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An ellipse obtained as the intersection

of a cone with an inclined plane.

The rings of Saturn are circular, but

when seen partially edge on, as in this

image, they appear to be ellipses. In

addition, the planet itself is an

ellipsoid, flatter at the poles than the

equator. Picture by ESO

EllipseFrom Wikipedia, the free encyclopedia

In mathematics, an ellipse is a curve on a plane surrounding two focalpoints such that a straight line drawn from one of the focal points to anypoint on the curve and then back to the other focal point has the samelength for every point on the curve. As such, it is a generalization of acircle which is a special type of an ellipse that has both focal points at thesame location. The shape of an ellipse (how 'elongated' it is) isrepresented by its eccentricity which for an ellipse can be any numberfrom 0 (the limiting case of a circle) to arbitrarily close to but less than 1.

Ellipses are the closed type of conic section: a plane curve that resultsfrom the intersection of a cone by a plane. (See figure to the right.)Ellipses have many similarities with the other two forms of conic sections:the parabolas and the hyperbolas, both of which are open andunbounded.

Analytically, an ellipse can also be defined as the set of points such thatthe ratio of the distance of each point on the curve from a given point(called a focus or focal point) to the distance from that same point on thecurve to a given line (called the directrix) is a constant, called theeccentricity of the ellipse.

Ellipses are common in physics, astronomy and engineering. Forexample, the orbits of planets are ellipses with the Sun at one of the focalpoints. The same is true for moons orbiting planets and all other systemshaving two astronomical bodies. The shape of planets and stars are oftenwell described by ellipsoids. Ellipses also arise as images of a circleunder parallel projection and the bounded cases of perspectiveprojection, which are simply intersections of the projective cone with theplane of projection. It is also the simplest Lissajous figure, formed whenthe horizontal and vertical motions are sinusoids with the same frequency.A similar effect leads to elliptical polarization of light in optics.

The name ἔλλειψις was given by Apollonius of Perga in his Conics,emphasizing the connection of the curve with "application of areas".

Contents

1 Elements of an ellipse

2 Drawing ellipses2.1 Pins-and-string method

2.2 Trammel method

2.3 Parallelogram method

2.4 Approximations to ellipses

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2.4 Approximations to ellipses

3 Mathematical definitions and properties

3.1 In Euclidean geometry

3.1.1 Definition

3.1.2 Equations

3.1.3 Focus

3.1.4 Eccentricity3.1.5 Directrix

3.1.6 Circular directrix

3.1.7 Ellipse as hypotrochoid

3.1.8 Area

3.1.9 Circumference

3.1.10 Chords

3.1.10.1 Latus rectum

3.1.10.2 Curvature

3.2 In projective geometry

3.3 In analytic geometry

3.3.1 General ellipse

3.3.2 Canonical form3.4 In trigonometry

3.4.1 General parametric form3.4.2 Parametric form in canonical position

3.4.3 Polar form relative to center3.4.4 Polar form relative to focus3.4.5 General polar form

3.4.6 Angular eccentricity3.5 Degrees of freedom

4 Ellipses in physics4.1 Elliptical reflectors and acoustics

4.2 Planetary orbits4.3 Harmonic oscillators

4.4 Phase visualization4.5 Elliptical gears

4.6 Optics5 Ellipses in statistics and finance6 Ellipses in computer graphics

6.1 Drawing with Bézier spline paths7 Line segment as a type of degenerate ellipse

8 Ellipses in optimization theory9 See also

10 References11 Notes

12 External links

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The ellipse and some of its mathematical

properties.

Elements of an ellipse

See also: features of conic sections

Ellipses have two mutually perpendicular axes about which theellipse is symmetric These axes intersect at the center of theellipse due to this symmetry. The larger of these two axes,which corresponds to the largest distance between antipodalpoints on the ellipse, is called the major axis or transversediameter. (On the figure to the right it is represented by the linebetween the point labeled −a and the point labeled a.) Thesmaller of these two axes, and the smallest distance across the

ellipse, is called the minor axis or conjugate diameter.[1] (Onthe figure to the right it is represented by the line between thepoint labeled −b to the point labeled b.)

