passing through O and parallel to P. The notion of projective space was originally introduced by extending the Euclidean space, that is, by adding points at infinity to it, in order to define the projection for every point except O. Given another plane Q, which does not contain O, the restriction to Q of the above projection is called a perspectivity. With these definitions, a perspectivity is only a partial function, but it becomes a bijection if extended to projective spaces. Therefore, this notion is normally defined for projective spaces. The notion is also easily generalized to projective spaces of any dimension, over any field, in the following way: If f is a perspectivity from P to Q, and g a perspectivity from Q to P, with a different center, then is a homography from P to itself, which is called a central collineation, when the dimension of P is at least two. (See below and .) Originally, a homography was defined as the composition of a finite number of perspectivities. It is a part of the fundamental theorem of projective geometry (see below) that this definition coincides with the more algebraic definition sketched in the introduction and detailed below. Definition and expression in homogeneous coordinates A projective space P(V) of dimension n over a field K may be defined as the set of the lines through the origin in a K-vector space V of dimension . If a basis of V has been fixed, a point of V may be represented by a point of Kn+1. A point of P(V), being a line in V, may thus be represented by the coordinates of any nonzero point of this line, which are thus called homogeneous coordinates of the projective point. Given two projective spaces P(V) and P(W) of the same dimension, a homography is a mapping from P(V) to P(W), which is induced by an isomorphism of vector spaces . Such an isomorphism induces a bijection from P(V) to P(W), because of the linearity of f. Two such isomorphisms, f and g, define the same homography if and only if there is a nonzero element a of K such that . This may be written in terms of homogeneous coordinates in the following way: A homography φ may be defined by a nonsingular matrix [ai,j], called the matrix of the homography. This matrix is defined up to the multiplication by