed to parse (Missing texvc executable; please see math/README to configure.): Y(t)=Y_c + a\,\cos t\,\sin \varphi + b\,\sin t\,\cos\varphi

as the parameter t varies from 0 to 2π. Here Failed to parse (Missing texvc executable; please see math/README to configure.): (X_c,Y_c)

is the center of the ellipse, and Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi
is the angle between the Failed to parse (Missing texvc executable; please see math/README to configure.): X

-axis and the major axis of the ellipse.

Parametric form in canonical position[link]

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

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

Failed to parse (Missing texvc executable; please see math/README to configure.): X(t)=a\,\cos t

Failed to parse (Missing texvc executable; please see math/README to configure.): Y(t)=b\,\sin t

Note that the parameter t (called the eccentric anomaly in astronomy) is not the angle of Failed to parse (Missing texvc executable; please see math/README to configure.): (X(t),Y(t))

with the X-axis.

Formulae connecting a tangential angle Failed to parse (Missing texvc executable; please see math/README to configure.): \phi , the angle anchored at the ellipse's center Failed to parse (Missing texvc executable; please see math/README to configure.): \phi^\prime

(called also the polar angle from the ellipse center), and the parametric angle t[18] are[19][20][21]:

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan \phi=\frac {a}{b} \tan t=\frac {\tan \phi'}{(1-g)^2}=\frac {\tan \phi'}{1-e^2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan \phi^\prime=(1-f) \tan t

Failed to parse (Missing texvc executable; please see math/README to configure.): \tan t=\frac {b}{a} \tan \phi=\sqrt{(1-e^2)} \tan \phi=(1-g) \tan \phi=\frac {\tan \phi'}{\sqrt{(1-e^2)}}=\frac {a}{b} \tan \phi'

Polar form relative to center[link]

In polar coordinates, with the origin at the center of the ellipse and with the angular coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

measured from the major axis, the ellipse's equation is

Failed to parse (Missing texvc executable; please see math/README to configure.): r(\theta)=\frac{ab}{\sqrt{(b \cos \theta)^2 + (a\sin \theta)^2}}

Polar form relative to focus[link]

Polar coordinates centered at focus.

If instead we use polar coordinates with the origin at one focus, with the angular coordinate Failed to parse (Missing texvc executable; please see math/README to configure.): \theta = 0

still measured from the major axis, the ellipse's equation is

Failed to parse (Missing texvc executable; please see math/README to configure.): r(\theta)=\frac{a (1-e^{2})}{1 \pm e\cos\theta}

where the sign in the denominator is negative if the reference direction Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \phi , the polar form is

Failed to parse (Missing texvc executable; please see math/README to configure.): r=\frac{a (1-e^{2})}{1 - e\cos(\theta - \phi)}.

The angle Failed to parse (Missing texvc executable; please see math/README to configure.): \theta

in these formulas is called the true anomaly of the point.  The numerator Failed to parse (Missing texvc executable; please see math/README to configure.): a (1-e^{2})
of these formulas is the semi-latus rectum of the ellipse, usually denoted Failed to parse (Missing texvc executable; please see math/README to configure.): 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[link]

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:

Failed to parse (Missing texvc executable; please see math/README to configure.): r(\theta )=\frac{P(\theta )+Q(\theta )}{R(\theta )}

where

Failed to parse (Missing texvc executable; please see math/README to configure.): 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]

Failed to parse (Missing texvc executable; please see math/README to configure.): Q(\theta )=\sqrt{2} a b \sqrt{R(\theta )-2 r_0^2 \sin ^2\left(\theta -\theta_0\right)}

Failed to parse (Missing texvc executable; please see math/README to configure.): R(\theta )=\left(b^2-a^2\right) \cos (2 \theta -2 \varphi )+a^2+b^2

Angular eccentricity[link]

The angular eccentricity Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha

is the angle whose sine is the eccentricity e; that is,

Failed to parse (Missing texvc executable; please see math/README to configure.): \alpha=\sin^{-1}(e)=\cos^{-1}\left(\frac{b}{a}\right)=2\tan^{-1}\left(\sqrt{\frac{a-b}{a+b}}\right);\,\!

Degrees of freedom[link]

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 physics[link]

Elliptical reflectors and acoustics[link]

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[link]

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.)

For elliptical orbits, useful relations involving the eccentricity, Failed to parse (Missing texvc executable; please see math/README to configure.): e

are:

Failed to parse (Missing texvc executable; please see math/README to configure.): e={{r_\mathrm{ap}-r_\mathrm{per}}\over{r_\mathrm{ap}+r_\mathrm{per}}}={{r_\mathrm{ap}-r_\mathrm{per}}\over{2a}}

Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{ap}=(1+e)a\!\,              Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{per}=(1-e)a\!\,

where:

• Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{ap}\,\!

is radius at apoapsis (i.e., the farthest distance).

• Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{per}\,\!

is radius at periapsis (the closest distance).

