In order to calculate and sort things by distance from each other in a computer program I need to find an easy and fast distance metric to calculate. I can probably find one myself. However, first I need to know a term I can use to search for this stuff online. I want a term for a measurement of distance that can be totally different from the Euclidean metric but that has the same ordering as the Euclidean metric. For example, the metric r = Δx² + Δy² + Δz² is fine (notice that there is no square root.)
To be more precise, I want a term for two metrics d₁ and d₂ such that d₁(a, x) < d₁(a, y) ⟺ d₂(a, x) < d₂(a, y) for any a, x and y.