Projective frame
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 Template:Math, a projective frame is a Template:Math-tuple of points such that no hyperplane contains Template:Math of them. A projective frame is sometimes called a simplex,Template:Sfn although a simplex in a space of dimension Template:Math has at most Template:Math vertices.
In this article, only projective spaces over a field Template:Math are considered, although most results can be generalized to projective spaces over a division ring.
Let Template:Math be a projective space of dimension Template:Math, where Template:Math is a Template:Math-vector space of dimension Template:Math. Let <math>p:V\setminus\{0\}\to \mathbf P(V)</math> be the canonical projection that maps a nonzero vector Template:Mvar to the corresponding point of Template:Math, which is the vector line that contains Template:Mvar.
Every frame of Template:Math can be written as <math>\left(p(e_0), \ldots, p(e_{n+1})\right),</math> for some vectors <math>e_0, \dots, e_{n+1}</math> of Template:Mvar. The definition implies the existence of nonzero elements of Template:Math such that <math>\lambda_0e_0 + \cdots + \lambda_{n+1}e_{n+1}=0</math>. Replacing <math>e_i</math> by <math>\lambda_ie_i</math> for <math>i\le n</math> and <math>e_{n+1}</math> by <math>-\lambda_{n+1}e_{n+1}</math>, one gets the following characterization of a frame:
- Template:Math points of Template:Math form a frame if and only if they are the image by Template:Mvar of a basis of Template:Mvar and the sum of its elements.
Moreover, two bases define the same frame in this way, if and only if the elements of the second one are the products of the elements of the first one by a fixed nonzero element of Template:Math.
As homographies of Template:Math are induced by linear endomorphisms of Template:Mvar, it follows that, given two frames, there is exactly one homography 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 (where projective spaces are defined through axioms). It is sometimes called the first fundamental theorem of projective geometry. Template:Sfn
Every frame can be written as <math>(p(e_0), \ldots, p(e_n), p(e_0+\cdots+e_n)),</math> where <math>(e_0, \dots, e_n)</math> is basis of Template:Mvar. The projective coordinates or homogeneous coordinates of a point Template:Math over this frame are the coordinates of the vector Template:Mvar on the basis <math>(e_0, \dots, e_n).</math> If one changes the vectors representing the point Template:Math and the frame elements, the coordinates are multiplied by a fixed nonzero scalar.
Commonly, the projective space Template:Math is considered. It has a canonical frame consisting of the image by Template:Math of the canonical basis of Template:Math (consisting of the elements having only one nonzero entry, which is equal to 1), and Template:Math. On this basis, the homogeneous coordinates of Template:Math are simply the entries (coefficients) of Template:Math.
Given another projective space Template:Math of the same dimension Template:Mvar, and a frame Template:Math of it, there is exactly one homography Template:Math mapping Template:Math onto the canonical frame of Template:Math. The projective coordinates of a point Template:Math on the frame Template:Math are the homogeneous coordinates of Template:Math on the canonical frame of Template:Math.
In the case of a projective line, a frame consists of three distinct points. If Template:Math is identified with Template:Mvar with a point at infinity Template:Math added, then its canonical frame is Template:Math. Given any frame Template:Math), the projective coordinates of a point Template:Math are Template:Math, where Template:Mvar is the cross-ratio Template:Math. If Template:Math, the cross ratio is the infinity, and the projective coordinates are Template:Math.