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 , 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 $p:V\setminus\\to \mathbf P(V)$ 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 $\left(p(e_0), \ldots, p(e_)\right),$ for some vectors $e_0, \dots, e_$ of . The definition implies the existence of nonzero elements of such that $\lambda_0e_0 + \cdots + \lambda_e_=0$. Replacing $e_i$ by $\lambda_ie_i$ for $i\le n$ and $e_$ by $-\lambda_e_$, one gets the following characterization of a frame:
: points of form a frame if and only if they are the image by of a basis of 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 .

As homographies of are induced by linear endomorphisms of , 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''.

Every frame can be written as $(p(e_0), \ldots, p(e_n), p(e_0+\cdots+e_n)),$ where $(e_0, \dots, e_n)$ is basis of . The ''projective coordinates'' or homogeneous coordinates of a point over this frame are the coordinates of the vector on the basis $(e_0, \dots, e_n).$ 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 frame'' consisting of the image by of the canonical basis of (consisting of the elements having only one nonzero entry, which is equal to 1), and . On this basis, the homogeneous coordinates of are simply the entries (coefficients) 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 .

References

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* , translated from the 1977 French original by M. Cole and S. Levy, fourth printing of the 1987 English translation

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Projective geometry