Lesson plan
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\{0\}\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_{n+1})\right), for some vectors e_0, \dots, e_{n+1} of . The definition implies the existence of nonzero elements of such that \lambda_0e_0 + \cdots + \lambda_{n+1}e_{n+1}=0 . Replacing e_i by \lambda_ie_i for i\le n and e_{n+1} by -\lambda_{n+1}e_{n+1} , 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.…
Article summary
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\{0\}\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_{n+1})\right), for some vectors e_0, \dots, e_{n+1} of . The definition implies the existence of nonzero elements of such that \lambda_0e_0 + \cdots + \lambda_{n+1}e_{n+1}=0 . Replacing e_i by \lambda_ie_i for i\le n and e_{n+1} by -\lambda_{n+1}e_{n+1} , 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.…
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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