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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.

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    $\begingroup$ Are you saying two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ such that $\|x\|_1 < \|y\|_1 \iff \|x\|_2 < \|y\|_2$ for all $x, y$? $\endgroup$
    – muzzlator
    Jun 5, 2015 at 19:41
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    $\begingroup$ Using the term metric is confusing because a metric is a function on pairs and so there is no way to use that to define an ordering on the elements $\endgroup$
    – muzzlator
    Jun 5, 2015 at 19:44
  • $\begingroup$ @muzzlator That sounds right. $\endgroup$
    – Molly Stewart-Gallus
    Jun 5, 2015 at 19:44
  • $\begingroup$ He must mean norms, because "$r = x^{2} + y^{2} + z^{2}$" doesn't look like a metric. $\endgroup$
    – Jonathan Hebert
    Jun 5, 2015 at 19:46
  • $\begingroup$ @muzzlator One of the points is a constant. $\endgroup$
    – Molly Stewart-Gallus
    Jun 5, 2015 at 19:49

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