• Failed to parse (Missing texvc executable; please see math/README to configure.): a\,\!

is the length of the semi-major axis.

In addition, the semimajor axis Failed to parse (Missing texvc executable; please see math/README to configure.): a\,\!

is the arithmetic mean of Failed to parse (Missing texvc executable; please see math/README to configure.): r_\text{ap}\,\!
and Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{per}\,\!

, or Failed to parse (Missing texvc executable; please see math/README to configure.): a=\frac{{r_\text{ap}} + {r_\text{per}}}{2} , and the semiminor axis Failed to parse (Missing texvc executable; please see math/README to configure.): b\,\!

is the geometric mean of Failed to parse (Missing texvc executable; please see math/README to configure.): r_\text{ap}\,\!
and Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{per}\,\!

, or Failed to parse (Missing texvc executable; please see math/README to configure.): b=\, ^2 \! \! \! \! \sqrt{r_\mathrm{ap} \times r_\mathrm{per}}

. 

Also the semi-latus rectum (the distance from a focus to a point on the ellipse along a line parallel to the minor axis) is the harmonic mean of Failed to parse (Missing texvc executable; please see math/README to configure.): r_\text{ap}\,\!

and Failed to parse (Missing texvc executable; please see math/README to configure.): r_\mathrm{per}\,\!

, or Failed to parse (Missing texvc executable; please see math/README to configure.): \text {semi-latus rectum} = \frac{2}{\frac{1}{r_\text{ap}} + \frac{1}{r_\text{per}}}

.

Harmonic oscillators[link]

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[link]

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[link]

Two non-circular 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, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque from a constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed with inversely varying mechanical advantage.

Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.^[22]

An example gear 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.^[23]

Optics[link]

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.)

Ellipses in statistics and finance[link]

In statistics, a bivariate random vector (X, Y) is jointly elliptically distributed if its iso-density contours — loci of equal values of the density function — are ellipses. The concept extends to an arbitrary number of elements of the random vector, in which case in general the iso-density contours are ellipsoids. A special case is the multivariate normal distribution. The elliptical distributions are important in finance because if rates of return on assets are jointly elliptically distributed then all portfolios can be characterized completely by their mean and variance — that is, any two portfolios with identical mean and variance of portfolio return have identical distributions of portfolio return.^[24][25]

Ellipses in computer graphics[link]

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntosh QuickDraw API, and Direct2D on Windows. 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. M. L. V. Pitteway extended Bresenham's algorithm for lines to conics in 1967.^[26] Another efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken.^[27]

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm for drawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved them to have good properties.^[28] These algorithms need only a few multiplications and additions to calculate each vector.

It is beneficial to use a parametric formulation in computer graphics because 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.

Drawing with Bezier spline paths[link]

Multiple Bezier splines may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as an affine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since the constituent Bezier curves will behave appropriately under such transformations.

Line segment as a type of degenerate ellipse[link]

A line segment is a degenerate ellipse with semi-minor axis = 0 and eccentricity = 1, and with the focal points at the ends.^[29] 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.

Ellipses in optimization theory[link]

It is sometimes useful to find the minimum bounding ellipse on a set of points. The ellipsoid method is quite useful for attacking this problem.

See also[link]

References[link]

• Besant, W.H. (1907). "Chapter III. The Ellipse". Conic Sections. London: George Bell and Sons. p. 50. http://books.google.com/books?id=TRJLAAAAYAAJ&pg=PA50. 
• Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rd ed.). Scott Foresman/Little. p. 381. ISBN 0-673-38638-4. 
• Coxeter, H.S.M. (1969). Introduction to Geometry (2nd ed.). New York: Wiley. pp. 115–9. 
• Ellipse at Planetmath
• Weisstein, Eric W., "Ellipse" from MathWorld.

Notes[link]

External links[link]

Wikimedia Commons has media related to: Ellipses

• Video: How to draw Ellipse
• 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.
• Weisstein, Eric W., "Ellipse as hypotrochoid" from MathWorld.
• Ivanov, A.B. (2001), "Ellipse", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4, http://www.encyclopediaofmath.org/index.php?title=Ellipse&oldid=11394 

Imogen Heap
Background information
Birth name	Imogen Jennifer Jane Heap
Born	(1977-12-09) 9 December 1977 (age 34)
Origin	Havering, Essex, England
Genres	Electronica, alternative, indie, synthpop,[1] folktronica, ambient, dream pop, rock
Occupations	Musician, singer, songwriter, visual artist
Instruments	Vocals, keyboards, array mbira, cello, clarinet, guitar, drums, keytar, nail violin, vocal percussion, synthesizer, sampler, organ, Hang, vocoder
Years active	1997–present
Labels	Almo Sounds (1998–2001) Megaphonic (2005–present) RCA Victor (2006–present)
Associated acts	Frou Frou Urban Species IAMX MIKA
Website	imogenheap.com

Imogen Jennifer Jane Heap ( /ˈɪmədʒɨn ˈhiːp/; born 9 December 1977 in the London Borough of Havering)^[2] is a Grammy Award-winning English singer, composer and songwriter from Havering, Essex. She is known for her work as part of the musical duo Frou Frou and her solo albums, which she writes, produces, and mixes. She has produced three solo albums, the latest of which is 2009's Ellipse, which was a North American chart success and earned Heap two Grammy nominations, winning Best Engineered Album, Non-Classical.^[3]

Musical and songwriting style[link]

For her solo work (as well as her work with Frou Frou and Acacia), Heap plays heavily produced and arranged singer-songwriter pop incorporating elements of rock, dance and electronica. As a guest player and collaborator she has played rock (Jeff Beck), hip-hop (Urban Species) and theatre/film music.

