docs: Integrate KaTeX and add some formulas

7 files changed, 116 insertions, 18 deletions

diff --git a/src/chebyshev.rs b/src/chebyshev.rs
index fa8e92c..335b6f1 100644
--- a/src/chebyshev.rs
+++ b/src/chebyshev.rs
@@ -44,6 +44,13 @@ impl<T: Coordinates> Coordinates for Chebyshev<T> {
 }
 
 /// Compute the Chebyshev distance between two points.
+///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{chebyshev\_distance}(x, y) &= \|x - y\|_\infty \\
+/// &= \max_i |x_i - y_i|
+/// \end{aligned}
+/// ```
 pub fn chebyshev_distance<T, U>(x: T, y: U) -> T::Value
 where
     T: Coordinates,
diff --git a/src/cos.rs b/src/cos.rs
index 3d3219c..5d8f73f 100644
--- a/src/cos.rs
+++ b/src/cos.rs
@@ -12,8 +12,16 @@ use std::cmp::Ordering;
 ///
 /// Use [cosine_distance] instead if you are implementing [Proximity::distance()].
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{cosine\_similarity}(x, y) &= \frac{x \cdot y}{\|x\| \|y\|} \\
+/// &= \frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2} \sqrt{\sum_i y_i^2}} \\
+/// &= \cos \theta
+/// \end{aligned}
+/// ```
+///
 /// [cosine *similarity*]: https://en.wikipedia.org/wiki/Cosine_similarity
-/// [Proximity::distance()]: Proximity#method.distance
+/// [Proximity::distance()]: Proximity#tymethod.distance
 pub fn cosine_similarity<T, U>(x: T, y: U) -> T::Value
 where
     T: Coordinates,
@@ -39,6 +47,15 @@ where
 
 /// Compute the [cosine distance] between two points.
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{cosine\_distance}(x, y) &= 1 - \mathrm{cosine\_similarity}(x, y) \\
+/// &= 1 - \frac{x \cdot y}{\|x\| \|y\|} \\
+/// &= 1 - \frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2} \sqrt{\sum_i y_i^2}} \\
+/// &= 1 - \cos \theta
+/// \end{aligned}
+/// ```
+///
 /// [cosine distance]: https://en.wikipedia.org/wiki/Cosine_similarity
 pub fn cosine_distance<T, U>(x: T, y: U) -> T::Value
 where
@@ -97,8 +114,16 @@ where
 ///
 /// Use [prenorm_cosine_distance] instead if you are implementing [Proximity::distance()].
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{prenorm\_cosine\_similarity}(x, y) &= x \cdot y \\
+/// &= \sum_i x_i y_i \\
+/// &= \cos \theta
+/// \end{aligned}
+/// ```
+///
 /// [cosine *similarity*]: https://en.wikipedia.org/wiki/Cosine_similarity
-/// [`Proximity::distance()`]: Proximity#method.distance
+/// [`Proximity::distance()`]: Proximity#tymethod.distance
 pub fn prenorm_cosine_similarity<T, U>(x: T, y: U) -> T::Value
 where
     T: Coordinates,
@@ -118,6 +143,15 @@ where
 
 /// Compute the [cosine distance] between two pre-normalized (unit magnitude) points.
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{prenorm\_cosine\_distance}(x, y) &= 1 - \mathrm{prenorm\_cosine\_similarity}(x, y) \\
+/// &= 1 - x \cdot y \\
+/// &= 1 - \sum_i x_i y_i \\
+/// &= 1 - \cos \theta
+/// \end{aligned}
+/// ```
+///
 /// [cosine distance]: https://en.wikipedia.org/wiki/Cosine_similarity
 pub fn prenorm_cosine_distance<T, U>(x: T, y: U) -> T::Value
 where
@@ -175,6 +209,15 @@ where
 