The semi-major axis (denoted by a in the figure) and thesemi-minor axis (denoted by b in the figure) are one half ofthe 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 four points where these axes cross the ellipse are the vertices and are marked as a, −a, b, and −b. In additionto being at the largest and smallest distance from the center, these points are where the curvature of the ellipse is

maximum and minimum.[10]

The two foci (plural of focus and the term focal points is also used) of an ellipse are two special points F1 and F2

on the ellipse's major axis that are equidistant from the center point. The sum of the distances from any point P onthe ellipse to those two foci is constant and equal to the major axis (PF1 + PF2 = 2a). (On the figure to the right

this corresponds to the sum of the two green lines equaling the length of the major axis that goes from −a to a.)

The distance to the focal point from the center of the ellipse is sometimes called the linear eccentricity, f, of theellipse. Here it is denoted by f, but it is often denoted by c. Due to the Pythagorean theorem and the definition of

the ellipse explained in the previous paragraph: f2 = a2 −b2.

A second equivalent method of constructing an ellipse using a directrix is shown on the plot as the three blue lines.(See the Directrix section of this article for more information about this method). The dashed blue line is thedirectrix of the ellipse shown.

The eccentricity of an ellipse, usually denoted by ε or e, is the ratio of the distance between the two foci, to thelength of the major axis or e = 2f/2a = f/a. For an ellipse the eccentricity is between 0 and 1 (0 < e < 1). When theeccentricity 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 finitedistance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away. Theeccentricity is also equal to the ratio of the distance (such as the (blue) line PF2) from any particular point on an

ellipse to one of the foci to the perpendicular distance to the directrix from the same point (line PD), e = PF2/PD.

Drawing ellipses

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

Trammel of Archimedes (ellipsograph) animation

Pins-and-string method

The characterization of an ellipse as the locus of points sothat sum of the distances to the foci is constant leads to amethod of drawing one using two drawing pins, a length of

string, and a pencil.[11] In this method, pins are pushed intothe 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 penis used to pull the loop taut so as to form a triangle. The tipof the pen will then trace an ellipse if it is moved whilekeeping the string taut. Using two pegs and a rope, thisprocedure is traditionally used by gardeners to outline anelliptical flower bed; thus it is called the gardener's

ellipse.[12]

Trammel method

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 (M) and minor (N) axes of theellipse. Mark three points A, B, C on the ruler. A->C being the length of the semi-major axis and B->C the

length of the semi-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 mechanicaldevice that implements this principle. The ruler is replacedby a rod with a pencil holder (point C) at one end, and twoadjustable side pins (points A and B) that slide into two

perpendicular slots cut into a metal plate.[13] Themechanism can be used with a router to cut ellipses fromboard material. The mechanism is also used in a toy calledthe "nothing grinder".

Parallelogram method

In the parallelogram method, an ellipse is constructed pointby point using equally spaced points on two horizontal linesand equally spaced points on two vertical lines. It is basedon Steiner's theorem on the generation of conic sections.Similar methods exist for the parabola and hyperbola.

Approximations to ellipses

An ellipse of low eccentricity can be represented reasonably accurately by a circle with its centre offset. To drawthe orbit with a pair of compasses the centre of the circle should be offset from the focus by an amount equal to theeccentricity multiplied by the radius.

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Ellipse construction applying the parallelogram

method

Mathematical definitions and properties

In Euclidean geometry

Definition

In Euclidean geometry, the ellipse is usually defined as the bounded case of a conic section, or as the set of pointssuch that the sum of the distances to two fixed points (the foci) is constant. The ellipse can also be defined as the setof points such that the distance from any point in that set to a given point in the plane (a focus) is a constant positivefraction less than 1 (the eccentricity) of the perpendicular distance of the point in the set to a given line (called thedirectrix). Yet another equivalent definition of the ellipse is that it is the set of points that are equidistant from onepoint in the plane (a focus) and a particular circle, the directrix circle (whose center is the other focus).

