The ellipse and some of its mathematical properties.

An ellipse obtained as the intersection of a cone with a plane. A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex

In mathematics, an ellipse (from the Greek ἔλλειψις, literally absence) is a locus of points in a plane such that the sum of the distances to two fixed points is a constant. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Polish ( język polski, polszczyzna) is the Official language of Poland. In Mathematics, a locus ( Latin for "place" plural loci) is a collection of points which share a property Distance is a numerical description of how far apart objects are The two fixed points are called foci (singular- focus). In Geometry, the foci (singular focus) are a pair of special points used in describing Conic sections The four types of conic sections are the Circle An alternate definition would be that an ellipse is the path traced out by a point whose distance from a fixed point, called the focus, maintains a constant ratio less than one with its distance from a straight line not passing through the focus, called the directrix. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

Overview

An ellipse is a type of conic section: if a conical surface is cut with a plane which does not intersect the cone's base, the intersection of the cone and plane is an ellipse. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface In Geometry, a ( general) conical surface is the unbounded Surface formed by the union of all the straight lines that pass through a fixed For a short elementary proof of this, see Dandelin spheres. In Geometry, a nondegenerate Conic section formed by a plane intersecting a cone has one or two Dandelin spheres characterized thus Each

Algebraically, an ellipse is a curve in the Cartesian plane defined by an equation of the form

such that B² < 4AC, where all of the coefficients are real, and where more than one solution, defining a pair of points (x, y) on the ellipse, exists. Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object In Mathematics, the Cartesian coordinate system (also called rectangular coordinate system) is used to determine each point uniquely in a plane

The line segment AB, that passes through the foci and terminates on the ellipse, is called the major axis. In Geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its end points The major axis is the longest segment that can be obtained by joining two points on the ellipse. The line segment CD, which passes through the center (halfway between the foci), perpendicular to the major axis, and terminates on the ellipse, is called the minor axis. In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent The semimajor axis (denoted by a in the figure) is one half the major axis: the line segment from the center, through a focus, and to the edge of the ellipse. In Geometry, the semi-major axis (also semimajor axis) is used to describe the dimensions of ellipses and hyperbolae Likewise, the semiminor axis (denoted by b in the figure) is one half the minor axis. In Geometry, the semi-minor axis (also semiminor axis) is a Line segment associated with most Conic sections (that is with ellipses and

If the two foci coincide, then the ellipse is a circle; in other words, a circle is a special case of an ellipse, one where the eccentricity is zero. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section.

An ellipse centered at the origin can be viewed as the image of the unit circle under a linear map associated with a symmetric matrix A = PDP^T, D being a diagonal matrix with the eigenvalues of A, both of which are real positive, along the main diagonal, and P being a real unitary matrix having as columns the eigenvectors of A. In Mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference In Mathematics, a unit circle is In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes Then the axes of the ellipse will lie along the eigenvectors of A, and the (square root of the) eigenvalues are the lengths of the semimajor and semiminor axes.

An ellipse can be produced by multiplying the x coordinates of all points on a circle by a constant, without changing the y coordinates. This is equivalent to stretching the circle out in the x-direction.

Eccentricity

The shape of an ellipse can be expressed by a number called the eccentricity of the ellipse, conventionally denoted . In Mathematics, the eccentricity, denoted e or \varepsilon is a parameter associated with every conic section. The eccentricity is a non-negative number less than 1 and greater than or equal to 0. A negative number is a Number that is less than zero, such as −2 It is the value of the constant ratio of the distance of a point on an ellipse from a focus to that from the corresponding directrix. An eccentricity of 0 implies that the two foci occupy the same point and that the ellipse is a circle. Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the It can also be expressed as the sine of the angular eccentricity, . In the study of ellipses and related geometry various parameters in the distortion of a circle into an ellipse are identified and employed Aspect ratio Flattening and eccentricity For an ellipse with semimajor axis a and semiminor axis b, the eccentricity is

The greater the eccentricity is, the larger the ratio of a to b, and therefore the more elongated the ellipse. A ratio is an expression which compares quantities relative to each other

If c equals the distance from the center to either focus, then

The distance c is known as the linear eccentricity of the ellipse. The distance between the foci is 2c or 2aε.

