In ordinary 3-dimensional Euclidean geometry we are used to computing distances by summing squares of orthogonal coordinate distances (e.g. in
Pythagoras' Theorem). At the turn of the last century Einstein dramatically raised the profile of 4-dimensional geometry (space plus time)
in which the distance in time contributes negatively to the "distance".
In 2000, myself and Wilhelm Klingenberg discovered a geometry on the space of oriented lines in 3-dimensional Euclidean space which has two
negative contributions - mathematically we say that the metric has signature (2,2), or is neutral.
Such geometries have not been investigated much, probably because few good examples were known. We have therefore had to build up our ideas from scratch: linear algebra, tensor analysis and submanifold theory. The situation turns out to be much more complex than the positive definite case. Our publications on this are here.