The equivalence of these definitions can be proved using the Dandelin spheres.

Equations

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The equation of an ellipse whose major and minor axes coincide with the Cartesian axes is

This means any noncircular ellipse is a squashed circle. If we draw an ellipse twice as long as it is wide, and drawthe circle centered at the ellipse's center with diameter equal to the ellipse's longer axis, then on any line parallel tothe shorter axis the length within the circle is twice the length within the ellipse. So the area enclosed by an ellipse iseasy to calculate—it's the lengths of elliptic arcs that are hard.

Focus

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

Eccentricity

The eccentricity of the ellipse (commonly denoted as either e or ) is

(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

Other formulas for the eccentricity of an ellipse are listed in the article on eccentricity of conic sections. Formulasfor the eccentricity of an ellipse that is expressed in the more general quadratic form are described in the articlededicated to conic sections.

Directrix

Each focus F of the ellipse is associated with a line parallel to the minor axis called a directrix. Refer to theillustration on the right, in which the ellipse is centered at the origin. The distance from any point P on the ellipse tothe 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 usingthe Dandelin spheres) can be taken as another definition of the ellipse.Besides the well-known ratio e = f/a, where f is the distance from the center to the focus and a is the distance fromthe center to a vertex (most sharply curved point of the ellipse), it is also true that e = a/d, where d is the distancefrom the center to the directrix.

Circular directrix

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An ellipse (in red) as a special case of

the hypotrochoid with R = 2r.

The ellipse can also be defined as the set of points that are equidistant from one focus and a circle, the directrixcircle, that is centered on the other focus. The radius of the directrix circle equals the ellipse's major axis, so thefocus and the entire ellipse are inside the directrix circle.

Ellipse as hypotrochoid

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

Area

The area enclosed by an ellipse is:

where a and b are one-half of the ellipse's major and minoraxes respectively.

An ellipse defined implicitly by

has area .

The area formula πab is easy to understand: start with a

circle of radius b (so its area is πb2) and stretch it by afactor a/b to make an ellipse. This increases the area by the same factor:

πb2(a/b) = πab.

For the ellipse in standard form, , and hence

, with horizontal intercepts at ± a, the area

can be computed as twice the integral of the positive square

root:

The second integral is the area of a circle of radius , i.e., ; thus we have:

The area formula can also be proven in terms of polar coordinates using the coordinate transformation

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Any point inside the ellipse with x-intercept a and y-intercept b can be defined in terms of r and , where and .

To define the area differential in such coordinates we use the Jacobian matrix of the coordinate transformation times:

We now integrate over the ellipse to find the area:

Circumference

The circumference of an ellipse is:

where again a is the length of the semi-major axis and e is the eccentricity and where the function is the complete

elliptic integral of the second kind. This may be evaluated directly using the Carlson symmetric form.[14] This gives a

succinct and rapidly converging method for evaluating the circumference.[15]

The exact infinite series is:

or

where is the double factorial. Unfortunately, this series converges rather slowly; however, by expanding in

terms of , Ivory[16] and Bessel[17] derived an expression which converges much more

rapidly,

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A good approximation is Ramanujan's:

and a better approximation is

For the special case where the minor axis is half the major axis, these become:

or, as an estimate of the better approximation,

More generally, the arc length of a portion of the circumference, as a function of the angle subtended, is given by anincomplete elliptic integral.

See also: Meridian arc#Meridian distance on the ellipsoid

The inverse function, the angle subtended as a function of the arc length, is given by the elliptic

functions.[citation needed]

Chords

The midpoints of a set of parallel chords of an ellipse are collinear.[18]:p.147

Latus rectum

The chords of an ellipse which are perpendicular to the major axis and pass through one of its foci are called the

latera recta of the ellipse. The length of each latus rectum is 2b2

a .

