Re: [math-fun] comparing homogeneous vectors
I was going to answer this until I noticed Fred had written: << [N]onzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Suddenly I have no idea what this means. One of the axioms of a metric (or rather a norm on which a metric is based) is ||X|| = 0 <=> X = 0. So I am confused as to what sort of "metric" is being considered here, or else what the notation ||X||, er |X|, means. --Dan Fred wrote: << Suppose we have a space with points represented by homogeneous coordinate vectors --- that is, {F - 0} factored out, where F is some continuous ground field, in practice \R or \C. How do we tell whether two points are (approximately) the same? If the metric is Euclidean (e.g. real projective), a slightly clumsy solution is to rescale both vectors X,Y to length unity, and check for either |X-Y| and |X+Y| small; unless |X| or |Y| is small to start with, when it can be snapped back to zero (representing no point). But when the metric is more general (e.g. complex projective, conformal inversive), nonzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Ignoring the local metric, performing the comparison using instead a Euclidean metric, offends against my aesthetic scruples; in particular failing to be invariant under local isometries. This apparently trivial algorithmic conundrum has irritated me for years. Has anybody got a better idea?
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Good point --- I don't think I'd ever actually noticed this disparity before! Probably the most obvious case of what I have in mind is conformal / Moebius / Minkowski / Poincare geometry, where the quadratic form defining the metric is x^2+y^2+z^2-t^2; we'd like to restrict ourselves to using this, but then how can isotropic vectors such as [1,0,0,1] or [3,0,4,5] be rescaled for comparison? WFL On 1/12/10, Dan Asimov <dasimov@earthlink.net> wrote:
I was going to answer this until I noticed Fred had written:
<< [N]onzero vectors may possibly be isotropic --- |X| = 0 with X /= 0.
Suddenly I have no idea what this means. One of the axioms of a metric (or rather a norm on which a metric is based) is
||X|| = 0 <=> X = 0.
So I am confused as to what sort of "metric" is being considered here, or else what the notation ||X||, er |X|, means.
--Dan
Fred wrote:
<< Suppose we have a space with points represented by homogeneous coordinate vectors --- that is, {F - 0} factored out, where F is some continuous ground field, in practice \R or \C. How do we tell whether two points are (approximately) the same?
If the metric is Euclidean (e.g. real projective), a slightly clumsy solution is to rescale both vectors X,Y to length unity, and check for either |X-Y| and |X+Y| small; unless |X| or |Y| is small to start with, when it can be snapped back to zero (representing no point).
But when the metric is more general (e.g. complex projective, conformal inversive), nonzero vectors may possibly be isotropic --- |X| = 0 with X /= 0. Ignoring the local metric, performing the comparison using instead a Euclidean metric, offends against my aesthetic scruples; in particular failing to be invariant under local isometries.
This apparently trivial algorithmic conundrum has irritated me for years. Has anybody got a better idea?
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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Quoting Dan Asimov <dasimov@earthlink.net>:
Suddenly I have no idea what this means. One of the axioms of a metric (or rather a norm on which a metric is based) is
||X|| = 0 <=> X = 0.
There's norms and then there's metrics. Minkowsky space has its light cone - non-zero vectors of length zero, and phase space has its symplectic metric with coordinates and momenta. - hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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Dan Asimov -
Fred lunnon -
mcintosh＠servidor.unam.mx