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In mathematics, an ellipse is a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve. As such, it is a generalization of a circle, which is a special type of an ellipse that has both focal points at the same location. The shape of an ellipse (how 'elongated' it is) is represented by its eccentricity, which for an ellipse can be any number from 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 results from 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 and unbounded. The cross section of a cylinder is an ellipse, unless the section is parallel to the axis of the cylinder. Analytically, an ellipse can also be defined as the set of points such that the 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 the curve to a given line (called the directrix) is a constant, called the eccentricity of the ellipse. Ellipses are common in physics, astronomy and engineering. For example, the orbit of each planet in the solar system is an ellipse with the barycenter of the planet-Sun pair at one of the focal points. The same is true for moons orbiting planets and all other systems having two astronomical bodies. The shape of planets and stars are often well described by ellipsoids. Ellipses also arise as images of a circle under parallel projection and the bounded cases of perspective projection, which are simply intersections of the projective cone with the plane of projection. It is also the simplest Lissajous figure, formed when the horizontal and vertical motions are sinusoids with the same frequency. A similar effect leads to elliptical polarization of light in optics. The name, ἔλλειψις (élleipsis, "omission"), was given by Apollonius of Perga in his ''Conics'', emphasizing the connection of the curve with "application of areas". ==Elements of an ellipse== Ellipses have two perpendicular axes about which the ellipse is symmetric. These axes intersect at the center of the ellipse due to this symmetry. The larger of these two axes, which corresponds to the largest distance between antipodal points on the ellipse, is called the major axis. (On the figure to the right it is represented by the line segment between the point labeled ''−a'' and the point labeled ''a''.) The smaller of these two axes, and the smallest distance across the ellipse, is called the minor axis.〔The "major axis" and "minor axis" are sometimes called the "transverse diameter" and "conjugate diameter"; see This usage is now rare.〕 (On the figure to the right it is represented by the line segment between the point labeled ''−b'' to the point labeled ''b''.) The semi-major axis (denoted by ''a'' in the figure) and the semi-minor axis (denoted by ''b'' in the figure) are one half of the major and minor axes, respectively. These are sometimes called (especially in technical fields) the major and minor semi-axes,〔 〕〔 〕 the major and minor semiaxes,〔 〕〔 〕 or major radius and minor radius.〔 〕〔 〕〔 〕〔 The Mathematical Association of America (1976), (The American Mathematical Monthly, vol. 83, page 207 ) 〕 The four points where these axes cross the ellipse are the vertices and are marked as ''a'', ''−a'', ''b'', and ''−b''. In addition to being at the largest and smallest distance from the center, these points are where the curvature of the ellipse is maximum and minimum.〔 . 〕 The two foci (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 on the ellipse to those two foci is constant and equal to the major axis (''PF1'' + ''PF2'' = 2''a''). (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 the ellipse. 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: ''f''2 = ''a''2 −''b''2. 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 the directrix 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 the length of the major axis or ''e'' = 2''f''/2''a'' = ''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 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 finite distance apart and a parabola if one focus is kept fixed as the other is allowed to move arbitrarily far away. The eccentricity 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''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ellipse」の詳細全文を読む スポンサード リンク
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