There are two possible different metrics for a plane:
-The usual Euclidean metric, with the circle as the main geometric feature (the locus of all points at the same distance from a given one).
-The spacetime plane of modern physics with its Minkowskian metric, where the circle is substituted by a straight equilateral hyperbola as the locus for all poits at a given distance from a center.
The following applet allows you to try the "beta" version for a GeoGebra Environment where the euclidean tools were substituted by their minkowskian counterparts (some tools, being the same in both cases, remain unchanged).
You can check and download HERE this GeoGebra environment to use it freely at your computer.
You can also read our proposal at the GeoGebra Forum to build entirely from scratch a new version for this environment. It would allow to work efficiently into the minkowskian spacetime plane of modern physics and establish its interesting relations, which follow entirely their euclidean counterparts.
You can see how this works in two examples from basic triangle geometry:
You can see both copnstructions together as well as the ampliated forum proposal in this link.