Matches in DBpedia 2016-04 for { ?s ?p "In mathematics, an affine coordinate system is a coordinate system on an affine space where each coordinate is an affine map to the number line. In other words, it is an injective affine map from an affine space A to the coordinate space Kn, where K is the field of scalars, for example, the real numbers R. The most important case of affine coordinates in Euclidean spaces is real-valued Cartesian coordinate system. Orthogonal affine coordinate systems are rectangular, and others are referred to as oblique.A system of n coordinates on n-dimensional space is defined by a (n+1)-tuple (O, R1, … Rn) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula: (x1, … xn) ↦ O + x1 (R1 − O) + … + xn (Rn − O) .Note that Rj − O are difference vectors with the origin in O and ends in Rj .An affine space cannot have a coordinate system with n less than its dimension, but n may indeed be greater, which means that the coordinate map is not necessary surjective.Examples of n-coordinate system in an (n−1)-dimensional space are barycentric coordinates and affine \"homogeneous\" coordinates (1, x1, … , xn−1). In the latter case the x0 coordinate is equal to 1 on all space, but this \"reserved\" coordinate allows for matrix representation of affine maps similar to one used for projective maps."@en }
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- Affine_coordinate_system abstract "In mathematics, an affine coordinate system is a coordinate system on an affine space where each coordinate is an affine map to the number line. In other words, it is an injective affine map from an affine space A to the coordinate space Kn, where K is the field of scalars, for example, the real numbers R. The most important case of affine coordinates in Euclidean spaces is real-valued Cartesian coordinate system. Orthogonal affine coordinate systems are rectangular, and others are referred to as oblique.A system of n coordinates on n-dimensional space is defined by a (n+1)-tuple (O, R1, … Rn) of points not belonging to any affine subspace of a lesser dimension. Any given coordinate n-tuple gives the point by the formula: (x1, … xn) ↦ O + x1 (R1 − O) + … + xn (Rn − O) .Note that Rj − O are difference vectors with the origin in O and ends in Rj .An affine space cannot have a coordinate system with n less than its dimension, but n may indeed be greater, which means that the coordinate map is not necessary surjective.Examples of n-coordinate system in an (n−1)-dimensional space are barycentric coordinates and affine \"homogeneous\" coordinates (1, x1, … , xn−1). In the latter case the x0 coordinate is equal to 1 on all space, but this \"reserved\" coordinate allows for matrix representation of affine maps similar to one used for projective maps.".