Curvature

The curvature is .

In projective geometry

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In projective geometry, an ellipse can be defined as the set of all points of intersection between corresponding linesof two pencils of lines which are related by a projective map (see Steiner's theorem). By projective duality, anellipse can be defined also as the envelope of all lines that connect corresponding points of two lines which arerelated by a projective map.

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

An ellipse is also the result of projecting a circle, sphere, or ellipse in three dimensions onto a plane, by parallellines. It is also the result of conical (perspective) projection of any of those geometric objects from a point O onto aplane P, provided that the plane Q that goes through O and is parallel to P does not cut the object. The image of anellipse 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 of the Cartesian plane that, in non-

degenerate cases, satisfy the implicit equation[19][20]

provided

To distinguish the degenerate cases from the non-degenerate case, let ∆ be the determinant

that is,

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.[21]:p.63

Canonical form

Let . Through change of coordinates (a rotation of axes and a translation) the general ellipse can bedescribed by the canonical implicit equation

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Here are the point coordinates in the canonical system, whose origin is the center of the ellipse,

whose -axis is the unit vector coinciding with the major axis, and whose -axis is the perpendicular

vector coinciding with the minor axis. That is, and

.

In this system, the center is the origin and the foci are and .

Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters.Translation of an ellipse centered at is expressed as

Moreover, any canonical ellipse can be obtained by scaling the unit circle of , defined by the equation

by factors a and b along the two axes.

For an ellipse in canonical form, we have

The distances from a point on the ellipse to the left and right foci are and ,

respectively.

In trigonometry

General parametric form

An ellipse in general position can be expressed parametrically as the path of a point , where

as the parameter t varies from 0 to 2π. Here is the center of the ellipse, and is the angle between the

-axis and the major axis of the ellipse.

Parametric form in canonical position

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

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Parametric equation for the ellipse

(red) in canonical position. The

eccentric anomaly t is the angle of

the blue line with the X-axis.

Polar coordinates centered at the

center.

Note that the parameter t (called the eccentric anomaly in astronomy) isnot the angle of with the X-axis.

For a given point on an ellipse, formulae connecting the tangential angle ,

the polar angle from the ellipse center , and the parametric angle t[22]

are:[23][24][25][26]

Polar form relative to center

In polar coordinates, with the origin at the center of the ellipse and with theangular coordinate measured from the major axis, the ellipse's equation is

REFLEXIVE PROPERTY;

The ray of light emitting from one focus after reflection from inner surface ofellipse passes thorough the other focus

.

Polar form relative to focus

If instead we use polar coordinates with the origin at one focus, with the angular coordinate still measuredfrom the major axis, the ellipse's equation is

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Polar coordinates centered at

focus.

Semi-latus rectum.

where the sign in the denominator is negative if the reference direction points towards the center (asillustrated 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 , the polar form is

The angle in these formulas is called the true anomaly of the point. Thenumerator of these formulas is the semi-latus rectum of the

ellipse, usually denoted . It is the distance from a focus of the ellipse to theellipse itself, measured along a line perpendicular to the major axis.

General polar form

The following equation on the polar coordinates (r, θ) describes a generalellipse with semidiameters a and b, centered at a point (r0, θ0), with the a

axis rotated by φ relative to the polar axis:[citation needed]

where

Angular eccentricity

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

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), whileparabolae have four. Said another way, the set of all ellipses in the plane, with any natural metric (such as the

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Hausdorff distance) is a five-dimensional manifold. These degrees can be identified with, for example, thecoefficients A,B,C,D,E of the implicit equation, or with the coefficients Xc, Yc, φ, a, b of the general parametric

form.

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 thatdisturbance, 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 arereflected to the second focus. Since no other smooth curve has such a property, it can be used as an alternativedefinition of an ellipse. (In the special case of a circle with a source at its center all light would be reflected back tothe center.) If the ellipse is rotated along its major axis to produce an ellipsoidal mirror (specifically, a prolatespheroid), 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 insome document scanners.

