Trait acap::distance::Distance[−][src]

pub trait Distance where
    Self: Copy,
    Self: Into<Self::Value>,
    Self: PartialOrd<Self::Value>,
    Self: PartialOrd, {
    type Value: Value + PartialOrd<Self>;
    fn value(self) -> Self::Value { ... }
}

Expand description

A distance between two points.

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. Implementors must satisfy, for all distances $x$ and $y$ :

\begin{aligned}
x.\mathrm{value}() &< y.\mathrm{value}() & &\iff& x.\mathrm{value}() &< y \\
& & &\iff& x &< y.\mathrm{value}() \\
& & &\iff& x &< y
\end{aligned}

Any monotonically increasing function can be used to create an order embedding. For example, EuclideanDistance holds a squared distance, rather than the distance itself. Comparisons still behave correctly because $x \mapsto x^2$ is an increasing function. This lets us avoid computing relatively expensive square roots until we need the value() itself, at which point the inverse function $x \mapsto \sqrt{x}$ must be applied.