Made group action "future concepts"
parent
d9d43731d7
commit
566390dc60
|
@ -65,32 +65,6 @@ We do *not* at this time support more than one composition operator per type. Al
|
|||
|
||||
Also, a type should provide either multiplication or addition operators depending on the flavor of the operation. To distinguish between the two, we will use a tag (see below).
|
||||
|
||||
Group Action
|
||||
------------
|
||||
|
||||
Group actions are concepts in and of themselves that can be concept checked (see below).
|
||||
In particular, a group can *act* on another space.
|
||||
For example, the [cyclic group of order 6](http://en.wikipedia.org/wiki/Cyclic_group) can rotate 2D vectors around the origin:
|
||||
|
||||
q = R(i)*p
|
||||
where R(i) = R(60)^i, where R(60) rotates by 60 degrees
|
||||
|
||||
Hence, we formalize by the following extension of the concept:
|
||||
|
||||
* valid expressions:
|
||||
* `q = traits<T>::Act(g,p)`, for some instance, *p*, of a space *S*, that can be acted upon by the group element *g* to produce *q* in *S*.
|
||||
* `q = traits<T>::Act(g,p,Hp)`, if the space acted upon is a continuous differentiable manifold. *
|
||||
|
||||
In the latter case, if *S* is an n-dimensional manifold, *Hp* is an output argument that should be
|
||||
filled with the *nxn* Jacobian matrix of the action with respect to a change in *p*. It typically depends
|
||||
on the group element *g*, but in most common example will *not* depend on the value of *p*. For example, in
|
||||
the cyclic group example above, we simply have
|
||||
|
||||
Hp = R(i)
|
||||
|
||||
Note there is no derivative of the action with respect to a change in g. That will only be defined
|
||||
for Lie groups, which we introduce now.
|
||||
|
||||
Lie Group
|
||||
---------
|
||||
|
||||
|
@ -129,22 +103,6 @@ However, the exponential map is unnecessarily expensive for use in optimization.
|
|||
|
||||
Most Lie groups we care about are *Matrix groups*, continuous sub-groups of *GL(n)*, the group of *n x n* invertible matrices. In this case, a lot of the derivatives calculations needed can be standardized, and this is done by the `LieGroup` superclass. You only need to provide an `AdjointMap` method.
|
||||
|
||||
Lie Group Action
|
||||
----------------
|
||||
|
||||
When a Lie group acts on a space, we have two derivatives to care about:
|
||||
|
||||
* `gtasm::manifold::traits<T>::act(g,p,Hg,Hp)`, if the space acted upon is a continuous differentiable manifold.
|
||||
|
||||
An example is a *similarity transform* in 3D, which can act on 3D space, like
|
||||
|
||||
q = s*R*p + t
|
||||
|
||||
Note that again the derivative in *p*, *Hp* is simply *s R*, which depends on *g* but not on *p*.
|
||||
The derivative in *g*, *Hg*, is in general more complex.
|
||||
|
||||
For now, we won't care about Lie groups acting on non-manifolds.
|
||||
|
||||
Vector Space
|
||||
------------
|
||||
|
||||
|
@ -199,3 +157,50 @@ Boost provides a nice way to check whether a given type satisfies a concept. For
|
|||
BOOST_CONCEPT_ASSERT(IsVectorSpace<Point2>)
|
||||
|
||||
asserts that Point2 indeed is a model for the VectorSpace concept.
|
||||
|
||||
Future Concepts
|
||||
===============
|
||||
|
||||
|
||||
Group Action
|
||||
------------
|
||||
|
||||
Group actions are concepts in and of themselves that can be concept checked (see below).
|
||||
In particular, a group can *act* on another space.
|
||||
For example, the [cyclic group of order 6](http://en.wikipedia.org/wiki/Cyclic_group) can rotate 2D vectors around the origin:
|
||||
|
||||
q = R(i)*p
|
||||
where R(i) = R(60)^i, where R(60) rotates by 60 degrees
|
||||
|
||||
Hence, we formalize by the following extension of the concept:
|
||||
|
||||
* valid expressions:
|
||||
* `q = traits<T>::Act(g,p)`, for some instance, *p*, of a space *S*, that can be acted upon by the group element *g* to produce *q* in *S*.
|
||||
* `q = traits<T>::Act(g,p,Hp)`, if the space acted upon is a continuous differentiable manifold. *
|
||||
|
||||
In the latter case, if *S* is an n-dimensional manifold, *Hp* is an output argument that should be
|
||||
filled with the *nxn* Jacobian matrix of the action with respect to a change in *p*. It typically depends
|
||||
on the group element *g*, but in most common example will *not* depend on the value of *p*. For example, in
|
||||
the cyclic group example above, we simply have
|
||||
|
||||
Hp = R(i)
|
||||
|
||||
Note there is no derivative of the action with respect to a change in g. That will only be defined
|
||||
for Lie groups, which we introduce now.
|
||||
|
||||
|
||||
Lie Group Action
|
||||
----------------
|
||||
|
||||
When a Lie group acts on a space, we have two derivatives to care about:
|
||||
|
||||
* `gtasm::manifold::traits<T>::act(g,p,Hg,Hp)`, if the space acted upon is a continuous differentiable manifold.
|
||||
|
||||
An example is a *similarity transform* in 3D, which can act on 3D space, like
|
||||
|
||||
q = s*R*p + t
|
||||
|
||||
Note that again the derivative in *p*, *Hp* is simply *s R*, which depends on *g* but not on *p*.
|
||||
The derivative in *g*, *Hg*, is in general more complex.
|
||||
|
||||
For now, we won't care about Lie groups acting on non-manifolds.
|
||||
|
|
Loading…
Reference in New Issue