Sound waves are reflected in a similar way, so in a large elliptical room a person standing at one focus can hear aperson standing at the other focus remarkably well. The effect is even more evident under a vaulted roof shaped asa section of a prolate spheroid. Such a room is called a whisper chamber. The same effect can be demonstratedwith 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 haveused this property for eavesdropping on political matters); the Mormon Tabernacle at Temple Square in Salt LakeCity, Utah; at an exhibit on sound at the Museum of Science and Industry in Chicago; in front of the University ofIllinois at Urbana-Champaign Foellinger Auditorium; and also at a side chamber of the Palace of Charles V, in theAlhambra.

Planetary orbits

Main article: Elliptic orbit

In the 17th century, Johannes Kepler discovered that the orbits along which the planets travel around the Sun areellipses with the Sun at one focus, in his first law of planetary motion. Later, Isaac Newton explained this as acorollary 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 totalenergy 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 thereference 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 inverselyproportional to the square of the distance. Thus, in principle, the motion of two oppositely charged particles inempty space would also be an ellipse. (However, this conclusion ignores losses due to electromagnetic radiationand quantum effects which become significant when the particles are moving at high speed.)

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For elliptical orbits, useful relations involving the eccentricity are:

where

is the radius at apoapsis (the farthest distance) is the radius at periapsis (the closest distance)

is the length of the semi-major axis

Also, in terms of and , the semi-major axis is their arithmetic mean, the semi-minor axis is their geometricmean, and the semi-latus rectum is their harmonic mean. In other words,

.

Harmonic oscillators

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse. Such is the case, forinstance, of a long pendulum that is free to move in two dimensions; of a mass attached to a fixed point by aperfectly elastic spring; or of any object that moves under influence of an attractive force that is directly proportionalto its distance from a fixed attractor. Unlike Keplerian orbits, however, these "harmonic orbits" have the center ofattraction 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 andhorizontal inputs of an oscilloscope. If the display is an ellipse, rather than a straight line, the two signals are out ofphase.

Elliptical gears

Two non-circular gears with the same elliptical outline, each pivoting around one focus and positioned at the properangle, will turn smoothly while maintaining contact at all times. Alternatively, they can be connected by a link chainor timing belt, or in the case of a bicycle the main chainring may be elliptical, or an ovoid similar to an ellipse in

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form. Such elliptical gears may be used in mechanical equipment to produce variable angular speed or torque froma constant rotation of the driving axle, or in the case of a bicycle to allow a varying crank rotation speed withinversely varying mechanical advantage.

Elliptical bicycle gears make it easier for the chain to slide off the cog when changing gears.[27]

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.[28]

Optics

In a material that is optically anisotropic (birefringent), the refractive index depends on the direction of thelight. The dependency can be described by an index ellipsoid. (If the material is optically isotropic, this

ellipsoid is a sphere.)In lamp-pumped solid-state lasers, elliptical cylinder-shaped reflectors have been used to direct light from thepump lamp (coaxial with one ellipse focal axis) to the active medium rod (coaxial with the second focal

axis).[29]

In laser-plasma produced EUV light sources used in microchip lithography, EUV light is generated byplasma positioned in the primary focus of an ellipsoid mirror and is collected in the secondary focus at the

input of the lithography machine.[30]

Ellipses in statistics and finance

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

return.[31][32]

Ellipses in computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the MacIntoshQuickDraw API, and Direct2D on Windows. Jack Bresenham at IBM is most famous for the invention of 2Ddrawing 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.[33] Another efficient

generalization to draw ellipses was invented in 1984 by Jerry Van Aken.[34]

In 1970 Danny Cohen presented at the "Computer Graphics 1970" conference in England a linear algorithm fordrawing ellipses and circles. In 1971, L. B. Smith published similar algorithms for all conic sections and proved

them to have good properties.[35] These algorithms need only a few multiplications and additions to calculate eachvector.

