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In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work ''Der barycentrische Calcül'',〔 〕 are a system of coordinates used in projective geometry, as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If the homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than the dimension of the projective space being considered. For example, two homogeneous coordinates are required to specify a point on the projective line and three homogeneous coordinates are required to specify a point in the projective plane. ==Introduction== The real projective plane can be thought of as the Euclidean plane with additional points added, which are called points at infinity, and are considered to lie on a new line, the line at infinity. There is a point at infinity corresponding to each direction (numerically given by the slope of a line), informally defined as the limit of a point that moves in that direction away from the origin. Parallel lines in the Euclidean plane are said to intersect at a point at infinity corresponding to their common direction. Given a point on the Euclidean plane, for any non-zero real number ''Z'', the triple is called a ''set of homogeneous coordinates'' for the point. By this definition, multiplying the three homogeneous coordinates by a common, non-zero factor gives a new set of homogeneous coordinates for the same point. In particular, is such a system of homogeneous coordinates for the point . For example, the Cartesian point can be represented in homogeneous coordinates as or . The original Cartesian coordinates are recovered by dividing the first two positions by the third. Thus unlike Cartesian coordinates, a single point can be represented by infinitely many homogeneous coordinates. The equation of a line through the origin may be written where ''n'' and ''m'' are not both 0. In parametric form this can be written . Let ''Z'' = 1/''t'', so the coordinates of a point on the line may be written . In homogeneous coordinates this becomes . In the limit, as ''t'' approaches infinity, in other words, as the point moves away from the origin, ''Z'' approaches 0 and the homogeneous coordinates of the point become . Thus we define as the homogeneous coordinates of the point at infinity corresponding to the direction of the line . As any line of the Euclidean plane is parallel to a line passing through the origin, and since parallel lines have the same point at infinity, the infinite point on every line of the Euclidean plane has been given homogeneous coordinates. To summarize: *Any point in the projective plane is represented by a triple , called the homogeneous coordinates or projective coordinates of the point, where ''X'', ''Y'' and ''Z'' are not all 0. *The point represented by a given set of homogeneous coordinates is unchanged if the coordinates are multiplied by a common factor. *Conversely, two sets of homogeneous coordinates represent the same point if and only if one is obtained from the other by multiplying all the coordinates by the same nonzero constant. *When ''Z'' is not 0 the point represented is the point in the Euclidean plane. *When ''Z'' is 0 the point represented is a point at infinity. Note that the triple is omitted and does not represent any point. The origin is represented by .〔For the section: 〕 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「Homogeneous coordinates」の詳細全文を読む スポンサード リンク
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