diff options
Diffstat (limited to 'src/distance.rs')
| -rw-r--r-- | src/distance.rs | 37 |
1 files changed, 24 insertions, 13 deletions
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 |