A skilled multi-instrumentalist, Heap extensively uses manipulated electronic sounds as an integral part of her music. She also mixes ambient sound into her music (such as the sound of a frying pan in use cooking food, in the background of her song "My Secret Friend") and has commented that "certain sounds give the music a width and a space, and that's important."^[4]

Heap states that her song lyrics come from personal experience, but are not straightforwardly confessional. She has stated "Most of the time, the lyrics are kind of like my secret messages to my friends or my boyfriend or my mum or my dad. I would never tell them that these songs are about them or which specific lyric is about somebody. Often, when I sit down to write a lyric, it is in the heat of the moment, and something has just happened."^[4]

Biography[link]

Early life[link]

Heap played music from an early age, becoming classically trained in several instruments including piano, cello and clarinet. She attended Friends School, a private, Quaker-run boarding school in Saffron Walden. Heap's mother (an art therapist) and her father (a construction rock retailer) separated when she was twelve. By the age of thirteen, she had begun writing songs.

Heap did not get along well with the music teacher at her boarding school, so she principally taught herself sequencing, music engineering, sampling and production (on Atari computers). She also taught herself to play the guitar and drums, and subsequently two percussion/idiophone instruments, the array mbira and the Hang.^[5] After school, she went on to study at the BRIT School for Performing Arts & Technology in Croydon, South London.

Early work as musician (including work with Acacia)[link]

After being introduced to Nik Kershaw by his manager Mickey Modern, Heap and Kershaw recorded four demos that Mickey Modern took to Rondor Music. Consequently, a few months later she signed her first record contract at 18 to independent record label Almo Sounds^[6]. Modern and Wood formed Modernwood Management, and managed Heap until 2006, when Modernwood was dissolved. Wood continues to manage the artist via his new company, Radius Music^[7].

During 1996, Heap began working with an experimental pop band called Acacia, which featured her future collaborator Guy Sigsworth and was fronted by the singer Alexander Nilere. While never a full member of the band, Heap was a guest vocalist (as a counterpart to Nilere) and contributed to various Acacia single and album tracks. One Acacia song, "Maddening Shroud", would later be covered by Frou Frou^[8].

Mickey Modern asked Dennis Arnold to place Imogen in the line up in the 1996 Prince's Trust Concert in Hyde Park, London organized by Harvey Goldsmith. Heap performed four songs between sets by The Who and Eric Clapton.

[edit] 1998–2000: iMegaphone

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Heap's debut album, iMegaphone (an anagram of "Imogen Heap") was a mixture of self-penned and self-produced tracks, alongside tracks co-written and produced with established producers such as David Kahne, former Eurythmic Dave Stewart and Guy Sigsworth. The album was released in 1998 internationally via Almo Sounds, to favorable reviews comparing Heap's angst-filled songs to work by PJ Harvey, Kate Bush and Annie Lennox. Promotion for the record included a tour of America and performances around Europe. Three singles were commercially released in the UK: "Getting Scared", "Shine" and "Come Here Boy". "Oh Me, Oh My" was sent to US radio stations in place of "Shine".

Heap's early success was soon replaced by problems. Almo Sounds cut funding for UK promotion and gave Heap a deadline to deliver songs for her second album. Upon delivery of the songs, she was told that they lacked "hit potential". It was announced that the record label would be sold to Universal and its artists moved to other labels or released. Heap was one of the artists who was dropped from the label, leaving her without a record contract. iMegaphone had, however, been licensed from Almo Sounds to Aozora Records in Japan, who eventually re-released and re-promoted the album in January 2002, featuring "Blanket" and "Aeroplane" (a Frou Frou remix/remake of one of her B-sides, "Airplane" of the Shine single released in 1998). The album featured new packaging, all-new artwork, and a previously unavailable hidden track, entitled "Kidding", recorded live during her 1999 tour.

Copies of the original Almo Sounds release remain rare. A Brazilian label, Trama Records, currently claims to hold the license to the record and has started re-printing copies of the album in limited quantities. The album was released digitally on the US iTunes Music Store in early 2006. After achieving commercial success with her work with Guy Sigsworth as the duo Frou Frou and her second solo album, Speak for Yourself, Heap was able to secure the re-release of iMegaphone.

