Plücker coordinates: Difference between revisions
→See also: if anything the original page is more mathematically advanced lol |
m →Projective distance: Typo R^n |
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{{Expert}} | {{Expert|Wedgie}} | ||
[[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]] | [[File:Plucker_embedding.png|thumb|600px|right|Schematic illustration of the Plücker embedding. Linear subspaces of <math>\mathbb{R}^n</math> (here lines) get mapped to points on a quadric surface in projective space.]] | ||
{{Wikipedia|Plücker embedding}} | {{Wikipedia|Plücker embedding}} | ||
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In Euclidean space, one usually takes advantage of the dot product to measure angles. | In Euclidean space, one usually takes advantage of the dot product to measure angles. | ||
Given vectors <math>a, b \in \mathbb{R^n | Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have | ||
:<math> | :<math> | ||
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In projective space, there is an analogous formula, using the wedge product instead. | In projective space, there is an analogous formula, using the wedge product instead. | ||
Given some real point <math>j \in \mathbb{R^n | Given some real point <math>j \in \mathbb{R}^n</math> with homogeneous coordinates <math>y</math>, and a linear subspace <math>P \in \mathrm{Gr} (k, n)</math> with Plücker coordinates <math>X</math>, we define the projective distance as | ||
:<math> | :<math> | ||