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.]]

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.

In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the '''wedgie''') are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.

The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.

The definition here is given in terms of temperament matrices, but by duality, we can also embed interval spaces in the same way.

More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.

These two representations are related via the [[Hodge dual]].

== Definition ==

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\begin{align}

\iota: \mathrm{Gr} (k, n)

& \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\

& \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\

\operatorname {span} (m_1, \ldots, m_k)

& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .

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</math>

Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.

Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.

While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.

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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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Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.

== See also ==

* [[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

* [[Hodge dual]] - acts on wedgies

[[Category:Exterior algebra]]

@@ Line 1: / Line 1: @@
-{{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}}
-In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the [[wedgie]]) are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.
+In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the '''wedgie''') are a way to assign coordinates to abstract temperaments, by viewing them as elements of some projective space.
 The usual way to write down an abstract temperament is via its mapping matrix, but Plücker coordinates give us a unique description that is useful for some calculations.
+The definition here is given in terms of temperament matrices, but by duality, we can also embed interval spaces in the same way.
+More specifically, the interval subspace spanned by the commas of some temperament can also be used to give unique coordinates to that temperament.
+These two representations are related via the [[Hodge dual]].
 == Definition ==
@@ Line 19: / Line 22: @@
 \begin{align}
 \iota: \mathrm{Gr} (k, n)
-	& \to \mathbf{P}\left(\Lambda^{k} \mathbb{R}^n \right) \\
+	& \to \mathbf{P}\left(\Lambda^{k} \, \mathbb{R}^n \right) \\
 \operatorname {span} (m_1, \ldots, m_k)
 	& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, .
@@ Line 25: / Line 28: @@
 </math>
-Here, <math>\Lambda^{k} \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
+Here, <math>\Lambda^{k} \, \mathbb{R}^n</math> is the k-th exterior power (the subspace containing all k-vectors). This construction is independent of the basis we choose.
 While the original space of temperaments has dimension <math>k(n-k)</math>, the space of Plücker coordinates is typically larger, with dimension <math>\binom{n}{k} - 1</math>.
@@ Line 116: / 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 123: / 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 133: / Line 136: @@
 Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.
+== See also ==
+* [[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
+* [[Hodge dual]] - acts on wedgies
 [[Category:Exterior algebra]]