Drawing

An ellipse can be inscribed within a rectangle using two pins, a loop of string, and a pencil. In Geometry, a rectangle is defined as a Quadrilateral where all four of its angles are Right angles A rectangle with vertices ABCD would be denoted as The pins are placed at the foci and the pins and pencil are enclosed inside the loop. The pencil is placed on the paper inside the loop and the string made taut. The string will form a triangle. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line If the pencil is moved around with the string kept taut, the sum of the distances from the pencil to the pins will remain constant, thus satisfying the definition of an ellipse.

Consider the center of the rectangle to be the origin and the lengths of its sides to be 2a and 2b, with a being larger than b. The major axis then passes through the origin and is parallel to the longer side. The two pins are placed the distance c away from the origin in each direction along the major axis. The required length of the string used to form the loop is 2a + 2c.

Equations

An ellipse with a semimajor axis a and semiminor axis b, centered at the point (h,k) and having its major axis parallel to the x-axis may be specified by the equation

This ellipse can be expressed parametrically as

where t may be restricted to the interval . In Mathematics, parametric equations are a method of defining a curve

Parametric form of an ellipse rotated counterclockwise by an angle :

The ellipse (drawn in red) may be expressed as a special case of the hypotrochoid, R=2r. A hypotrochoid is a roulette traced by a point attached to a Circle of Radius r rolling around the inside of a fixed circle of radius R

The formula for the directrices is

If h = 0 and k = 0 (i. e. , if the center is the origin (0,0)), then we can express this ellipse in polar coordinates by the equation

With one focus at the origin, the ellipse's polar equation is

A Gauss-mapped form:

has normal (cosβ,sinβ). In Differential geometry, the Gauss map (named after Carl F Gauss) maps a Surface in Euclidean space R 3 to the unit

Semi-latus rectum and polar coordinates

The semi-latus rectum of an ellipse, usually denoted (lowercase L), is the distance from a focus of the ellipse to the ellipse itself, measured along a line perpendicular to the major axis. In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface Lower case (also lower-case or lowercase) minuscule, or small letters are the smaller form of letters as opposed to upper L is the twelfth letter of the Latin alphabet. Its name in English is el or occasionally ell (ɛl In Geometry, two lines or planes (or a line and a plane are considered perpendicular (or orthogonal) to each other if they form congruent It is related to and (the ellipse's semi-axes) by the formula or, if using the eccentricity,

In polar coordinates, an ellipse with one focus at the origin and the other on the negative x-axis is given by the equation

An ellipse can also be thought of as a projection of a circle: a circle on a plane at angle φ to the horizontal projected vertically onto a horizontal plane gives an ellipse of eccentricity sin φ, provided φ is not 90°. In Mathematics, the polar coordinate system is a two-dimensional Coordinate system in which each point on a plane is determined by

Area and circumference

The area enclosed by an ellipse is πab, where (as before) a and b are the ellipse's semimajor and semiminor axes. The area of a disk (the region inside a Circle) is &pi r 2 when the circle has Radius r.

The circumference C of an ellipse is , where the function E is the complete elliptic integral of the second kind. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse.

The exact infinite series is:

Or:

A good approximation is Ramanujan's:

or better approximation:

For the special case where the minor axis is half the major axis, we can use:

Or:

(better approximation). In Mathematics, a series is often represented as the sum of a Sequence of terms That is a series is represented as a list of numbers with An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful An approximation (represented by the symbol ≈ is an inexact representation of something that is still close enough to be useful

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. Determining the length of an irregular arc segment — also called Rectification of a Curve — was historically difficult In Integral calculus, elliptic integrals originally arose in connection with the problem of giving the Arc length of an Ellipse. The inverse function, the angle subtended as a function of the arc length, is given by the elliptic functions. In Mathematics, if &fnof is a function from A to B then an inverse function for &fnof is a function in the opposite direction from B In Complex analysis, an elliptic function is a function defined on the Complex plane which is periodic in two directions (a Doubly-periodic

