From Wikipedia, the free encyclopedia

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

Elements of an ellipse

The ellipse and some of its mathematical properties.

Drawing ellipses

The pins-and-string method

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

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

• http://gcsu211151.gcsu.edu/mathdemos/carpenterellipse/carpellipse_main.html 19 January 2010 18:39 UTC gcsu211151.gcsu.edu [Source type: FILTERED WITH BAYES]

^ With a fixed length of string connecting F1, P and F2, by placing a pencil at P and keeping the string taut an ellipse is traced as we move the pencil.

• http://gcsu211151.gcsu.edu/mathdemos/carpenterellipse/carpellipse_main.html 19 January 2010 18:39 UTC gcsu211151.gcsu.edu [Source type: FILTERED WITH BAYES]

^ The path of a heavenly body moving around another in a closed orbit in accordance with Newton’s gravitational law is an ellipse ( see Kepler’s laws of planetary motion ).

• ellipse (mathematics) -- Britannica Online Encyclopedia 19 January 2010 18:39 UTC www.britannica.com [Source type: Reference]

Let A,B,C,D be the corners of the rectangle, in clockwise order, with A-B being one of the long sides. .Draw a circle centered on A, whose radius is the short side A-D.^ It is annoying not to be able to draw a circle from center and radius, as in real drafting programs, but this can be worked around by putting the center of a square of side equal to the diameter at the desired center point.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

^ First, draw a circle with centre at P and radius PF. The intersection of this circle with an arc of a radius equal to the major axis (2a) is point E. Point R is the intersection of radius F'E and the ellipse.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

^ Great circle -- An imaginary circle on the surface of a sphere whose center is at the center of the sphere.

• Basics of Space Flight Editorial Page 19 January 2010 18:39 UTC www2.jpl.nasa.gov [Source type: Academic]

.From corner B draw a tangent to the circle.^ Tangents to a circle are perpendicular to the radius at the point of tangency, so it is easy to draw a tangent at a given point on a circle, or to construct the tangent from an external point.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

.The length L of this tangent is the distance between the foci.^ Reflections not passing through a focus will be tangent to a confocal hyperbola or ellipse, depending on whether the ray passes between the foci or not.

• Ellipse -- from Wolfram MathWorld 19 January 2010 18:39 UTC mathworld.wolfram.com [Source type: Academic]

^ The smaller the distance between the foci, the smaller is the eccentricity and the more closely the ellipse resembles a circle.

• ellipse (mathematics) -- Britannica Online Encyclopedia 19 January 2010 18:39 UTC www.britannica.com [Source type: Reference]

^ The two parameters in this case are the distance between the foci, 2c, and the sum of the radii, 2a.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

.Draw two perpendicular lines through the center of the rectangle and parallel to its sides; these will be the major and minor axes of the ellipse.^ A diameter is any chord through the center of the ellipse.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

^ Suppose an ellipse is given, and the major and minor axes are to be found.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

^ The box has sides parallel to the principal axes of the ELLIPSE .

• ACIS© R20 Library Reference - Spatial Corporation 19 January 2010 18:39 UTC doc.spatial.com [Source type: Reference]

.Place the foci on the major axis, symmetrically, at distance L/2 from the center.^ Eccentricity -- The distance between the foci of an ellipse divided by the major axis.

• Basics of Space Flight Editorial Page 19 January 2010 18:39 UTC www2.jpl.nasa.gov [Source type: Academic]

^ Perpendicular to the major axis through the centre, at the point on the major axis equidistant from the foci, is the minor axis.

• ellipse (mathematics) -- Britannica Online Encyclopedia 19 January 2010 18:39 UTC www.britannica.com [Source type: Reference]

^ The three eccentric circles in the second example are of different sizes, their centers placed at varying distances from each other.

• ArtLex's E-Em page 19 January 2010 18:39 UTC www.artlex.com [Source type: FILTERED WITH BAYES]

Other methods

Trammel of Archimedes (elpsograph) animation

Draw two perpendicular lines .M,N on the paper; these will be the major and minor axes of the ellipse.^ Suppose an ellipse is given, and the major and minor axes are to be found.

• Ellipse 19 January 2010 18:39 UTC www.du.edu [Source type: FILTERED WITH BAYES]

^ An ellipse rotated about its minor axis gives an oblate spheroid , while an ellipse rotated about its major axis gives a prolate spheroid .

• Ellipse -- from Wolfram MathWorld 19 January 2010 18:39 UTC mathworld.wolfram.com [Source type: Academic]

^ We have a cyclic trend, a major indication and a minor indication, and a right estimate of these factors will induce to correct apprehension of the Markets course.