2001: Interim work[link]

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In the gap between the end of promotion for iMegaphone internationally and the re-promotion, Heap had also begun to think about her second solo album, and had started writing songs, both solo, as well as working with Guy Sigsworth; however, as she was without a record deal, the songs were shelved. During the time when she was unsigned, Heap appeared on two UK singles, "Meantime" (a track written by her former Acacia colleagues Guy Sigsworth and Alexander Nilere for the soundtrack to the independent British film, G:MT – Greenwich Mean Time) and "Blanket" (a collaboration with Urban Species). In 2000, Heap sang on the album You Had It Coming by Jeff Beck.

2002–2003: Frou Frou[link]

Heap had kept in contact with Guy Sigsworth (who had co-written and produced "Getting Scared" from iMegaphone) and this led to the pair of them establishing the collaborative project Frou Frou.

The initial concept for Frou Frou was Sigsworth's, and the project was to have been an album written and produced by her with each track featuring a different singer, songwriter, poet or rapper. Heap explains that Sigsworth invited her over to his studio to write lyrics to a four-bar motif he had, with one condition – that she include the word "love" somewhere. The first line she came up with was "lung of love, leaves me breathless", and the Details album track, "Flicks" was born. A week later, Sigsworth phoned Heap up again, and together they wrote and recorded the future single "Breathe In".

Throughout the process, Frou Frou work was an equal partnership, with Heap and Sigsworth making equal contributions to writing, arrangement, production and instrumental performance and Heap handling all of the vocals.

In August 2002, they released the Details album and singles "Breathe In", "It's Good To Be In Love", and "Must Be Dreaming" (although the latter two were not commercially available). The album was critically acclaimed, but did not enjoy the commercial success that they had been hoping for.

In late 2003, after an extensive promotional tour of the UK, Europe and the US, the duo were told that their record label, Island Records would not be picking up the option for a second album.

Heap and Sigsworth remain firm friends, and have worked together since the project, including their temporary re-formation in late 2003, when they covered the Bonnie Tyler classic, "Holding Out for a Hero", which was featured during the credits of the movie Shrek 2 after Jennifer Saunders version in the film. Frou Frou saw a resurgence in popularity in 2004, when their album track "Let Go" was featured in the film Garden State.

In a 2005 interview Heap said of frou frou "(it) was really like a kind of little holiday from my own work. Guy and I, we have always worked together, and then over the years, it became clear that we wanted to do a whole album together. It was very organic and spontaneous - just one of those wonderful things that happens. But there was never a mention of a second record from either of us, and not uncomfortably. We're just both kind of free spirits. I love to work with a lot of different people, but I was also just gagging to see what I could do on my own. But I'm sure in the future, Guy and I will get back together to do another record, or to record a few songs together."^[4]

[edit] 2004–2005: The O.C. and Speak for Yourself

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Performing at Coachella Valley Music and Arts Festival

In December 2003, Heap announced on her Web site that she was going to write and produce her second solo album, using her site as a blog to publicise progress.

Heap set herself a deadline of one year to make the album, booking a session to master the album one year ahead in December 2004. She re-mortgaged her flat to fund production costs, including renting a studio at Atomic Studios, London (previously inhabited by UK grime artist, Dizzee Rascal), and purchasing instruments.

At the end of 2004, with the album completed, Heap premiered two album tracks online, selling them prior to the album's release – "Just for Now" and "Goodnight and Go".

In April 2005, The O.C. featured the vocoded-vocal track, "Hide and Seek" in the closing scenes of their season two finale. The track was released immediately to digital download services, such as iTunes, in the US, where it charted. The track was released to iTunes UK on 5 July 2005 (the same day as the UK airing of the season finale) and entered the official UK download chart.

Heap made a decision to put out the album on her own in the UK, starting her own record company, titled Megaphonic Records. The album was titled Speak for Yourself.

Speak for Yourself was released in the UK on 18 July 2005 on CD and iTunes UK, where it entered the top 10 chart. The initial 10,000 physical copies pressed sold out, distributed through large and independent record stores and Heap's own online shop.

In August 2005, Heap announced that she had licensed Speak for Yourself to Sony BMG imprint RCA Victor for the album release in the United States, Canada, and Mexico. The album was released in November 2005 and debuted at #144 in the Billboard Top 200 album chart. In concert, Heap performed solo, controlling the sound through her Apple PowerBook laptop, as well as singing and playing the piano and array mbira.

She returned from the US, already having sold over 120,000 copies.

2006–2007: The year of the 'White Rabbit'[link]

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Heap also announced, on her return to the UK, that she had signed a deal for the album to be released internationally, as well as re-promoted in the UK, with a new imprint of Sony BMG, White Rabbit, run by former Sony BMG UK A&R vice president Nick Raphael. The deal meant that the album could have the promotional backing provided by a respected major label, whilst Heap retained sole control and the team she established for Megaphonic Records.