 /// Compute the [angular distance] between two points.
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{angular\_distance}(x, y) &= \arccos(\mathrm{cosine\_similarity}(x, y)) \\
+/// &= \arccos \left( \frac{x \cdot y}{\|x\| \|y\|} \right) \\
+/// &= \arccos \left( \frac{\sum_i x_i y_i}{\sqrt{\sum_i x_i^2} \sqrt{\sum_i y_i^2}} \right) \\
+/// &= \theta
+/// \end{aligned}
+/// ```
+///
 /// [angular distance]: https://en.wikipedia.org/wiki/Cosine_similarity#Angular_distance_and_similarity
 pub fn angular_distance<T, U>(x: T, y: U) -> AngularDistance<T::Value>
 where
@@ -257,6 +300,15 @@ where
 
 /// Compute the [angular distance] between two points.
 ///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{prenorm\_angular\_distance}(x, y) &= \arccos(\mathrm{prenorm\_cosine\_similarity}(x, y)) \\
+/// &= \arccos(x \cdot y) \\
+/// &= \arccos \left( \sum_i x_i y_i \right) \\
+/// &= \theta
+/// \end{aligned}
+/// ```
+///
 /// [angular distance]: https://en.wikipedia.org/wiki/Cosine_similarity#Angular_distance_and_similarity
 pub fn prenorm_angular_distance<T, U>(x: T, y: U) -> AngularDistance<T::Value>
 where
diff --git a/src/distance.rs b/src/distance.rs
index 9ff9fd4..20d862b 100644
--- a/src/distance.rs
+++ b/src/distance.rs
@@ -15,11 +15,15 @@ impl<T: Num + NumAssign + Signed + Copy + PartialOrd> Value for T {}
 /// An implementation may be an actual numerical distance, or an [order embedding] of the true
 /// distance.  This allows for optimizations whenever distances can be compared more efficiently
 /// than their exact values can be computed, as is the case for [Euclidean distance].  Implementors
-/// must satisfy, for all distances `x` and `y`:
+/// must satisfy, for all distances `$x$` and `$y$`:
 ///
-/// * `x < y` iff `x.value() < y.value()`
-/// * `x.value() < y` iff `x.value() < y.value()`
-/// * `x < y.value()` iff `x.value() < y.value()`
+/// ```math
+/// \begin{aligned}
+/// x.\mathrm{value}() &< y.\mathrm{value}() & &\iff& x.\mathrm{value}() &< y \\
+/// & & &\iff& x &< y.\mathrm{value}() \\
+/// & & &\iff& x &< y
+/// \end{aligned}
+/// ```
 ///
 /// [order embedding]: https://en.wikipedia.org/wiki/Order_embedding
 /// [Euclidean distance]: crate::euclid::EuclideanDistance
@@ -79,19 +83,26 @@ impl<'k, 'v, K: Proximity<V>, V> Proximity<&'v V> for &'k K {
 
 /// Marker trait for [metric spaces].
 ///
-/// A metric must be symmetric and obey the [triangle inequality].  More precisely, let `x`, `y`,
-/// and `z` be any elements of a metric space, and let `d(x, y) = x.distance(y).value()`.  Then the
-/// following rules must hold:
+/// A metric must be symmetric and obey the [triangle inequality].  More precisely, let `$x$`,
+/// `$y$`, and `$z$` be any elements of a metric space, and let
+/// `$d(x, y) = x.\mathrm{distance}(y).\mathrm{value}()$`.  Then the following rules must hold:
 ///
-/// * `d(x, x) == 0`,
-/// * `d(x, y) == d(y, z)` (symmetry), and
-/// * `d(x, z) <= d(x, y) + d(y, z)` (triangle inequality).
+/// ```math
+/// \begin{aligned}
+/// d(x, x) &= 0 \\
+/// d(x, y) &= d(y, x) & \text{(symmetry)} \\
+/// d(x, z) &\le d(x, y) + d(y, z) & \text{(triangle inequality)}
+/// \end{aligned}
+/// ```
 ///
 /// Those conditions also imply the following condition:
 ///
-/// * `d(x, y) >= 0` (non-negativity)
-///
-/// Because we do not prohibit `d(x, y) == 0` for distinct `x` and `y`, these spaces are more
+/// ```math
+/// \begin{aligned}
+/// d(x, y) &\ge \rlap{0}\phantom{d(x, y) + d(y, z)} & \text{\phantom{(triangle inequality)}\llap{(non-negativity)}}
+/// \end{aligned}
+/// ```
+/// Because we do not prohibit `$d(x, y) = 0$` for distinct `$x$` and `$y$`, these spaces are more
 /// properly known as [pseudometric spaces].  This distinction is usually unimportant.
 ///
 /// [metric spaces]: https://en.wikipedia.org/wiki/Metric_space
diff --git a/src/euclid.rs b/src/euclid.rs
index 4f2309a..0f2281e 100644
--- a/src/euclid.rs
+++ b/src/euclid.rs
@@ -47,6 +47,13 @@ impl<T: Coordinates> Coordinates for Euclidean<T> {
 }
 
