Assume the receptor is fixed and consider the space of rotations of the ligand with respect to the receptor. This space, denoted by , can be specified by . We can represent this space as a subset of the 9-dimensional (9D) Euclidean space that satisfies the constraints and ; to optimize a function on this space we can add the constraints to the function via Lagrange multipliers and end up with an optimization problem on a 9D Euclidean space to which Euclidean optimization algorithms are directly applicable. Therefore, we end up with a 9D search space by fully disregarding the geometry of the original search space, namely .

By contrast, in a **manifold optimization** approach, we fully take into account that is a curved space, i.e., a nonlinear manifold of dimension 3 (3D). To develop optimization algorithms, we need **local parameterizations** of the manifold, of which many are available. A local parametrization corresponds to a local flattening/straightening of the manifold, a local deformation of the manifold to arrive at a subset of a Euclidean space. A desirable deformation is one that, informally speaking, deforms/scales the rotational angles similarly and does not distort the relationship between rotational angles. Think of a “reverse process” of looking at a curved mirror that distorts your image: some mirrors drastically distort the image producing an unrecognizable figure and some respect the original proportions and reflect some of the essential characteristics of the figure.

The **exponential coordinates parametrization** that locally maps the Euclidean tangent space onto the nonlinear manifold, is a parametrization of the latter kind, i.e., one that reflects essential features of the rotational movement. In this sense, it is a **natural parametrization** that contributes to the efficiency of local optimization algorithms and has been essential for our characterization of the funnels of the free-energy landscape.

For optimization on the rotational space , the search space of manifold optimization has a significantly lower 3D search space compared to the alternative we described above, however, we no longer can directly use the tried and tested Euclidean optimization algorithms in this case. For , this is not a serious drawback as efficient manifold optimization algorithms for this purpose exist, but for the more general problem of optimization on non-linear manifolds, developing effective algorithms is an issue that needs serious consideration and is the subject of active research.