Stretching and projection

An ellipse may be uniformly stretched along any axis, in or out of the plane of the ellipse, and it will still be an ellipse. The stretched ellipse will have different properties (perhaps changed eccentricity and semi-major axis length, for instance), but it will still be an ellipse (or a degenerate ellipse: a circle or a line). Similarly, any oblique projection onto a plane results in a conic section. This article discusses imaging of three-dimensional objects For an abstract mathematical discussion see Projection (linear algebra. If the projection is a closed curve on the plane, then the curve is an ellipse or a degenerate ellipse.

Reflection property

Assume an elliptic mirror with a light source at one of the foci. A mirror is an object with a surface that has good Specular reflection; that is it is smooth enough to form an Image. Then all rays are reflected to a single point — the second focus. Reflection is the change in direction of a Wave front at an interface between two different media so that the wave front returns into the medium from which Since no other curve has such a property, it can be used as an alternative definition of an ellipse. In a circle, all light would be reflected back to the center since all tangents are orthogonal to the radius. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

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 another focus remarkably well. Such a room is called a whisper chamber. Examples are the National Statuary Hall at the U.S. Capitol (where John Quincy Adams is said to have used this property for eavesdropping on political matters), at an exhibit on sound at the Museum of Science and Industry in Chicago, in front of the University of Illinois at Urbana-Champaign Foellinger Auditorium, and also at a side chamber of the Palace of Charles V, in the Alhambra. National Statuary Hall is a chamber in the United States Capitol devoted to Sculptures of prominent Americans. John Quincy Adams (July 11 1767 &ndash February 23 1848 was an American diplomat and politician who served as the sixth President of the United States The Museum of Science and Industry (MSI is located in Chicago, Illinois in Jackson Park, in the Hyde Park neighborhood adjacent to Chicago (ʃɪˈkɑːgoʊ is the largest City by population in the state of Illinois and the American Midwest of the United States. This article is about the flagship campus For other uses and locations of University of Illinois, see University of Illinois (disambiguation The University of This article is about the Alhambra in Granada Spain For other meanings see Alhambra (disambiguation.

Ellipses in physics

In the 17th century, Johannes Kepler explained that the orbits along which the planets travel around the Sun are ellipses in his first law of planetary motion. As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar Johannes Kepler (ˈkɛplɚ ( December 27 1571 &ndash November 15 1630) was a German Mathematician, Astronomer In Physics, an orbit is the gravitationally curved path of one object around a point or another body for example the gravitational orbit of a planet around a star In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. Later, Isaac Newton explained this as a corollary of his law of universal gravitation. Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Newton 's law of universal Gravitation is a physical law describing the gravitational attraction between bodies with mass

More generally, in the gravitational two-body problem, if the two bodies are bound to each other (i. In Classical mechanics, the two-body problem is to determine the motion of two point particles that interact only with each other 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.

The general solution for a harmonic oscillator in two or more dimensions is also an ellipse, but this time with the origin of the force located at the center of the ellipse. This article is about the harmonic oscillator in classical mechanics In mathematics the dimension of a Space is roughly defined as the minimum number of Coordinates needed to specify every point within it

In optics, an index ellipsoid describes the refractive index of a material as a function of the direction through that material. The index ellipsoid is a diagram of an Ellipsoid that depicts the orientation and relative magnitude of Refractive indices in a Crystal. The refractive index (or index of Refraction) of a medium is a measure for how much the speed of light (or other waves such as sound waves is reduced inside the medium This only applies to materials that are optically anisotropic. Anisotropy (pronounced with stress on the third syllable ˌænaɪˈsɒtrəpi is the property of being directionally dependent as opposed to Isotropy, which means homogeneity Also see birefringence. Birefringence, or double refraction, is the decomposition of a ray of Light into two rays (the ordinary ray and the extraordinary ray