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It is beneficial to use a parametric formulation in computer graphics because the density of points is greatest wherethere is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent"jaggedness" of the approximation.

Drawing with Bézier spline paths

Bézier splines may also be used to draw an ellipse to sufficient accuracy, since any ellipse may be construed as anaffine transformation of a circle. The spline methods used to draw a circle may be used to draw an ellipse, since theconstituent Bézier curves will behave appropriately under such transformations.

Line segment as a type of 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.[36] Although the eccentricity is 1 this is not a parabola. A radial elliptic trajectory is a non-trivial special caseof an elliptic orbit, where the ellipse is a line segment.

Ellipses in optimization theory

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

See also

Apollonius of Perga, the classical authorityCartesian oval, a generalization of the ellipse

Circumconic and inconicConic section

Ellipsoid, a higher dimensional analog of an ellipseElliptic coordinates, an orthogonal coordinate system based on families of ellipses and hyperbolae

Elliptical distribution, in statisticsElliptic partial differential equation

Great ellipseHyperbolaKepler's laws of planetary motion

Matrix representation of conic sectionsn-ellipse, a generalization of the ellipse for n foci

OvalParabola

Proofs involving the ellipseSpheroid, the ellipsoid obtained by rotating an ellipse about its major or minor axisSteiner circumellipse, the unique ellipse circumscribing a triangle and sharing its centroid

Steiner inellipse, the unique ellipse inscribed in a triangle with tangencies at the sides' midpointsSuperellipse, a generalization of an ellipse that can look more rectangular or more "pointy"

True, eccentric, and mean anomalyGeodesics on an ellipsoid

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References

Besant, W.H. (1907). "Chapter III. The Ellipse" (http://books.google.com/books?id=TRJLAAAAYAAJ&pg=PA50). Conic Sections. London: George Bell and Sons. p. 50.

Miller, Charles D.; Lial, Margaret L.; Schneider, David I. (1990). Fundamentals of College Algebra (3rded.). 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 (http://planetmath.org/encyclopedia/Ellipse2.html)Weisstein, Eric W., "Ellipse (http://mathworld.wolfram.com/Ellipse.html)", MathWorld.

Notes

1. ^ Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas(http://books.google.com/books?id=Uk4wAAAAMAAJ&pg=RA1-PA381&zoom=3). Harper & Brothers.

2. ^ Herschel, Sir John Frederick William (1842). A treatise on astronomy (http://books.google.com/books?id=hh0uNybw1ZUC&pg=PA256). Lea & Blanchard. p. 256.

3. ^ Lankford, John (1997). History of Astronomy: An Encyclopedia (http://books.google.com/books?id=berWESi5c5QC&pg=PA194). Taylor & Francis. p. 194. ISBN 978-0-8153-0322-0.

4. ^ Prasolov, Viktor Vasil evich; Tikhomirov, Vladimir Mikhaĭlovich (2001). Geometry(http://books.google.com/books?id=t7kbhDDUFSkC&pg=PA80). American Mathematical Society. p. 80.ISBN 978-0-8218-2038-4.

5. ^ Fenna, Donald (2007). Cartographic Science: A Compendium of Map Projections, With Derivations(http://books.google.com/books?id=8LZeu8RxOIsC&pg=PA24). CRC Press. p. 24. ISBN 978-0-8493-8169-0.

6. ^ AutoCAD release 13 command reference (http://books.google.com/books?id=q4hRAAAAMAAJ). Autodesk, Inc.1994. p. 216.

7. ^ Salomon, David (2006). Curves And Surfaces for Computer Graphics (http://books.google.com/books?id=m0Je92uycVAC&pg=PA365). Birkhäuser. p. 365. ISBN 978-0-387-24196-8.