 /// Compute the Euclidean distance between two points.
+///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{euclidean\_distance}(x, y) &= \|x - y\|_2 \\
+/// &= \sqrt{\sum_i (x_i - y_i)^2}
+/// \end{aligned}
+/// ```
 pub fn euclidean_distance<T, U>(x: T, y: U) -> EuclideanDistance<T::Value>
 where
     T: Coordinates,
diff --git a/src/hamming.rs b/src/hamming.rs
index c6822ee..da959b4 100644
--- a/src/hamming.rs
+++ b/src/hamming.rs
@@ -25,6 +25,13 @@ impl<T> Hamming<T> {
 }
 
 /// Compute the Hamming distance between two integers.
+///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{hamming\_distance}(x, y) &= |\{i \mid x_i \ne y_i\}| \\
+/// &= \mathrm{popcount}(x \wedge y)
+/// \end{aligned}
+/// ```
 pub fn hamming_distance<T: PrimInt>(x: T, y: T) -> i32 {
     (x ^ y).count_ones() as i32
 }
diff --git a/src/lp.rs b/src/lp.rs
index 80a6f8a..4afd209 100644
--- a/src/lp.rs
+++ b/src/lp.rs
@@ -1,4 +1,4 @@
-//! [L<sup>p</sup> spaces](https://en.wikipedia.org/wiki/Lp_space).
+//! [`$L^p$` spaces](https://en.wikipedia.org/wiki/Lp_space).
 
 use crate::coords::Coordinates;
 
@@ -25,9 +25,16 @@ pub use crate::chebyshev::Chebyshev as Linf;
 /// Compute the L<sup>∞</sup> distance between two points.
 pub use crate::chebyshev::chebyshev_distance as linf_distance;
 
-/// Compute the [L<sup>p</sup> distance] between two points.
+/// Compute the [`$L^p$` distance] between two points.
 ///
-/// [L<sup>p</sup> distance]: https://en.wikipedia.org/wiki/Lp_space
+/// ```math
+/// \begin{aligned}
+/// \mathrm{lp\_distance}(p, x, y) &= \|x - y\|_p \\
+/// &= \left( \sum_i |x_i - y_i|^p \right)^{1/p}
+/// \end{aligned}
+/// ```
+///
+/// [`$L^p$` distance]: https://en.wikipedia.org/wiki/Lp_space
 pub fn lp_distance<T, U>(p: T::Value, x: T, y: U) -> T::Value
 where
     T: Coordinates,
diff --git a/src/taxi.rs b/src/taxi.rs
index 83ebccf..f22afb0 100644
--- a/src/taxi.rs
+++ b/src/taxi.rs
@@ -44,6 +44,13 @@ impl<T: Coordinates> Coordinates for Taxicab<T> {
 }
 
 /// Compute the taxicab distance between two points.
+///
+/// ```math
+/// \begin{aligned}
+/// \mathrm{taxicab\_distance}(x, y) &= \|x - y\|_1 \\
+/// &= \sum_i |x_i - y_i|
+/// \end{aligned}
+/// ```
 pub fn taxicab_distance<T, U>(x: T, y: U) -> T::Value
 where
     T: Coordinates,