Made group action "future concepts"

release/4.3a0
dellaert 2014-12-28 13:21:54 +01:00
parent d9d43731d7
commit 566390dc60
1 changed files with 48 additions and 43 deletions

View File

@ -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.