8. ^ Kreith, Frank; Goswami, D. Yogi (2005). The CRC Handbook Of Mechanical Engineering(http://books.google.com/books?id=_wlZ5LHTyBIC&pg=SA11-PA8). CRC Press. pp. 11–8. ISBN 978-0-8493-0866-6. "Circles and Ellipses (11.3.2)"

9. ^ The Mathematical Association of America (1976), The American Mathematical Monthly, vol. 83, page 207(http://books.google.com.br/books?id=Xpk0AAAAIAAJ&q=ellipse+%22major+radius%22&dq=ellipse+%22major+radius%22&lr=&num=20&client=firefox-a&hl=en)

10. ^ Gibson, C. G. (2001), Elementary Geometry of Differentiable Curves: An Undergraduate Introduction,Cambridge University Press, p. 127, ISBN 9780521011075.

11. ^ Besant 1907, p. 57

12. ^ Armengaud, Aîné (1853). "Ovals, Ellipses, Parabolas, Volutes, etc. §53" (http://books.google.com/books?id=6skOAAAAQAAJ&pg=PA15#v=onepage&f=false). The Practical Draughtsman's Book of Industrial Design.Longman, Brown, Green, and Longmans. p. 16.

13. ^ Brown, Henry T. (1881). Five Hundred and Seven Mechanical Movements: Embracing All Those which areMost Important in Dynamics, Hydraulics, Hydrostatics, Pneumatics, Steam Engines, Mill and Other Gearing,Presses, Horology, and Miscellaneous Machinery; and Including Many Movements Never Before Published, andSeveral which Have Only Recently Come Into Use (http://books.google.com/books?id=TFwOAAAAYAAJ&pg=PA41). Brown & Brown. pp. 40–41 section 152.

14. ^ Carlson, B. C. (1995). "Numerical computation of real or complex elliptic integrals". Numerical Algorithms 10(1): 13–98. arXiv:math/9409227 (//arxiv.org/abs/math/9409227). Bibcode:1995NuAlg..10...13C(http://adsabs.harvard.edu/abs/1995NuAlg..10...13C). doi:10.1007/BF02198293(http://dx.doi.org/10.1007%2FBF02198293).

15. ^ Python code for the circumference of an ellipse in terms of the complete elliptic integral of the second kind

1/11/2014 Ellipse - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Ellipse 19/20

(http://paulbourke.net/geometry/ellipsecirc/python.code), retrieved 2013-12-28

16. ^ Ivory, J. (1798). "A new series for the rectification of the ellipsis" (http://books.google.com/books?

id=FaUaqZZYYPAC&pg=PA177). Transactions of the Royal Society of Edinburgh 4: 177–190.

17. ^ Bessel, F. W. (2010). "The calculation of longitude and latitude from geodesic measurements (1825)". Astron.

Nachr. 331 (8): 852–861. arXiv:0908.1824 (//arxiv.org/abs/0908.1824). doi:10.1002/asna.201011352

(http://dx.doi.org/10.1002%2Fasna.201011352). English translation of Astron. Nachr. 4, 241–254 (1825).

18. ^ Chakerian, G. D. "A Distorted View of Geometry." Ch. 7 in Mathematical Plums (R. Honsberger, editor).Washington, DC: Mathematical Association of America, 1979.

19. ^ Larson, Ron; Hostetler, Robert P.; Falvo, David C. (2006). "Chapter 10" (http://books.google.com/books?id=yMdHnyerji8C&pg=PA767). Precalculus with Limits (http://books.google.com/books?id=yMdHnyerji8C).Cengage Learning. p. 767. ISBN 0-618-66089-5.

20. ^ Young, Cynthia Y. (2010). "Chapter 9" (http://books.google.com/books?id=9HRLAn326zEC&pg=PA831).Precalculus (http://books.google.com/books?id=9HRLAn326zEC). John Wiley and Sons. p. 831. ISBN 0-471-75684-9.

21. ^ Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.

