Plücker coordinates: Difference between revisions

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[[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.]]

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In Euclidean space, one usually takes advantage of the dot product to measure angles.

Given vectors <math>a, b \in \mathbb{R^n}</math>, we famously have

Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have

:<math>

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In projective space, there is an analogous formula, using the wedge product instead.

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

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>

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== See also ==

* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies~~, including a more mathematically advanced description~~

* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies

* [[Exterior algebra]] - exterior product, which produces wedgies

* [[Interior product]] - interior product, dual of the exterior product

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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.]]
 {{Wikipedia|Plücker embedding}}
@@ Line 119: / Line 119: @@
 In Euclidean space, one usually takes advantage of the dot product to measure angles.
-Given vectors <math>a, b \in \mathbb{R^n}</math>, we famously have
+Given vectors <math>a, b \in \mathbb{R}^n</math>, we famously have
 :<math>
@@ Line 126: / Line 126: @@
 In projective space, there is an analogous formula, using the wedge product instead.
-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
+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>
@@ Line 139: / Line 139: @@
 == See also ==
-* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies, including a more mathematically advanced description
+* [[Wedgie supplement]] - Supplementary page going over additional information on wedgies
 * [[Exterior algebra]] - exterior product, which produces wedgies
 * [[Interior product]] - interior product, dual of the exterior product