mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, and more specifically in

projective geometry In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting (''p ...

, a projective frame or projective basis is a

tuple In mathematics, a tuple is a finite sequence or ''ordered list'' of numbers or, more generally, mathematical objects, which are called the ''elements'' of the tuple. An -tuple is a tuple of elements, where is a non-negative integer. There is o ...

of points in a

projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...

that can be used for defining

homogeneous coordinates In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. ...

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 geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. ...

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 In algebra, a division ring, also called a skew field (or, occasionally, a sfield), is a nontrivial ring in which division by nonzero elements is defined. Specifically, it is a nontrivial ring in which every nonzero element has a multiplicativ ...

. 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

\lambda_ie_i

for

i\le n

and

e_

-\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 Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unc ...

. This result is much more difficult in

synthetic geometry Synthetic geometry (sometimes referred to as axiomatic geometry or even pure geometry) is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulates ...

(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

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 In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defin ...

. If , the cross ratio is the infinity, and the projective coordinates are .

Notes

References

* * {{DEFAULTSORT:Projective Frame Projective geometry