[Merged by Bors] - feat(analysis/convex/between): betweenness in affine spaces #16191
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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.
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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.
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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_segmentthatduplicates
segmentin the affine space case (and is proved to equalsegmentwhen the affine space is a module considered as an affinespace over itself). However, the bulk of results concern betweenness,
not
affine_segment, and so would be just as relevant after such arefactor, even if some of the proofs would change, and indeed most of
the things stated about
affine_segmentinvolve+ᵥand-ᵥand sowould still be meaningful results, distinct from those already present
for
segment, after such a refactor (at which point they might applyfor whatever typeclass describes an
add_torsorfor a module that isalso an abstract affine combination space where the two affine
combination structures agree). So I think the actual duplication here
is minimal and defining
affine_segmentis a reasonable approach toallow 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_rayand thenprove appropriate results about how they relate to betweenness for
points.