Definition
A metric map is a term with two primary meanings depending on context.
In Mathematics (Metric Space Theory)
In mathematical analysis, a metric map is a function between metric spaces that does not increase any distance. Specifically, if $X$ and $Y$ are metric spaces with metrics $d_X$ and $d_Y$ respectively, a function $f: X \to Y$ is a metric map when, for any points $x$ and $y$ in $X$,
$$ d_Y(f(x), f(y)) \leq d_X(x, y). $$
Such functions are always continuous. They are also called Lipschitz functions with Lipschitz constant 1, nonexpansive maps, nonexpanding maps, weak contractions, or short maps.
Category of Metric Maps
Metric maps serve as the morphisms in the category of metric spaces, denoted Met. The composition of two metric maps is another metric map, and the identity map on a metric space is a metric map. A map between metric spaces is an isometry if and only if it is a bijective metric map whose inverse is also a metric map; thus the isomorphisms in Met are precisely the isometries.
Example
Consider the metric space $[0, 1/2]$ with the Euclidean metric. The function $f(x) = x^2$ is a metric map because for $x eq y$,
$$ |f(x) - f(y)| = |x + y|,|x - y| < |x - y|. $$
In Cartography and Surveying
In mapping and surveying, a metric map is a map on which measurements can be made to support mathematical computations. A critical attribute of a metric map is its scale, which allows map measurements to be related to their real-world counterparts. This contrasts with non-metric maps, which convey information visually without necessarily supporting precise quantitative measurements.
In Robotics
In autonomous robotics, a metric map is a representation of an environment that quantifies spatial relationships using distance measurements, typically Euclidean distances. It enables robots to navigate and localize themselves within a given space by relying on known distances to landmarks or features, facilitating path planning and obstacle avoidance.
