Commit 2022-09-30 13:10 699c2cab
View on Github →feat(analysis/convex/between): betweenness in affine spaces (#16191)
Define the notions of (weak and strict) betweenness for points in
affine spaces over an ordered ring, for use in describing geometrical
configurations.
Until convexity is refactored to support abstract affine combination
spaces, this means having a definition affine_segment
that
duplicates segment
in the affine space case (and is proved to equal
segment
when the affine space is a module considered as an affine
space over itself). However, the bulk of results concern betweenness,
not affine_segment
, and so would be just as relevant after such a
refactor, even if some of the proofs would change, and indeed most of
the things stated about affine_segment
involve +ᵥ
and -ᵥ
and so
would still be meaningful results, distinct from those already present
for segment
, after such a refactor (at which point they might apply
for whatever typeclass describes an add_torsor
for a module that is
also an abstract affine combination space where the two affine
combination structures agree). So I think the actual duplication here
is minimal and defining affine_segment
is a reasonable approach to
allow betweenness to be handled in affine spaces now rather than
making it depend on a possible future refactor.
There are certainly more things that could sensibly be stated about
betweenness (e.g. various forms of Pasch's axiom), but I think this is
a reasonable starting point.
One thing I definitely intend to add in a followup is notions of two
points being (weakly or strictly) on the same side or opposite sides
of an affine subspace (e.g. a line); I think it will probably be most
convenient to define those notions in terms of same_ray
and then
prove appropriate results about how they relate to betweenness for
points.