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Projective frame

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In <a href="Mathematics" title="Mathematics">mathematics</a>, and more specifically in <a href="Projective_geometry" title="Projective geometry">projective geometry</a>, a projective frame or projective basis is a <a href="Tuple" title="Tuple">tuple</a> of points in a <a href="Projective_space" title="Projective space">projective space</a> that can be used for defining <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> in this space. More precisely, in a projective space of dimension n, a projective frame is a n + 2-tuple of points such that no <a href="Hyperplane" title="Hyperplane">hyperplane</a> contains n + 1 of them. A projective frame is sometimes called a simplex,<a href="#cite_note-FOOTNOTEBaer200566-1">[1]</a> although a <a href="Simplex" title="Simplex">simplex</a> in a space of dimension n has at most n + 1 vertices.

In this article, only projective spaces over a field K are considered, although most results can be generalized to projective spaces over a <a href="Division_ring" title="Division ring">division ring</a>.

Let P(V) be a projective space of dimension n, where V is a K-vector space of dimension n + 1. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)}">

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<annotation encoding="application/x-tex">{\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)}</annotation>

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</math><img src="./398800fafbac541592c06b20a7ccfb3da9e8a771.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:19.703ex; height:2.843ex;" alt="{\displaystyle p:V\setminus \{0\}\to \mathbf {P} (V)}" loading="lazy"> be the canonical projection that maps a nonzero vector v to the corresponding point of P(V), which is the vector line that contains v.

Every frame of P(V) can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(p(e_{0}),\ldots ,p(e_{n+1})\right),}">

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<annotation encoding="application/x-tex">{\displaystyle \left(p(e_{0}),\ldots ,p(e_{n+1})\right),}</annotation>

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</math><img src="./86128093acf0edab6b2c269b566dbc92485db46b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.519ex; height:2.843ex;" alt="{\displaystyle \left(p(e_{0}),\ldots ,p(e_{n+1})\right),}" loading="lazy"> for some vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{0},\dots ,e_{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{0},\dots ,e_{n+1}}</annotation>

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</math><img src="./c6986df762d5b2ebe8fb274e997462e815e2718e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.718ex; height:2.009ex;" alt="{\displaystyle e_{0},\dots ,e_{n+1}}" loading="lazy"> of V. The definition implies the existence of nonzero elements of K such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{0}e_{0}+\cdots +\lambda _{n+1}e_{n+1}=0}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{0}e_{0}+\cdots +\lambda _{n+1}e_{n+1}=0}</annotation>

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</math><img src="./3b4b3c5d6626579547aca4e6e0c038ec88a5e1bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:26.289ex; height:2.509ex;" alt="{\displaystyle \lambda _{0}e_{0}+\cdots +\lambda _{n+1}e_{n+1}=0}" loading="lazy">. Replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

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</math><img src="./ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}e_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle \lambda _{i}e_{i}}</annotation>

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</math><img src="./5ffe6e632716927c61a636c56f94cdec3bb4179c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.038ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}e_{i}}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i\leq n}">

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<annotation encoding="application/x-tex">{\displaystyle i\leq n}</annotation>

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</math><img src="./1c372fbd81639c801a3397197b81a310bd207b1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.296ex; height:2.343ex;" alt="{\displaystyle i\leq n}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{n+1}}</annotation>

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</math><img src="./64e303c67926f259d6ce56a6c66e6168927be8c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.402ex; height:2.009ex;" alt="{\displaystyle e_{n+1}}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\lambda _{n+1}e_{n+1}}">

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<annotation encoding="application/x-tex">{\displaystyle -\lambda _{n+1}e_{n+1}}</annotation>

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</math><img src="./643288c9404b2d7f40167ad91ee6415c25c70ab2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.885ex; height:2.509ex;" alt="{\displaystyle -\lambda _{n+1}e_{n+1}}" loading="lazy">, one gets the following characterization of a frame:

<dl><dd>n + 2 points of P(V) form a frame if and only if they are the image by p of a basis of V and the sum of its elements.</dd></dl>

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 K.

As <a href="Homography" title="Homography">homographies</a> of P(V) are induced by linear endomorphisms of V, 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 <a href="Identity_map" class="mw-redirect" title="Identity map">identity map</a>. This result is much more difficult in <a href="Synthetic_geometry" title="Synthetic geometry">synthetic geometry</a> (where projective spaces are defined through axioms). It is sometimes called the first fundamental theorem of projective geometry.

Every frame can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),}">

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<annotation encoding="application/x-tex">{\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),}</annotation>

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</math><img src="./6aa38b2896f46fba8d26fe910c796df840e60942.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.888ex; height:2.843ex;" alt="{\displaystyle (p(e_{0}),\ldots ,p(e_{n}),p(e_{0}+\cdots +e_{n})),}" loading="lazy"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{0},\dots ,e_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle (e_{0},\dots ,e_{n})}</annotation>

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</math><img src="./592555aa7ba6b6558db5765ef7d603da6fd2f6de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.427ex; height:2.843ex;" alt="{\displaystyle (e_{0},\dots ,e_{n})}" loading="lazy"> is basis of V. The projective coordinates or <a href="Homogeneous_coordinates" title="Homogeneous coordinates">homogeneous coordinates</a> of a point p(v) over this frame are the coordinates of the vector v on the basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (e_{0},\dots ,e_{n}).}">

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<annotation encoding="application/x-tex">{\displaystyle (e_{0},\dots ,e_{n}).}</annotation>

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</math><img src="./1e434bca67cae992620c33cb8304c4c5872e65d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.074ex; height:2.843ex;" alt="{\displaystyle (e_{0},\dots ,e_{n}).}" loading="lazy"> If one changes the vectors representing the point p(v) and the frame elements, the coordinates are multiplied by a fixed nonzero scalar.

Commonly, the projective space Pn(K) = P(Kn+1) is considered. It has a canonical frame consisting of the image by p of the canonical basis of Kn+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 of p(v) are simply the entries (coefficients) of v.

Given another projective space P(V) of the same dimension n, and a frame F of it, there is exactly one homography h mapping F onto the canonical frame of P(Kn+1). The projective coordinates of a point a on the frame F are the homogeneous coordinates of h(a) on the canonical frame of Pn(K).

In the case of a projective line, a frame consists of three distinct points. If P1(K) is identified with K with a point at infinity ∞ added, then its canonical frame is (∞, 0, 1). Given any frame (a0, a1, a2), the projective coordinates of a point a ≠ a0 are (r, 1), where r is the <a href="Cross-ratio" title="Cross-ratio">cross-ratio</a> (a, a2; a1, a0). If a = a0, the cross ratio is the infinity, and the projective coordinates are (1,0).

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-FOOTNOTEBaer200566-1"><a href="#cite_ref-FOOTNOTEBaer200566_1-0">^</a> <a href="#CITEREFBaer2005">Baer 2005</a>, p. 66.

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<li id="cite_note-FOOTNOTEBerger2009chapter_6-2"><a href="#cite_ref-FOOTNOTEBerger2009chapter_6_2-0">^</a> <a href="#CITEREFBerger2009">Berger 2009</a>, chapter 6.

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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFBaer2005" class="citation book cs1">Baer, Reinhold (2005). Linear Algebra and Projective Geometry. Courier Corporation. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-44565-6</bdi>.</cite></li>

<li><cite id="CITEREFBerger2009" class="citation book cs1">Berger, Marcel (2009). Geometry I. Berlin Heidelberg: Springer Science & Business Media. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-540-11658-5</bdi>.</cite></li></ul></div><div><div class="zim-footer">

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