[Manifolds](http://en.wikipedia.org/wiki/Manifold#Charts.2C_atlases.2C_and_transition_maps) and [charts](http://en.wikipedia.org/wiki/Manifold#Charts.2C_atlases.2C_and_transition_maps) are intimately linked concepts. We are only interested here in [differentiable manifolds](http://en.wikipedia.org/wiki/Differentiable_manifold#Definition), continuous spaces that can be locally approximated *at any point* using a local vector space, called the [tangent space](http://en.wikipedia.org/wiki/Tangent_space). A *chart* is an invertible map from the manifold to that tangent space.
In GTSAM we assume that a manifold type can yield such a *Chart* at any point, and we require that a functor `defaultChart` is available that, when called for any point on the manifold, returns a Chart type. Hence, the functor itself can be seen as an *Atlas*.
In detail, we ask the following are defined for a MANIFOLD type:
* values:
*`dimension`, an int that indicates the dimensionality *n* of the manifold. In Eigen-fashion, we also support manifolds whose dimenionality is only defined at runtime, by specifying the value -1.
* functors:
*`defaultChart`, returns the default chart at a point p
* types:
*`TangentVector`, type that lives in tangent space. This will almost always be an `Eigen::Matrix<double,n,1>`.
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.
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.
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.
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).
A group can *act* on another space. For example, a *similarity transform* in 3D can act on 3D space, like
q = s*R*p + t
Even finite groups can act on continuous entities. 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:
*`group::act(g,t)`, for some instance of a space T, that can be acted upon by the group
A Lie group is both a manifold *and* a group. Hence, a LIE_GROUP type should implements both MANIFOLD and GROUP concepts. However, we now also need to be able to evaluate the derivatives of compose and inverse. Hence, we have the following extra valid expressions:
where above the `H` arguments stand for optional Jacobian arguments. That makes it possible to create factors implementing priors (PriorFactor) or relations between two instances of a Lie group type (BteweenFactor).
GTSAM Types start with Uppercase, e.g., `gtsam::Point2`, and are models of the TESTABLE, MANIFOLD, GROUP, LIE_GROUP, and VECTOR_SPACE concepts.
`gtsam::traits` is our way to associate these concepts with types, and we also define a limited number of `gtsam::tags` to select the correct implementation of certain functions at compile time (tag dispatching).
We will not use Eigen-style or STL-style traits, that define *many* properties at once. Rather, we use boost::mpl style meta-programming functions to facilitate meta-programming, which return a single type or value for every trait. Some rationale/history can be found [here](http://www.boost.org/doc/libs/1_55_0/libs/type_traits/doc/html/boost_typetraits/background.html).
Traits allow us to play with types that are outside GTSAM control, e.g., `Eigen::VectorXd`. However, for GTSAM types, it is perfectly acceptable (and even desired) to define associated types as internal types, as well, rather than having to use traits internally.
Finally, note that not everything that makes a concept is defined by traits. For example, although a CHART type is supposed to have a `retract` function, there is no trait for this: rather, the
* Values: `gtsam::traits::someValue<T>::value`, i.e., they are mixedCase starting with a lowercase letter and define a `value`, *and* a `value_type`. For example:
* Functors: `gtsam::traits::someFunctor<T>::type`, i.e., they are mixedCase starting with a lowercase letter and define a functor (i.e., no *type*). The functor itself should define a `result_type`. A contrived example
By *inherting* the trait from the functor, we can just use the [currying](http://en.wikipedia.org/wiki/Currying) style `gtsam::traits::manhattan<Point2>℗(q)`. Note that, although technically a functor is a type, in spirit it is a free function and hence starts with a lowercase letter.
A tag can be used for [tag dispatching](http://www.boost.org/community/generic_programming.html#tag_dispatching),
e.g., below is a generic compose:
```
#!c++
namespace detail {
template <classT>
T compose(const T& p, const T& q, additive_group_tag) {
return p + q;
}
template <classT>
T compose(const T& p, const T& q, multiplicative_group_tag) {
return p * q;
}
}
template <T>
T compose(const T& p, const T& q) {
return detail::compose(p, q, traits::group_flavor<T>::type);
}
```
Tags also facilitate meta-programming. Taking a leaf from [The boost Graph library](http://www.boost.org/doc/libs/1_40_0/boost/graph/graph_traits.hpp),
tags can be used to create useful meta-functions, like `is_lie_group`, below.
