In mathematics, an ellipse (from the Greek ἔλλειψις, literally absence) is a conic section, the locus of points in a plane such that the sum of the distances to two fixed points is a constant. The two fixed points are called foci (singular- focus). 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.
The line segment AB, that passes through the foci and terminates on the ellipse, is called the major axis. 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. 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. Likewise, the semiminor axis (denoted by b in the figure) is one half the minor axis.
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 , being a diagonal matrix with the eigenvalues of , both of which are real positive, along the main diagonal, and being a real unitary matrix having as columns the eigenvectors of . Then the axes of the ellipse will lie along the eigenvectors of , and the (1 over the square root of the) eigenvalues are the lengths of the semimajor and semiminor axes, which are one-half of the lengths of the major and minor axes respectively.
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.
The greater the eccentricity is, the larger the ratio of a to b, and therefore the more elongated the ellipse.
If c equals the distance from the center to either focus, then
An ellipse can be inscribed within a rectangle using two pins, a loop of string, and a pencil. 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. 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 can be seen in the particular case of the point of the ellipse lying on the major axis a. Here the length of the string is the distance from the first focus to the center c plus the distance from the center to the point on the ellipse a plus the distance from that point back to the second focus a-c, all together 2a.
This ellipse can be expressed parametrically as
Parametric form of an ellipse rotated counterclockwise by an angle :
The formula for the directrices is
If = 0 and = 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:
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°.
The exact infinite series is:
or better approximation:
For the special case where the minor axis is half the major axis, we can use:
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.
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. 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.
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.
In optics, an index ellipsoid describes the refractive index of a material as a function of the direction through that material. This only applies to materials that are optically anisotropic. Also see birefringence.
Drawing an ellipse as a graphics primitive is common in standard display libraries, such as the Macintosh QuickDraw API, the Windows Graphics Device Interface (GDI) and the Windows Presentation Foundation (WPF). Often such libraries are limited 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. An efficient generalization to draw ellipses was invented in 1984 by Jerry Van Aken (IEEE CG&A, Sept. 1984).
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.
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