More indentation fixes.
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				|  | @ -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 | ||||
| ============== | ||||
|  |  | |||
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