22. ^ If the ellipse is illustrated as a meridional one for the earth, the tangential angle is equal to geodetic latitude, theangle is the geocentric latitude, and parametric angle t is a parametric (or reduced) latitude of auxiliary circle

23. ^ Ellipse at MathWorld, derived from formula (58) and (60) (http://mathworld.wolfram.com/Ellipse.html)

24. ^ clarifies problems with MathWorld formula (60) (http://math.stackexchange.com/a/410339/41584)

25. ^ Auxiliary circle and various ellipse formulas(http://www.math.uoc.gr/~pamfilos/eGallery/problems/Auxiliary.html)

26. ^ Meeus, J. (1991). "Ch. 10: The Earth's Globe". Astronomical Algorithms. Willmann-Bell. p. 78. ISBN 0-943396-35-2.

27. ^ David Drew. "Elliptical Gears". [1](http://jwilson.coe.uga.edu/emt668/EMAT6680.2003.fall/Drew/Emat6890/Elliptical%20Gears.htm)

28. ^ Grant, George B. (1906). A treatise on gear wheels (http://books.google.com/books?id=fPoOAAAAYAAJ&pg=PA72). Philadelphia Gear Works. p. 72.

29. ^ http://www.rp-photonics.com/lamp_pumped_lasers.html

30. ^ http://www.cymer.com/plasma_chamber_detail/

31. ^ Chamberlain, G. (February 1983). "A characterization of the distributions that imply mean—Variance utility

functions" (http://www.sciencedirect.com/science/article/pii/0022053183901291). Journal of Economic Theory 29(1): 185–201. doi:10.1016/0022-0531(83)90129-1 (http://dx.doi.org/10.1016%2F0022-0531%2883%2990129-1).

32. ^ Owen, J.; Rabinovitch, R. (June 1983). "On the class of elliptical distributions and their applications to the theory

of portfolio choice". Journal of Finance 38: 745–752. JSTOR 2328079 (//www.jstor.org/stable/2328079).

33. ^ Pitteway, M.L.V. (1967). "Algorithm for drawing ellipses or hyperbolae with a digital plotter"

(http://comjnl.oxfordjournals.org/content/10/3/282.abstract). The Computer Journal 10 (3): 282–9.doi:10.1093/comjnl/10.3.282 (http://dx.doi.org/10.1093%2Fcomjnl%2F10.3.282).

34. ^ Van Aken, J.R. (September 1984). "An Efficient Ellipse-Drawing Algorithm". IEEE Computer Graphics and

Applications 4 (9): 24–35. doi:10.1109/MCG.1984.275994 (http://dx.doi.org/10.1109%2FMCG.1984.275994).

35. ^ Smith, L.B. (1971). "Drawing ellipses, hyperbolae or parabolae with a fixed number of points"

(http://comjnl.oxfordjournals.org/content/14/1/81.short). The Computer Journal 14 (1): 81–86.doi:10.1093/comjnl/14.1.81 (http://dx.doi.org/10.1093%2Fcomjnl%2F14.1.81).

36. ^ Seligman, Courtney (1993–2010). "Orbital Motions Ellipses and Other Conic Sections"(http://cseligman.com/text/history/ellipses.htm). Online Astronomy eText.

External links

Video: How to draw Ellipse (http://www.youtube.com/watch?v=7UD8hOs-vaI)

Apollonius' Derivation of the Ellipse (http://mathdl.maa.org/convergence/1/?pa=content&sa=viewDocument&nodeId=196&bodyId=203) at Convergence

1/11/2014 Ellipse - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Ellipse 20/20

(http://mathdl.maa.org/convergence/1/)The Shape and History of The Ellipse in Washington, D.C. (http://faculty.evansville.edu/ck6/ellipse.pdf) byClark KimberlingCollection of animated ellipse demonstrations. (http://www.mathopenref.com/tocs/ellipsetoc.html) Ellipse,

axes, semi-axes, area, perimeter, tangent, foci.Weisstein, Eric W., "Ellipse as (http://mathworld.wolfram.com/Hypotrochoid.html.html) hypotrochoid",

MathWorld.Ivanov, A.B. (2001), "Ellipse" (http://www.encyclopediaofmath.org/index.php?title=Ellipse&oldid=11394),

in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

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