Ellipses in computer graphics

Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the Macintosh QuickDraw API, the Windows Graphics Device Interface (GDI) and the Windows Presentation Foundation (WPF). The term geometric primitive in Computer graphics and CAD systems is used in various senses with common meaning of atomic geometric objects the system can handle Quickdraw also refers to equipment used for Rock climbing. QuickDraw is the 2D graphics library and associated The Graphics Device Interface (GDI is one of the three core components or "subsystems" together with the kernel and the Windows API for the user interface The Windows Presentation Foundation (or WPF) formerly code-named Avalon, is a graphical subsystem in. Often such libraries are limited and can only draw an ellipse with either the major axis or the minor axis horizontal. 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. Jack Elton Bresenham (born in 1937 was a professor of Computer science. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984).

The following is example JavaScript code using the parametric formula for an ellipse to calculate a set of points. The ellipse can be then approximated by connecting the points with lines.

/*
* This functions returns an array containing 36 points to draw an
* ellipse. 
*
* @param x {double} X coordinate
* @param y {double} Y coordinate
* @param a {double} Semimajor axis
* @param b {double} Semiminor axis
* @param angle {double} Angle of the ellipse
*/
function calculateEllipse(x, y, a, b, angle, steps) 
{
  if (steps == null)
    steps = 36;
  var points = [];
 
  var beta = -angle / 180 * Math. PI;
  var sinbeta = Math. sin(beta);
  var cosbeta = Math. cos(beta);
 
  for (var i = 0; i < 360; i += 360 / steps) 
  {
    var alpha = i / 180 * Math. PI;
    var sinalpha = Math. sin(alpha);
    var cosalpha = Math. cos(alpha);
 
    var X = x + (a * cosalpha * cosbeta - b * sinalpha * sinbeta);
    var Y = y + (a * cosalpha * sinbeta + b * sinalpha * cosbeta);
 
    points. push(new OpenLayers. Geometry. Point(X, Y));
   }
 
  return points;
}

One beneficial consequence of using the parametric formula is that the density of points is greatest where there is the most curvature. Thus, the change in slope between each successive point is small, reducing the apparent "jaggedness" of the approximation.

See also

References

• Charles D. The superellipse (or Lamé curve) is the geometric figure defined in the Cartesian coordinate system as the set of all points ( x, y) with In Geometry, a hyperbola ( Greek, "over-thrown" has several equivalent definitions In Mathematics, the parabola (pəˈræbələ from the Greek παραβολή) is a Conic section, the intersection of a right circular In geometry an oval or ovoid (from Latin ovum, 'egg' is any Curve resembling an egg or an Ellipse. In Astronomy, the true anomaly \nu\\! (Greek nu also written \theta\ or f\) is the angle between the direction z-s of The eccentric anomaly is the angle between the direction of Periapsis and the current position of an object on its Orbit, projected onto the ellipse's circumscribing In the study of orbital dynamics the mean anomaly of an orbiting body is the Angle the body would have traveled about the center of the orbit's Auxiliary circle In Mathematics, the matrix representation of conic sections is one way of studying a Conic section, its axis, vertices, foci, In Astronomy, Kepler's Laws of Planetary Motion are three mathematical laws that describe the motion of Planets in the Solar System. The derivation of the cartesian form for an Ellipse is simple and instructive Miller, Margaret L. Lial, David I. Schneider: Fundamentals of College Algebra. 3rd Edition Scott Foresman/Little 1990. ISBN 0-673-38638-4. Page 381
• Coxeter, H. S. M. : Introduction to Geometry, 2nd ed. New York: Wiley, pp. 115-119, 1969.
• Ellipse at the Encyclopedia of Mathematics (Springer)
• Ellipse at Planetmath
• Eric W. Weisstein, Ellipse at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W

External links

• Apollonius' Derivation of the Ellipse at Convergence
• Ellipse & Hyperbola Construction - An interactive sketch showing how to trace the curves of the ellipse and hyperbola. (Requires Java. )
• Ellipse Construction - Another interactive sketch, this time showing a different method of tracing the ellipse. (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.
• Woodworking videos showing how to work with ellipses in wood.
• Ellipse as hypotrochoid