Plücker coordinates: Difference between revisions

Line 72:

A rational point <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>.

The same relations as above can be derived, where the ~~homogeneous~~ coordinates have entries in <math>\mathbb{Z}</math> instead.

The same relations as above can be derived, where we represent M as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.

~~Note that~~ in the ~~corresponding equivalence relation~~, the ~~matrices~~ <math>g \~~in GL_k (\mathbb~~{Z})</math> ~~have~~ <math>|\det (g)| = 1</math>.

Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.

== Height ==

A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated.

We can define the height of a rational point simply as the Euclidean norm on the Plücker coordinates.

$$

H(P) = \left\| P \right\| = \left\| m_1 \wedge \ldots \wedge m_n \right\| \\

$$

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.

$$

\begin{align}

\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\

\det(\mathrm{G}) &= \det(M^{\mathsf{T}} M) \\

\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| P \right\| \, .

\end{align}

$$

In regular temperament theory, this height is usually known as simply the [[Tenney-Euclidean temperament_measures #TE complexity|complexity]].

== Projective distance ==

Given a temperament, we want to have some notion of distance, so that we can measure how well the temperament approximates JI. Since we are talking about linear subspaces (which all intersect at the origin), the only thing that is sensible to measure is the angle between them.

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

$$

\frac{a \cdot b}{\left\| a \right\| \left\| b \right\| } = \cos (\theta)

$$

~~Because~~ the Plücker coordinates ~~are homogeneous~~, we can ~~always put them in a 'canonical' form by dividing all entries by their GCD and ensuring~~ the ~~first element is non~~-~~negative~~.

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

$$

d(P, j) = \frac{X \wedge y}{\left\| X \right\| \left\| y \right\| } = \sin (\theta)

$$

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>.

Unlike the dot product formula, this works for subspaces of any dimension.

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

@@ Line 72: / Line 72: @@
 A rational point <math>P</math> on <math>\mathrm{Gr}(k, n)</math> is a k-dimensional subspace such that <math>P \cap \mathbb{Z}^n</math> is a rank k sublattice of <math>\mathbb{Z}^n</math>.
-The same relations as above can be derived, where the homogeneous coordinates have entries in <math>\mathbb{Z}</math> instead.
+The same relations as above can be derived, where we represent M as integer matrix <math>M \in \mathbb{Z} ^ {k \times n}</math> and the projective coordinates similarly have entries in <math>\mathbb{Z}</math> instead.
-Note that in the corresponding equivalence relation, the matrices <math>g \in GL_k (\mathbb{Z})</math> have <math>|\det (g)| = 1</math>.
+Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.
+== Height ==
+A height function is a way to measure the 'arithmetic complexity' of a rational point. For example, the rational numbers <math>\frac{3}{2}</math> and <math>\frac{3001}{2001}</math> are close to eachother, but intuitively the second is much more complicated.
+We can define the height of a rational point simply as the Euclidean norm on the Plücker coordinates.
+$$
+	H(P) = \left\| P \right\| = \left\| m_1 \wedge \ldots \wedge m_n \right\| \\
+$$
+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.
+$$
+	\begin{align}
+	\mathrm{G}_{ij} &= \left\langle m_i, m_j \right\rangle \\
+	\det(\mathrm{G}) &= \det(M^{\mathsf{T}} M) \\
+	\sqrt{\det(\mathrm{G})} &= \left\| m_1 \wedge \ldots \wedge m_n \right\| = \left\| P \right\| \, .
+	\end{align}
+$$
+In regular temperament theory, this height is usually known as simply the [[Tenney-Euclidean temperament_measures #TE complexity|complexity]].
+== Projective distance ==
+Given a temperament, we want to have some notion of distance, so that we can measure how well the temperament approximates JI. Since we are talking about linear subspaces (which all intersect at the origin), the only thing that is sensible to measure is the angle between them.
+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
+$$
+	\frac{a \cdot b}{\left\| a \right\|  \left\| b \right\| } = \cos (\theta)
+$$
-Because the Plücker coordinates are homogeneous, we can always put them in a 'canonical' form by dividing all entries by their GCD and ensuring the first element is non-negative.
+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
+$$
+	d(P, j) = \frac{X \wedge y}{\left\| X \right\|  \left\| y \right\| } = \sin (\theta)
+$$
+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>.
+Unlike the dot product formula, this works for subspaces of any dimension.
+Since for any decent temperament this angle will be extremely small, we can take <math>\sin (\theta) \approx \theta</math>.