a nonzero element of K. The homogeneous coordinates of a point and the coordinates of its image by φ are related by When the projective spaces are defined by adding points at infinity to affine spaces (projective completion) the preceding formulas become, in affine coordinates, which generalizes the expression of the homographic function of the next section. This defines only a partial function between affine spaces, which is defined only outside the hyperplane where the denominator is zero. Homographies of a projective line The projective line over a field K may be identified with the union of K and a point, called the "point at infinity" and denoted by ∞ (see projective line). With this representation of the projective line, the homographies are the mappings which are called homographic functions or linear fractional transformations. In the case of the complex projective line, which can be identified with the Riemann sphere, the homographies are called Möbius transformations. These correspond precisely with those bijections of the Riemann sphere that preserve orientation and are conformal. In the study of collineations, the case of projective lines is special due to the small dimension. When the line is viewed as a projective space in isolation, any permutation of the points of a projective line is a collineation, since every set of points are collinear. However, if the projective line is embedded in a higher-dimensional projective space, the geometric structure of that space can be used to impose a geometric structure on the line. Thus, in synthetic geometry, the homographies and the collineations of the projective line that are considered are those obtained by restrictions to the line of collineations and homographies of spaces of higher dimension. This means that the fundamental theorem of projective geometry (see below) remains valid in the one-dimensional setting. A homography of a projective line may also be properly defined by insisting that the mapping preserves cross-ratios. Projective frame and coordinates A projective frame or projective basis of a projective space of dimension is an ordered set of points such that no hyperplane contains of them. A projective frame is sometimes called a simplex, although a simplex in a space of dimension has at most vertices. Projective spaces over a commutative field are considered in this section, although most results may be generalized to projective spaces over a division ring. Let be a projective space of dimension , where is a