Inmathematics, and more specifically inprojective geometry, aprojective frame orprojective basis is atuple of points in aprojective space that can be used for defininghomogeneous coordinates in this space. More precisely, in a projective space of dimensionn, a projective frame is an + 2-tuple of points such that nohyperplane containsn + 1 of them. A projective frame is sometimes called asimplex,^[1] although asimplex in a space of dimensionn has at mostn + 1 vertices.

In this article, only projective spaces over a fieldK are considered, although most results can be generalized to projective spaces over adivision ring.

LetP(V) be a projective space of dimensionn, whereV is aK-vector space of dimensionn + 1. Let${\displaystyle p:V\setminus \{0\}\to \mathbf{P}(V)}$ be the canonical projection that maps a nonzero vectorv to the corresponding point ofP(V), which is the vector line that containsv.

Every frame ofP(V) can be written as${\displaystyle \left(p(e_0),\dots ,p(e_{n+1})\right),}$ for some vectors${\displaystyle e_0,\dots ,e_{n+1}}$ ofV. The definition implies the existence of nonzero elements ofK such that${\displaystyle {\lambda }_0{e}_0+\cdots +{\lambda }_{n+1}{e}_{n+1}=0}$. Replacing${\displaystyle e_i}$ by${\displaystyle {\lambda }_i{e}_i}$ for${\displaystyle i\le n}$ and${\displaystyle e_{n+1}}$ by${\displaystyle -{\lambda }_{n+1}{e}_{n+1}}$, one gets the following characterization of a frame:

n + 2 points ofP(V) form a frame if and only if they are the image byp of a basis ofV 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 ofK.

Ashomographies ofP(V) are induced by linear endomorphisms ofV, 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 theidentity map. This result is much more difficult insynthetic geometry (where projective spaces are defined through axioms). It is sometimes called thefirst fundamental theorem of projective geometry.^[2]

Every frame can be written as${\displaystyle (p(e_0),\dots ,p(e_n),p(e_0+\cdots +e_n)),}$ where${\displaystyle (e_0,\dots ,e_n)}$ is basis ofV. Theprojective coordinates orhomogeneous coordinates of a pointp(v) over this frame are the coordinates of the vectorv on the basis${\displaystyle (e_0,\dots ,e_n).}$ If one changes the vectors representing the pointp(v) and the frame elements, the coordinates are multiplied by a fixed nonzero scalar.

Commonly, the projective spaceP_n(K) =P(Kⁿ⁺¹) is considered. It has acanonical frame consisting of the image byp of the canonical basis ofKⁿ⁺¹ (consisting of the elements having only one nonzero entry, which is equal to 1), and(1, 1, ..., 1). On this basis, the homogeneous coordinates ofp(v) are simply the entries (coefficients) ofv.

Given another projective spaceP(V) of the same dimensionn, and a frameF of it, there is exactly one homographyh mappingF onto the canonical frame ofP(Kⁿ⁺¹). The projective coordinates of a pointa on the frameF are the homogeneous coordinates ofh(a) on the canonical frame ofP_n(K).

In the case of a projective line, a frame consists of three distinct points. IfP₁(K) is identified withK with a point at infinity∞ added, then its canonical frame is(∞, 0, 1). Given any frame(a₀,a₁,a₂), the projective coordinates of a pointa ≠a₀ are(r, 1), wherer is thecross-ratio(a,a₂;a₁,a₀). Ifa =a₀, the cross ratio is the infinity, and the projective coordinates are(1,0).

• Baer, Reinhold (2005).Linear Algebra and Projective Geometry. Courier Corporation.ISBN 978-0-486-44565-6.
• Berger, Marcel (2009).Geometry I. Berlin Heidelberg: Springer Science & Business Media.ISBN 978-3-540-11658-5.

• Projective geometry

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