Since the concept of distance in Euclidean space is not invariant in projective space, that is , distance is invariant under Euclidean transformations but not under projective transformations, is it possible to define a distance from point to lines in 3D projective space which is invariant under projective transformations?
2026-02-23 18:59:46.1771873186
How to define a "distance" from point to line in 3D projective space which is projectively invariant?
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Without additional reference objects, this is impossible: any pair of a point and a line can be mapped to any other such pair by a projective transformation, so any projectively invariant quantity you assign to that has to be constant, i.e. be the same for all pairs, in which case I wouldn't call it a distance.
If you introduce additional reference objects, then you can measure with respect to these, but you'll have to ensure that these reference objects remain fixed under (your subset of) projective transformations. One common example is using a conic as the fundamental domain of a Cayley-Klein metric.