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Terminology.dox
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Terminology.dox
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/*
Terminology between usages (e.g. different countries, different disciplines)
can vary somewhat. For example, should "conjugate" be interpreted to mean
one of the "complex conjugate" type operations, or "Clifford conjugate".
Important to know what a function does from its name. Important to not
think it does something different - even if that means to use a nonstandard
name.
To avoid potential confusion in words that are used differently in different
contexts, engabra strives to use unambiguous terms/words - even if that
requires to introduce a few new or unusual words. This includes:
Direction vs Chirality vs Orientation vs Permutation
Algebraic concepts include - in engabra terminology:
\arg Direction - associated with a "B wrt A" relationship
\arg Chirality - associated with disambiguation standard (arbitrary)
\arg Orientation - associated with in/co-variance to geometric transformation
\arg Permutation - associated with ordering of factors
Orientation: wrt relationship - sca,vec,biv,tri
\arg Scalar - has Orientation - plus/minus difference between two
\arg Vector - has Orientation - tip/tail relationship
\arg BiVector - has Orientation - sweep ordering
\arg TriVector - has Orientation - plus/minus (chirality entity in 3D)
Permutation - cyclic vs acyclic ordering of factors - biv,tri
\arg Scalar - does not have Permutation (zero factors)
\arg Vector - does not have Permutation (one factor)
\arg BiVector - has permutation (two factors)
\arg TriVector - has permutation (three factors)
Chrality: handedness in algebraic space or subspace - vec,tri
\arg Scalar -does not have Chirality
\arg Vector - has Chirality
\arg BiVector - does not have Chirality
\arg TriVector - has Chirality
Direction: in/co-variance to (geo)spatial transformation - vec,biv
\arg Scalar - does not have Direction - invariant to geometric transform
\arg Vector - has Direction - changes under rotation/reflection
\arg BiVector - has Direction - changes under rotation/reflection
\arg TriVector - does not have Direction - invariant to geometric transform
Inversion of:
\arg Negation - unitary-() - sign flip: sca,vec,biv,tri
\arg Orientation - oriverse() - sign flip: [sca],vec,biv,tri
\arg Chirality - chiverse() - sign flip: vec,tri
\arg Direction - dirverse() - sign flip: vec,biv
\arg Permutation - reverse() - sign flip: biv,tri
Orientation of field (chirality of scalar)
Scalar chirality selection happens before generating the GA space.
Interp: scalar used to build vectors in other directions
----
-+--
--+-
---+
Interp: Use this "postive-scalar field" interpretation
+--- : oriverse()
++-- : reverse() [reversion]
+-+- : opposite() [space inversion]
+--+ : dirverse() [clifford conjugation - reverse*opposite]
Conjugation in the sense of pairwise relationships (wrt flip vs noflip)
\arg Complex Conjugation : inverse of permutation
\arg Clifford Conjugation : inverse of direction
Complex Conjugates - change sign on items that square to -1
+OO- : Complex - classic complex conjugate operations
+O-O : Spinor - quaternion conjugate operations
O+O- : ImSpin - Imaginary * conjugate of spinor
++-- : MultiVector - useful for defining inner product
Clifford Conjugates - change sign on directed entities (geometry co-variant)
Is related to geometry inversion
Composite of reverse*opposite
+--+ : MultiVector - useful for defining amplitude and inverse
Direction is interpreted to be associated with space. E.g. in
multidimensional contexts. Vector and Bivectors are therefore are entities
with intrinsic directions.
Orientation is interpreted to be associated with chirality or "handedness".
In GA, all entities have an orientation. The operation of unitary negation
is associated with inverting the orientation.
In physical terms, orientation is interpreted as a "with respect to"
association. E.g. Given two scalar values associated with observation A
and observation B, have opposite orientation if the difference is interpreted
as B wrt A (BwA) or if it is interpreted as AwB.
Scalars don't really have a direction, since they are invariant to
geometric changes such as translation, rotation, and reflection.
The orientation of Vectors is similar. E.g. for BwA, the ending "tip" is at B
relative to the starting tail at A.
Direction is associated with rotation of this vector. E.g. where is it
"pointing" relative to some other reference object. The orientation is
associated with the tip/tail interpretation, while the direction is associated
with some external object (e.g. another GA entity, a coordinate frame basis,
etc).
BiVectors have orientation associated with the order in which two generating
vectors are defined. E.g. vector BwA or AwB (TODO sweep direction?).
The Direction of BiVectors is interpreted with respect to some external
entity.
The G3 TriVector is the pseudo-scalar for algebra in 3D space. Therefore,
in 3D space, trivectors, like scalars, have orientation, but do not have
direction.
E.g.
Direction - relative to external reference. Can be changed.
Is an extrinsic property.
Orientation - internal property.
Pose?
Reference WRT?
E.g.
\arg Scalar: orientation is attached to counting direction. E.g. positive
values have one orientation, negative values have another. Unitary negation
changes the orientation of a scalar.
\arg Vector: orientation is
Complex - type consisting of scalar and pseudo-scalar
Dirplex - type consisting of Vector and BiVector
*/