Plücker coordinates: Difference between revisions
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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> | |||
\begin{align} | \begin{align} | ||
\iota: \mathrm{Gr} (k, n) | \iota: \mathrm{Gr} (k, n) | ||
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& \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | & \mapsto \left[ m_1 \wedge \ldots \wedge m_k \right] \, . | ||
\end{align} | \end{align} | ||
</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. | ||
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The simplest non-trivial case is <math>\mathrm{Gr} (2, 4)</math>. | The simplest non-trivial case is <math>\mathrm{Gr} (2, 4)</math>. | ||
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> | |||
\begin{equation} | \begin{equation} | ||
\begin{bmatrix} | \begin{bmatrix} | ||
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\end{bmatrix} \, . | \end{bmatrix} \, . | ||
\end{equation} | \end{equation} | ||
</math> | |||
These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span. | These are not 'proper' coordinates, as doing row operations on this matrix preserves the row-span. | ||
The projective coordinates can be calculated by taking the determinants of all <math>2 \times 2</math> sub-matrices | The projective coordinates can be calculated by taking the determinants of all <math>2 \times 2</math> sub-matrices | ||
:<math> | |||
p_{ij} = | p_{ij} = | ||
\begin{vmatrix} | \begin{vmatrix} | ||
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y_i & y_j | y_i & y_j | ||
\end{vmatrix} \, , | \end{vmatrix} \, , | ||
</math> | |||
which finally gives us | which finally gives us | ||
:<math> | |||
\begin{equation} | \begin{equation} | ||
\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] \, . | ||
\end{equation} | \end{equation} | ||
</math> | |||
Note the use of colons to signify that these coordinates are homogeneous. | Note the use of colons to signify that these coordinates are homogeneous. | ||
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For the example above on <math>\mathrm{Gr} (2, 4)</math>, the Plücker relation is | For the example above on <math>\mathrm{Gr} (2, 4)</math>, the Plücker relation is | ||
:<math> | |||
p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0 \, . | p_{12} p_{34} - p_{13} p_{24} + p_{14} p_{23} = 0 \, . | ||
</math> | |||
Note that in this case, there is only one such relation, but in higher dimensions there will be many. | Note that in this case, there is only one such relation, but in higher dimensions there will be many. | ||
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We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>. | We can define the height of a rational point simply as the Euclidean norm on its Plücker coordinates <math>X = \iota (P)</math>. | ||
:<math> | |||
H(P) = \left\| X \right\| = \left\| p_1 \wedge \ldots \wedge p_n \right\| \\ | H(P) = \left\| X \right\| = \left\| p_1 \wedge \ldots \wedge p_n \right\| \\ | ||
</math> | |||
This is equivalent to the covolume of the lattice defined by P (also know as the lattice determinant), which can be easily computed using the Gram matrix. | This is equivalent to the covolume of the lattice defined by P (also know as the lattice determinant), which can be easily computed using the Gram matrix. | ||
:<math> | |||
\begin{align} | \begin{align} | ||
\mathrm{G}_{ij} &= \left\langle p_i, p_j \right\rangle \\ | \mathrm{G}_{ij} &= \left\langle p_i, p_j \right\rangle \\ | ||
\sqrt{\det(\mathrm{G})} &= \left\| p_1 \wedge \ldots \wedge p_n \right\| = \left\| X \right\| \, . | \sqrt{\det(\mathrm{G})} &= \left\| p_1 \wedge \ldots \wedge p_n \right\| = \left\| X \right\| \, . | ||
\end{align} | \end{align} | ||
</math> | |||
In regular temperament theory, this height is usually known as simply the [[Tenney-Euclidean temperament_measures #TE complexity|complexity]]. | In regular temperament theory, this height is usually known as simply the [[Tenney-Euclidean temperament_measures #TE complexity|complexity]]. | ||
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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> | |||
\frac{a \cdot b}{\left\| a \right\| \left\| b \right\| } = \cos (\theta) \, . | \frac{a \cdot b}{\left\| a \right\| \left\| b \right\| } = \cos (\theta) \, . | ||
</math> | |||
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}</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> | |||
d(P, j) = \frac{ \left\| X \wedge y \right\| }{\left\| X \right\| \left\| y \right\| } = \sin (\theta) \, . | d(P, j) = \frac{ \left\| X \wedge y \right\| }{\left\| X \right\| \left\| y \right\| } = \sin (\theta) \, . | ||
</math> | |||
Where we can take <math>j</math> to be the usual n-limit vector of log primes, so that <math> y = \left[ 1 : \log_2 (3) : \ldots : \log_2 (p_n) \right] </math>. | Where we can take <math>j</math> to be the usual n-limit vector of log primes, so that <math> y = \left[ 1 : \log_2 (3) : \ldots : \log_2 (p_n) \right] </math>. | ||