Speak for Yourself was re-released on the label on 24 April 2006, ahead of a full promotional push on 15 May, a week after the second single, "Goodnight and Go", was commercially released in the UK.

In August 2006, Heap performed a set at the V Festival, where it was announced that "Headlock" was to be the third single to be lifted from the album, and released on 16 October 2006 in the UK.

In late September and early October, Heap embarked on a tour of the UK, holding a competition on MySpace for different support acts for each venue, before touring throughout Canada and the US in November and December. This was her first tour of North America that included a band, incorporating upright bass, percussion, and support acts Kid Beyond and Levi Weaver on beatbox and guitar, respectively. In December 2006, Heap was featured on the front page of The Green Room magazine.

On 7 December 2006, Heap received two Grammy nominations for the 49th Annual Grammy Awards, one for Best New Artist and the other for Best Song Written For Motion Picture, Television Or Other Visual Media for "Can't Take It In".

[edit] 2007–2009: Ellipse

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Throughout the creation of her album Ellipse, Heap posted vlogs, or VBlogs as she called them, through YouTube.^[9] She used these to comment on the album as well as update on its release. The album's release was pushed back multiple times. These included Heap being asked to perform at the annual event PopTech in October 2008. During the event, she premiered one of her album's songs, "Wait it Out".

Heap announced on her Twitter page that Ellipse's first single would be "First Train Home". On 17 August 2009 Heap made the entire album Ellipse available for live streaming via her webpage.

Ellipse was released in the United Kingdom on 24 August, and in the United States on 25 August.

[edit] 2011–present: Heapsongs

On 14 March 2011 Heap officially started work on a new record as fans sent in nearly 900 "sound seeds", or samples of everyday sounds such as a "dishwasher door", a "bicycle" or a "burning match". Heap has stated that the concept for this record will be to record one track over a two week period every 2 months. Each song and video will be released immediately.^[10] According to her website, the album will be completed in roughly 1.5 years.

The first song, initially entitled #heapsong1 and later retitled "Lifeline", premiered worldwide on 28 March 2011 via Ustream^[11] along with a live remix by Tim Exile. "Lifeline" was released on 30 March 2011 as a digital download from Imogen's website and via iTunes, Amazon and other digital retailers. Released alongside this was a 12 page 3DiCD package (a 3D virtual CD) including crowd sourced (and paid for) images, the instrumental version of the song, the "seeds and solos only" version and "heap speaks seeds and solos" - an-18 minute commentary by Heap on how the sounds and solos were used in "Lifeline".

On 6 May 2011 Heap tweeted that she and Deadmau5 were working on a collaboration. The song is tentatively titled "Telemisscommunications".^[12][13][14]

List of heapsongs so far[link]

1. heapsong1 is entitled "Lifeline" and was released on 30 March 2011.
2. heapsong2 is entitled "Propeller Seeds" and was released on 5 July 2011.
3. heapsong3 is entitled "Neglected Space" and was released on 17 October 2011.
4. heapsong4 is entitled "Minds Without Fear" and was released on 21 October 2011. It was the first song to be featured on 2011 Indian show, The Dewarists.
5. heapsong5 is entitled "Xizi She Knows" and was released on 24 January 2012. It is the fifth song off the forthcoming album overall.^[15]
6. heapsong6 is entitled "Me The Machine" and was performed live by Heap on 22 April 2012. The song premiered along with the film, Love the Earth, as part of Earth Day 2012.

Recordings for television and film[link]

Heap has recorded several songs for films, including a cover of the Classics IV hit "Spooky" for the soundtrack to the Reese Witherspoon film Just Like Heaven. Her song "Hide and Seek" was featured in The Last Kiss, starring Zach Braff (who used her former band Frou Frou's "Let Go" in his 2004 film Garden State), and was also used in a 2007 episode of Saturday Night Live, hosted by Shia LaBeouf. "The Moment I Said It" was also used in the episode "Seven Seconds" of the CBS crime drama Criminal Minds.

In 2004, while recording her second solo album, she was commissioned to record a cover of a short nursery rhyme for the HBO television series, Six Feet Under, entitled "I'm A Lonely Little Petunia (In An Onion Patch)".

In late 2005, Heap was asked to write a track for the soundtrack of The Chronicles of Narnia: The Lion, the Witch and the Wardrobe entitled "Can't Take It In", when a track that fellow Brit singer Dido submitted was deemed unfitting. Heap's track is played at the end of the film in an orchestral version produced by Heap and Harry Gregson Williams, who scored the movie. In addition, she composed a track for the film The Chronicles of Narnia: Prince Caspian, but it was deemed to be too dark in tone for the film. Instead, it was included in her album Ellipse as "2-1". 2-1 has also featured in CSI Miami (Season 8 Episode 9), as well as promotional trailers for the film The Lovely Bones.

In March 2006, Heap completed a track about locusts, entitled "Glittering Cloud", for a CD of music about the plagues of Egypt entitled Plague Songs, accompanying The Margate Exodus project, for musical director Brian Eno.

