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FOUR LINES IN SPACE JAUME AGUADE
 

Summary: FOUR LINES IN SPACE
JAUME AGUAD┤E
Abstract.We introduce an invariant of sets of four lines in three-dimensional pro-
jective space which classifies these sets up to linear equivalence, and we relate this
invariant to the representation theory of the infinite dihedral group.
Mathematics Subject Classification (2000): 51Axx, 16G20, 20C20, 15A21.
Key words: four subspace problem, pairs of involutions, representations, invariants.
1. Introduction
The modern development of computer vision has produced an increased interest in
some problems in classic geometry related to joint invariants (see, for instance, [11]
and [14]). For example, the invariants of n points in space are the volume cross-
ratio, as noted in the classic book by Veblen and Young ([15], section 27; see [14] for
a modern point of view): to determine if two sets of n points are equivalent under
a linear transformation we just need to compute their volume cross-ratios and see
if they agree for the two sets of points. Here we address a similar problem, namely
to obtain the invariants of four lines in space. Given four lines in space we define a
cross-ratio-like invariant --which can be easily computed-- such that two sets of
four lines are equivalent under a linear transformation if and only if the invariant
has the same value for the two sets of lines. After some preliminaries to fix the
notation, the invariant is defined in section 3 (see definition 2). The work in this

  

Source: AguadÚ, Jaume - Departament de MatemÓtiques, Universitat Aut˛noma de Barcelona
Aut˛noma de Barcelona, Universitat - Departament de MatemÓtiques

 

Collections: Mathematics