In mathematics, and more specifically in projective geometry, a projective frame or projective basis is a tuple of points in a projective space that can be used for defining homogeneous coordinates in this space. More precisely, in a projective space of dimension, a projective frame is a -tuple 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. In this article, only projective spaces over a field are considered, although most results can be generalized to projective spaces over a division ring. Let be a projective space of dimension, where is a -vector space of dimension. Let be the canonical projection that maps a nonzero vector to the corresponding point of, which is the vector line that contains. Every frame of can be written as for some vectors of. The definition implies the existence of nonzero elements of such that. Replacing by for and by, one gets the following characterization of a frame: Moreover, two bases define the same frame in this way, if and only ifthe elements of the second one are the products of the elements of the first one by a fixed nonzero element of. As homographies of are induced by linear endomorphisms of, it follows that, given two frames, there is exactly onehomography mapping the first one onto the second one. In particular, the only homography fixing the points of a frame is the identity map. This result is much more difficult in synthetic geometry. It is sometimes called the first fundamental theorem of projective geometry. Every frame can be written as where is basis of. The projective coordinates or homogeneous coordinates of a point over this frame are the coordinates of the vector on the basis If one changes the vectors representing the point and the frame elements, the coordinates are multiplied by a fixed nonzero scalar. Commonly, the projective space is considered. It has a canonical frameconsisting of the image by of the canonical basis of , and. On this basis, the homogeneous coordinates of are simply the entries of. Given another projective space of the same dimension, and a frame of it, there is exactly one homography mapping onto the canonical frame of. The projective coordinates of a point on the frame are the homogeneous coordinates of on the canonical frame of. In the case of a projective line, a frame consists of three distinct points. If is identified with with a point at infinity added, then its canonical frame is. Given any frame ), the projective coordinates of a point are, where is the cross-ratio. If, the cross ratio is the infinity, and the projective coordinates are.