Heap recorded an a cappella version of the Leonard Cohen track "Hallelujah", for the season three finale of The O.C., and her "Not Now But Soon" was included on the original soundtrack for the NBC show, Heroes.

Song usage[link]

Imogen Heap and Frou Frou songs have been featured in various TV shows, movies, advertisements, and marching band productions, notably including CSI, The OC, SNL, Garden State, and So You Think You Can Dance.

Also notable is the sampling of Heap's song "Hide & Seek" in Jason DeRulo's single "Whatcha Say",^[16] which peaked at #1 on the Billboard Hot 100.

The Bluecoats Drum and Bugle Corps used Heap's song "Aha" in their 2010 production titled, "Metropolis: The Future is Now"

The Pack rapper Young L's 2001 mixtape 'As I Float: The Great John Nash' makes extensive use of samples of Imogen Heap songs on nearly every track.

Collaborations[link]

Heap has collaborated as a guest vocalist, co-writer, remixer or producer with many various artists throughout her career. Among them co-writing and producing By The Time for Mika and Now or Never for Josh Groban. The diverse range of other musicians Heap has worked with include IAMX, Jeff Beck, Temposhark, LHB, J. Peter Schwalm, Way Out West, Jon Bon Jovi, Mich Gerber, Sean Lennon, Urban Species, Matt Willis, Jon Hopkins, MIKA, Acacia, Britney Spears, Nik Kershaw, Blue October, Joshua Radin, Nitin Sawhney, I Fight Dragons and Deadmau5.^[12][13][14]

Use of technology[link]

Heap is an outspoken advocate of using new technology to interact and collaborate with her fans. In August 2009 she used Vokle, an online auditorium, to take questions from listeners over video chat.^[17]

Imogen also teamed up with Vokle to hold open cello auditions for her North American tour. She provided sheet music for “Aha” on her website and encouraged local fans to learn the part and audition live via Vokle. Imogen would then pick the cellist to accompany her for that particular city - sometimes with the help of viewers and her puppet Lion, Harold.

In 2010 Imogen opened her online auditions to singers and choirs and invited them to audition via submitted YouTube videos to accompany her on stage as she performed the song "Earth" from Ellipse. The winner of each local show was also invited to do a 15 minute gig of their own. In the studio, the official album recording of "Earth" was made up entirely of numerous tracks of vocals.

July 2011 saw Imogen unveil a pair of in development, high-tech musical gloves at the TEDGlobal conference in Edinburgh, Scotland.^[18] Inspired by the VAMP ^[19] system developed by Elly Jessop at MIT’s media lab, Imogen set out to develop the musical gloves in collaboration with Thomas Mitchell, a lecturer in music systems at the University of the West of England, Bristol. The gloves combine sensors developed by 5DT,^[20] x-io Technologies ^[21] with Shure microphones.

Using nothing but hand gestures, Imogen is currently able to amplify/record/loop acoustic instruments and her voice, play virtual instruments and manipulate these sounds live. Imogen has for many years been working toward a less constricting live set up which enables her to be mobile while performing live multiple musical production tasks, songs and improvising spontaneously without the need 'to go back to basecamp'. The gloves enable the audience to instinctively understand and connect with the process of what's going on on stage - the 'hidden' 50% of her performance. This is part of a larger audio/visual performance project Imogen has been working towards for a couple of years with the aim of touring in 2013.

Charity[link]

In 2008 she participated in a music album called Songs for Tibet: The Art of Peace, which is an initiative to support Tibet, Dalai Lama Tenzin Gyatso and to underline the human rights situation in Tibet. The album was issued on 5 August via iTunes and on 19 August in music stores around the world.^[22] On 12 October 2008, Heap also participated in "Run 10k: Cancer Research UK," placing fifth of the women in the actual run and raising over £1000 for the cause with the help of her fans.

In 2008 Imogen was asked to perform at POP!Tech in Camden, Maine (US). There she performed selections from her then forthcoming album Ellipse. After her set and an encouraging plea for another performance later in the conference by the audience and organizers, Imogen agreed. Having nothing else prepared though, she decided to improvise a song on the spot with parameters (tempo, key) suggested by the audience. After the show, Imogen was asked by a Poptech attendee if she would give the newly created piece of music to his charity. A ‘lightbulb’ moment occurred in Heap’s head and she saw the potential in doing these improvised pieces for local charities at each show during the tour she would soon begin.

The first of these songs materialized at Imogen’s show at Shepherd’s Bush Empire, in London on the 19th of February 2010. Using the same parameters and audience participation from POP!Tech, Imogen improvised a song titled, "The Shepherdess". After the show, Imogen made the song available worldwide as a digital download on her website asking for donations per download. All proceeds went to the Great Ormond Street Hospital where Imogen was diagnosed with osteomyelitis and underwent life-saving surgery as a little girl. Loving the concept, Imogen rolled this out for her North American Tour, donating all the proceeds for each song to a local charity from that city.

