Introduce a class which represent a set of GSplines

[rafaelrojasmiliani]

Motivation

We are working with two different sets of GSpline

• Set 1: (Is a Manifold) The set of GSplines that passes through a series of waypoints. Is defined by
  • Sequence of waypoints
  • Domain
  • Basis
• Set 2: (Is a Vector Space) The set of Gsplines with unique domain interval lengths. Is defined by
  • Interval lengths
  • Domain
  • Basis
• Remark On space of functions and domain information Note that the Set 2 is a vector space for a given domain. Function with different domain cannot be added. In the case of Set 1, we have that all problems that involve it have a fixed domain.
• Remark on different domains We underline that the GSplines defined in different domains are isomorphic. We need to apply a linear dilation if the proportion between the lengths of the intervals rest the same.

A GSpline set can be a GSpline factory

[rafaelrojasmiliani]

Why this is necessary

This will be necessary to facilitate the conversion between vectors and GSplines. Both sets allow to represent a GSpline as a vector of differnt nature, i.e. the interval lengths and the basis coefficients (points in Lagrange polynomial representation).

• Pros
  • factory for GSplines alternative to Interpolator
  • factory for Functionals, derivative, integral, sobolev norm
• Cons
  • Adds an additional data structure
  • Do not solves the automatic cast between GSplines and vectors.
  • Do not allow to have a functional that returns a unique type that can behavie as a double, an Eigen::VectorXd and a GSpline.

[rafaelrojasmiliani]

Possible strategy, decouple memory management from the factory

In the case of the vector space of functions with same interval lengths, basis and dimension of the codomain, we have the following fact. Regardless the domain of the space, the size of the matrices is for different operators is the same.
In fact, although different domains induce different spaces (the space of functions in {0,1} is not the space of functions in {0,2}), the memory management is the same.
Following this idea, we can create a singleton class SpaceBuffer which is stored in a map with std::tupe<double,double,double> key. Each one of these SpaceBuffer's have their own space with the derivative matrix, integral operator, transpose operator and matrix multiplication. The the GsplineSpace class, has a pointer to a SpaceBuffer and copies from here the required memory.