More indentation fixes.

release/4.3a0
Paul Furgale 2014-12-08 08:55:14 +00:00
parent 21596e1894
commit 2b1d907a74
1 changed files with 16 additions and 16 deletions

View File

@ -38,15 +38,15 @@ Chart
A given chart is implemented using a small class that defines the chart itself (from manifold to tangent space) and its inverse.
* types:
* `ManifoldType`, a pointer back to the type
* `ManifoldType`, a pointer back to the type
* valid expressions:
* `v = Chart::Local(p,q)`, the chart, from manifold to tangent space, think of it as *q (-) p*
* `p = Chart::Retract(p,v)`, the inverse chart, from tangent space to manifold, think of it as *p (+) v*
* `v = Chart::Local(p,q)`, the chart, from manifold to tangent space, think of it as *q (-) p*
* `p = Chart::Retract(p,v)`, the inverse chart, from tangent space to manifold, think of it as *p (+) v*
For many differential manifolds, an obvious mapping is the `exponential map`, which associates straight lines in the tangent space with geodesics on the manifold (and it's inverse, the log map). However, there are two cases in which we deviate from this:
* Sometimes, most notably for *SO(3)* and *SE(3)*, the exponential map is unnecessarily expensive for use in optimization. Hence, the `defaultChart` functor returns a chart that is much cheaper to evaluate.
* While vector spaces (see below) are in principle also manifolds, it is overkill to think about charts etc. Really, we should simply think about vector addition and subtraction. Hence, while a `defaultChart` functor is defined by default for every vector space, GTSAM will never call it.
* Sometimes, most notably for *SO(3)* and *SE(3)*, the exponential map is unnecessarily expensive for use in optimization. Hence, the `defaultChart` functor returns a chart that is much cheaper to evaluate.
* While vector spaces (see below) are in principle also manifolds, it is overkill to think about charts etc. Really, we should simply think about vector addition and subtraction. Hence, while a `defaultChart` functor is defined by default for every vector space, GTSAM will never call it.
Group
@ -54,15 +54,15 @@ Group
A [group](http://en.wikipedia.org/wiki/Group_(mathematics)) should be well known from grade school :-), and provides a type with a composition operation that is closed, associative, has an identity element, and an inverse for each element.
* values:
* `group::identity<G>()`
* `group::identity<G>()`
* valid expressions:
* `group::compose(p,q)`
* `group::inverse(p)`
* `group::between(p,q)`
* `group::compose(p,q)`
* `group::inverse(p)`
* `group::between(p,q)`
* invariants (using namespace group):
* `compose(p,inverse(p)) == identity`
* `compose(p,between(p,q)) == q`
* `between(p,q) == compose(inverse(p),q)`
* `compose(p,inverse(p)) == identity`
* `compose(p,between(p,q)) == q`
* `between(p,q) == compose(inverse(p),q)`
We do *not* at this time support more than one composition operator per type. Although mathematically possible, it is hardly ever needed, and the machinery to support it would be burdensome and counter-intuitive.
@ -83,8 +83,8 @@ Even finite groups can act on continuous entities. For example, the [cyclic grou
Hence, we formalize by the following extension of the concept:
* valid expressions:
* `group::act(g,t)`, for some instance of a space T, that can be acted upon by the group
* `group::act(g,t,H)`, if the space acted upon is a continuous differentiable manifold
* `group::act(g,t)`, for some instance of a space T, that can be acted upon by the group
* `group::act(g,t,H)`, if the space acted upon is a continuous differentiable manifold
Group actions are concepts in and of themselves that can be concept checked (see below).
@ -133,8 +133,8 @@ Testable
Unit tests heavily depend on the following two functions being defined for all types that need to be tested:
* valid expressions:
* `print(p,s)` where s is an optional string
* `equals(p,q,tol)` where tol is an optional tolerance
* `print(p,s)` where s is an optional string
* `equals(p,q,tol)` where tol is an optional tolerance
Implementation
==============