In 2011 Imogen was set to play a benefit concert in New Zealand's Christchurch city to help rebuild the Unlimited Paenga Tawhiti High School, following a severe 6.3 aftershock in February originating from the 7.1 earthquake that struck the Canterbury region in September 2010. The concert was held at the Burnside High Aurora Centre, also featuring performances from Roseanna Gamlen-Greene, and The Harbour Union including The Eastern, Lindon Puffin, Delaney Davidson and The Unfaithful Ways.^[23]

Live 4 X[link]

In 2010, Imogen Heap partnered with Thomas Ermacora of Bubbletank^[24] to organize a series of online charitable events called Live 4 X.

The initial event was inspired by the 2010 Pakistan floods. Triggered by monsoon rains, the floods left approximately one-fifth of the country of Pakistan underwater, affecting over 14 million people and damaging or destroying over 900,000 homes. Teaming up with Richard Branson’s Virgin Unite and Vokle.com, Heap and Ermacorda create an online webcast/fundraiser to raise awareness and money for the flood stricken. Hosted by comedian, creative, and internet personality Ze Frank, the webcast included a series of conversations with Cameron Sinclair of Architecture for Humanity, Gary Slutkin, and Anders Wilhelmson, (and later Richard Branson and Mary Robinson) with live performances by musicians Ben Folds, Amanda Palmer, Kate Havnevik, KT Tunstall, Josh Groban, Kaki King, Zoe Keating and Mark Isham.

The premise of Live 4 X thus established, Heap has since continued to refine the model, organize, host, and perform a number of charitable, streaming-live, concert events. By integrating live entertainment with educated discussion and technology, Live 4 X became an effective charitable outreach tool.

Following the devastating Great East Japan Earthquake and tsunami of 2011, Heap told Washington Times Communities journalist and recording artist Jennifer Grassman, that she intended to continue organizing Live 4 X events to benefit various charitable causes.^[25]

Catalog of Live 4 X events to date:

• August 31, 2010 – Live 4 Pakistan raised funds for flood relief and recovery in that region. Musicians included Ben Folds, Amanda Palmer, Kate Havnevik, KT Tunstall, Josh Groban, and Zoe Keating. In an ironic turn of events, Heap, was kept from appearing on Live 4 Pakistan due to Hurricane Earl which at the time was progressing along the US eastern seaboard. Heap, stranded and unable to get an internet connection, later posted a video message as well as a performance of her song “Wait It Out” from Ellipse.^[26]
• February 3, 2011 – Live 4 Capetown^[27]
• April 11, 2011 – Live 4 Sendai raised funds for Japanese tsunami recovery following the disastrous Great East Japan Earthquake of 2011. The event was also used to solicit rebuilding design ideas on behalf of Architecture for Humanity. Performers included Amanda Palmer, Ben Folds, KT Tunstall and Jamie Cullum and hosted by Ze Frank.^[28]

Film[link]

After touring for nearly two years straight for her album Speak for Yourself Imogen continued her travels, this time with only a laptop and video camera on hand as she began her writing trip for her next album. Nine weeks later she returned to the UK with the beginnings of the award-winning Ellipse and footage (as requested by a fan to film the making of the album) from its quiet beginning. Back in Essex, Imogen sought the talents of long-time friend and film maker Justine Pearsall to continue documenting the creation of the album. The film documents every moment of joy, excitement, frustration, and even the renovation of the Imogen’s childhood home including turning her old playroom into her new home studio. Everything In-Between: The Story of Ellipse was released in November 2010.

On November 5, 2010 at the Royal Albert Hall, Heap conducted an orchestra (including friends and family) as they performed an original composition by Imogen herself orchestrated by Andrew Skeet. Heap also worked with London Contemporary Voices at this time, a scratch choir formed for this concert, which continues as a new choir in its own right. It was the score to the concept film Love The Earth - in creative partnership and co-production with Thomas Ermacora again for another Bubbletank production - in which fans were invited to submit video footage highlighting all of the breathtaking qualities of nature to be selected and edited into a film. This performance was broadcasted live worldwide.

In March for the Birds' Eye Festival at the Birds Eye View Film Festival at the Southbank Centre, Imogen composed in collaboration with Andrew Skeet an a cappella choral score to the first ever surrealist film ‘The Seashell and the Clergyman’ (Germaine Dulac, 1927) with the Holst Singers, a programme repeated at the Reverb Festival at the Roundhouse in February 2012 and in the Sage, Gateshead.

Heap also performed in the Film and Music Arena at Latitude Festival in 2011.

