User:Dave3457/Sandbox/Plane wave
Elements of an ellipse[edit]
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 semimajor axis (denoted by a in the figure) and the semiminor axis (denoted by b in the figure) are one half of the major and minor diameters, 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 diameter ( 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 closed curve 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).
- ^ Haswell, Charles Haynes (1920). Mechanics' and Engineers' Pocket-book of Tables, Rules, and Formulas. Harper & Brothers. Retrieved 2007-04-09.
- ^ John Herschel (1842) A Treatise on Astronomy, page 256
- ^ John Lankford (1996), History of Astronomy: An Encyclopedia, page 194
- ^ V. Prasolov and V. Tikhomirov (2001), Geometry, page 80
- ^ Donald Fenna (2006), Cartographic science: a compendium of map projections, with derivations, page 24
- ^ Autocad release 13: command reference, page 216
- ^ David Salomon (2006), Curves and surfaces for computer graphics, page 365
- ^ CRC Press (2004), The CRC handbook of mechanical engineering, page 11-8
- ^ The Mathematical Association of America (1976), The American Mathematical Monthly, vol. 83, page 207