• The Gann Ellipse - A Newly Discovered Trading Tool... 19 January 2010 18:39 UTC www.sacredscience.com [Source type: FILTERED WITH BAYES]

Mark three points A, B, C on the ruler. .With one hand, move the ruler onto the paper, turning and sliding it so as to keep point A always on line M, and B on line N.^ If the endpoints of a segment are moved along two intersecting lines, a fixed point on the segment (or on the line that prolongs it) describes an arc of an ellipse.

• Ellipse -- from Wolfram MathWorld 19 January 2010 18:39 UTC mathworld.wolfram.com [Source type: Academic]

^ One of the critical assumptions of Euclidean geometry is that given any straight line and a point not on that line, there is exactly one line that can be drawn through that point parallel to the first line.

• Astronomical Glossary 19 January 2010 18:39 UTC nedwww.ipac.caltech.edu [Source type: Academic]

^ It may be defined as the path of a point moving in a plane so that the ratio of its distances from a fixed point (the focus ) and a fixed straight line (the directrix ) is a constant less than one.

• ellipse (mathematics) -- Britannica Online Encyclopedia 19 January 2010 18:39 UTC www.britannica.com [Source type: Reference]

.With the other hand, keep the pencil's tip on the paper, following point C of the ruler.^ It consists of all the points in a plane that satisfy the following: a+b=(twice the length of the semi-major axis), where a is the distance from one focus to the point on the ellipse, and b is the distance from the other focus to the same point on the ellipse.

• Zoom Astronomy Glossary: E 19 January 2010 18:39 UTC www.enchantedlearning.com [Source type: FILTERED WITH BAYES]

The tip will trace out an ellipse.

Approximations to ellipses

Ellipses in physics

Elliptical reflectors and acoustics

Planetary orbits

Harmonic oscillators

Phase visualization

Elliptical gears

Optics

Mathematical definitions and properties

In Euclidean geometry

Definition

Eccentricity

The distance from the center to either focus is ae, or simply

Directrix

Ellipse as hypotrochoid

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

Area

.If the ellipse is given by the implicit equation Ax² + Bxy + Cy² + 1 = 0, then the area is .^ She more rigorously defines an equation for the approximation of the area of the ellipse, extended from the circle approximation.

• The Luxor Ellipse 19 January 2010 18:39 UTC www.egyptorigins.org [Source type: FILTERED WITH BAYES]

Circumference

.The circumference C of an ellipse is , where the function E is the complete elliptic integral of the second kind.^ Two different definitions of the function E (the so-called complete elliptic integral of the second kind ) have been given, which correspond to two different exact expressions for the perimeter of an ellipse: Either P = 4 a  E( e 2  ) or P = 4 a  E( e ) .

• Circumference/Perimeter of an Ellipse: Formula(s) - Numericana 19 January 2010 18:39 UTC home.att.net [Source type: Academic]

^ Exact Expansions for the Perimeter of an Ellipse : Some of the above exact formulas for the circumference of an ellipse may be expressed using Gauss's (1812) hypergeometric function F  (also denoted 2 F 1  ).

• Circumference/Perimeter of an Ellipse: Formula(s) - Numericana 19 January 2010 18:39 UTC home.att.net [Source type: Academic]

^ Keywords for further research are "elliptic integrals" (which are to an ellipse what inverse trigonometric functions are to a circle) and "elliptic functions" (the elliptic equivalents of trigonometric functions).

• Circumference/Perimeter of an Ellipse: Formula(s) - Numericana 19 January 2010 18:39 UTC home.att.net [Source type: Academic]

The exact infinite series is:

For computational purposes a much faster series where the denominators vanish at a rate is given by:

Q = b⁴ + 60ab³ + 134a²b² + 60a³b + a⁴

v₁ = ba(15b⁴ + 68ab³ + 90a²b² + 68a³b + 15a⁴)

v₂ = − a⁶ − b⁶ + 126ab⁵ + 1041a²b⁴ + 1764a³b³ + 1041a⁴b² + 126a⁵b

In projective geometry

In analytic geometry

General ellipse

with

Canonical form

.Any ellipse can be obtained by rotation and translation of a canonical ellipse with the proper semi-diameters.^ With the CircleAndEllipseProblem , it is assumed that the circles and ellipses in question are mutable - one can stretch them, rotate them, translate them, etc...

