Plücker coordinates: Difference between revisions
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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}} | ||
In [[exterior algebra]] applied to [[regular temperament theory]], '''Plücker coordinates''' (also known as the | 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 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. | ||
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The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors: | The Plücker map takes a temperament and embeds it into a projective space by taking the wedge product of the basis vectors: | ||
:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\iota: \mathrm{Gr} (k, n) | \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) | \operatorname {span} (m_1, \ldots, m_k) | ||
& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | & \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | ||
\end{ | \end{aligned} | ||
</math> | </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>. | 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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An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix | An element <math>M</math> spanned by two lines <math>x, y</math>, can be represented as the matrix | ||
:<math> | :<math> | ||
\begin{ | \begin{bmatrix} | ||
x_{1} & x_{2} & x_{3} & x_{4} \\ | |||
y_{1} & y_{2} & y_{3} & y_{4} | |||
\end{bmatrix} \, . | |||
</math> | </math> | ||
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:<math> | :<math> | ||
\iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{23} : p_{24} : p_{34} \right] \, . | \iota (M) = \left[ x \wedge y \right] = \left[ p_{12} : p_{13} : p_{14} : p_{23} : p_{24} : p_{34} \right] \, . | ||
</math> | </math> | ||
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The height can be easily computed using the {{w|Gram matrix}}: | The height can be easily computed using the {{w|Gram matrix}}: | ||
:<math> | :<math> | ||
\begin{ | \begin{aligned} | ||
\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\ | \mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\ | ||
\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, . | \sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| X \right\| \, . | ||
\end{ | \end{aligned} | ||
</math> | </math> | ||
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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> | ||
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Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>. | 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]] | [[Category:Exterior algebra]] | ||