Discography[link]

• 1998: iMegaphone
• 2005: Speak for Yourself
• 2009: Ellipse
• 2012: Heapsongs

Frou Frou[link]

• Details (2002 • Island Records/MCA/Universal) (UK: #128)
• "Breathe In" (single) (2002 • Island Records/Universal) (UK: #44)
• "It's Good to be in Love" (single) (2002 • Island Records/Universal)
• "Must be Dreaming" (single) (2002 • Island Records/Universal)
• "Let Go" (Promotional single)

Guest appearances[link]

• Hate EP by Acacia (1996 • Radar Records)
• Sway EP by Acacia (1996 • Radar Records)
• Maddening Shroud EP by Acacia (1997 • WEA)
• Cradle by Acacia – all tracks except "Wire" (1997 • WEA)
• Blanket by Urban Species – "Blanket" and "Predictably Unpredictable" (1998)
• Amor Fati by Mich Gerber – "Embers of Love", "Sirens Call (Qishm)" and "Mare" (2000)
• ¡Viva Nueva! by Rustic Overtones – "Valentine's Day Massacre" (2001 • Tommy Boy Records)
• You Had It Coming by Jeff Beck – "Dirty Mind" and "Rollin' and Tumblin'" (2001 • Sony Music)
• Tell 'Em Who We Are by LHB^{[disambiguation needed ]} – "Coming Up For Air" (2003 • Telstar TV)
• Contact Note by Jon Hopkins – "Second Sense" (2004 • Just Music)
• It's Better To Have Loved EP by Temposhark – "Not That Big (Metronomy Remix)" (2005 • Paper & Glue)
• Foiled by Blue October – "Congratulations" (2006 • Universal Records)
• Musikain by J. Peter Schwalm – "P.I.N." (2006 • Musikain Records)
• The Invisible Line by Temposhark – "Not That Big" (2007 • Paper & Glue)
• London Undersound by Nitin Sawhney – "Bring It Home" (2008 • Cooking Vinyl)
• Kingdom of Welcome Addiction by IAMX – "My Secret Friend" (2009 • Metropolis Records)
• The Boy Who Knew Too Much by Mika – "By The Time" (2009 • Casablanca Records)
• Live at Ronnie Scott's - Jeff Beck - (2008 • Eagle Records) - (2009 • DVD)

Compilation appearances[link]

• New Dawn (Class of '94) – "Missing You" – (1994)
• I Still Know What You Did Last Summer soundtrack – "Getting Scared" – (1998 • Warner Music Group)
• Virtual Sexuality soundtrack – "Come Here Boy" – (1998 • EMI)
• G:MT – Greenwich Mean Time soundtrack – "Mean Time" (with GMT) – (1999 • Island Records/Universal)
• Serve Chilled – "Blanket" (with Urban Species) – (1999 • Hed Kandi)
• Women Talking Dirty soundtrack – "Getting Scared" – (2001 • Polygram International)
• American Psycho 2: All American Girl soundtrack – "Angry Angel" – (2002 • Rondor Music (London) Ltd)
• Garden State – "Let Go" – (2004 • Epic Records/Sony BMG)
• Nearing Grace Soundtrack – "The Walk" – (2005 • Sony BMG)
• Music from the OC: Mix 4 – "Goodnight and Go" – (2005 • Warner Music Group)
• Six Feet Under, Vol. 2: Everything Ends – "I'm A Lonely Little Petunia (In An Onion Patch)" – (2005 • Astralwerks/EMI)
• Just Like Heaven Soundtrack – "Spooky" – (2005 • Sony BMG)
• Music from the OC: Mix 5 – "Hide and Seek" – (2005 • Warner Music Group)
• The Chronicles of Narnia: The Lion, the Witch and the Wardrobe – "Can't Take It In" – (2005 • Disney Records/EMI)
• The Last Kiss soundtrack – "Hide and Seek" – (2005 • Lakeshore Pictures)
• Plague Songs – "Glittering Clouds (Locusts)" – (2006 • 4AD)
• So You Think You Can Dance (US) – "Hide and Seek" – (2006)
• So You Think You Can Dance (US) – "Let Go" – (2006)
• The Black Donnellys – "The Moment I Said It" – (2007)
• In Search of Sunrise 6 – "Hide and Seek" (Tiëstos' In Search Of Sunrise Remix) – (2007)
• The Holiday – "Just For Now" and "Let Go" (Under 'Frou Frou') – (2006)
• So You Think You Can Dance – "The Moment I Said It" – (2007)
• Heroes Original Soundtrack – "Not Now, But Soon" – (2008 • NBC Records)
• Criminal Minds – "The Moment I Said It" – (2008)
• Songs For Tibet – The Art of Peace' – "Hide & Seek 2" – (2008)
• Greys Anatomy – "Bad Body Double" – (2009)
• So You Think You Can Dance (US) - "Aha!" - (2009)
• The Vampire Diaries - "Wait it Out" - (2009)
• Melrose Place - "Aha!" - (2009)
• Powder Blue - "Blanket" - (2009)
• Chuck – "Wait it Out" – (2010)
• CSI: NY - "Wait it Out" - (2010)
• Made in Chelsea - "Goodnight and Go" (2011)

References[link]

External links[link]

Wikimedia Commons has media related to: Imogen Heap

• Official website
• London Contemporary Voices Choir