• Circle And Ellipse Problem 19 January 2010 18:39 UTC c2.com [Source type: Original source]

^ Semi-major axis -- Half the distance of an ellipse's maximum diameter, the distance from the center of the ellipse to one end.

• Basics of Space Flight Editorial Page 19 January 2010 18:39 UTC www2.jpl.nasa.gov [Source type: Academic]

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

In trigonometry

General parametric form

Parametric form in canonical position

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

Polar form relative to center

Polar form relative to focus

Polar coordinates centered at focus.

.The angle θ in these formulas is called the true anomaly of the point.^ True anomaly -- The angular distance of a point in an orbit past the point of periapsis, measured in degrees.

• Basics of Space Flight Editorial Page 19 January 2010 18:39 UTC www2.jpl.nasa.gov [Source type: Academic]

.The numerator of these formulas is the semi-latus rectum of the ellipse, usually denoted l.^ Exact Expansions for the Perimeter of an Ellipse : Some of the above exact formulas for the circumference of an ellipse may be expressed using Gauss's (1812) hypergeometric function F  (also denoted 2 F 1  ).

• Circumference/Perimeter of an Ellipse: Formula(s) - Numericana 19 January 2010 18:39 UTC home.att.net [Source type: Academic]

.It is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis.^ In undisturbed elliptic motion, the angle measured at the center of the ellipse from pericenter to the point on the circumscribing auxiliary circle from which a perpendicular to the major axis would intersect the orbiting body.

• Astronomical Glossary 19 January 2010 18:39 UTC nedwww.ipac.caltech.edu [Source type: Academic]

^ The distance from one focus to a point on the ellipse plus the distance from that point to the other focus, remains constant - see http://mathworld.wolfram.com/Ellipse.html .

• Circle And Ellipse Problem 19 January 2010 18:39 UTC c2.com [Source type: Original source]

^ If you draw a right angle triangle with one side along the axis of an ellipse, the triangle will not be circumscribed by the ellipse (unless the ellipse happens to be a circle at the time).

• Circle And Ellipse Problem 19 January 2010 18:39 UTC c2.com [Source type: Original source]

Semi-latus rectum.

General polar form

Gauss-mapped form

Angular eccentricity

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

Degrees of freedom

Ellipses in computer graphics

/*
* This functions returns an array containing 36 points to draw an
* ellipse.^ The Expanded Drawing values are from the half-span distances to the respective ellipse maximum vertical points.
 The Luxor Ellipse 19 January 2010 18:39 UTC www.egyptorigins.org [Source type: FILTERED WITH BAYES]



*
* @param x {double} .X coordinate
* @param y {double} Y coordinate
* @param a {double} Semimajor axis
* @param b {double} Semiminor axis
* @param angle {double} Angle of the ellipse
*/
function calculateEllipse(x, y, a, b, angle, steps) 
{
  if (steps == null)
    steps = 36;
  var points = [];
 
  // Angle is given by Degree Value
  var beta = -angle * (Math.^ Unit Circle A circle of radius=1 around the origin of (x,y) coordinates, used for extending the definition of trigonometric functions to angles larger than 90 degrees.
 Glossary 19 January 2010 18:39 UTC www-spof.gsfc.nasa.gov [Source type: FILTERED WITH BAYES]


^ Usually the triangle is drawn with resting on one of its shorter sides, and these functions are viewed as depending on the bottom acute angle (angle smaller than 90 degrees).
 Glossary 19 January 2010 18:39 UTC www-spof.gsfc.nasa.gov [Source type: FILTERED WITH BAYES]


^ Or else , in polar coordinates (r, f ), as the curve whose points satisfy a relation r = a(1 - e)/(1 + e cos f ) where a is the semi-major axis, half the width in the direction through the two foci.
 Glossary 19 January 2010 18:39 UTC www-spof.gsfc.nasa.gov [Source type: FILTERED WITH BAYES]


PI / 180); //(Math.PI/180) converts Degree Value into Radians
  var sinbeta = Math.sin(beta);
  var cosbeta = Math.cos(beta);
 
  for (var i = 0; i < 360; i += 360 / steps) 
  {
    var alpha = i * (Math.PI / 180) ;
    var sinalpha = Math.sin(alpha);
    var cosalpha = Math.cos(alpha);
 
    var X = x + (a * cosalpha * cosbeta - b * sinalpha * sinbeta);
    var Y = y + (a * cosalpha * sinbeta + b * sinalpha * cosbeta);
 
    points.push(new OpenLayers.Geometry.Point(X, Y));
   }
 
  return points;
}

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