Group.h

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/* ----------------------------------------------------------------------------
 
 * GTSAM Copyright 2010, Georgia Tech Research Corporation,
 * Atlanta, Georgia 30332-0415
 * All Rights Reserved
 * Authors: Frank Dellaert, et al. (see THANKS for the full author list)
 
 * See LICENSE for the license information
 
 * -------------------------------------------------------------------------- */

 
#pragma once
 
#include <gtsam/base/Testable.h>
 
#include <utility>
#include <type_traits>
 
namespace gtsam {

struct group_tag {};

struct multiplicative_group_tag {};
struct additive_group_tag {};
 
template <typename T> struct traits;

template<typename G>
class IsGroup {
public:
  typedef typename traits<G>::structure_category structure_category_tag;
  typedef typename traits<G>::group_flavor flavor_tag;
  //typedef typename traits<G>::identity::value_type identity_value_type;
 
  GTSAM_CONCEPT_USAGE(IsGroup) {
    static_assert(
        (std::is_base_of_v<group_tag, structure_category_tag>),
        "This type's structure_category trait does not assert it as a group (or derived)");
    e = traits<G>::Identity();
    e = traits<G>::Compose(g, h);
    e = traits<G>::Between(g, h);
    e = traits<G>::Inverse(g);
 
    if constexpr (std::is_same_v<flavor_tag, multiplicative_group_tag>) {
      e = g * h;
      //e = -g; // todo this should work, but it is failing for Quaternions
    } else if constexpr (std::is_same_v<flavor_tag, additive_group_tag>) {
      e = g + h;
      e = h - g;
      e = -g;
    }
    // todo: how do we test the act concept? or do we even need to?
  }
 
private:
  G e, g, h;
  bool b;
};

template<typename G>
GTSAM_CONCEPT_REQUIRES(IsGroup<G>,bool) //
check_group_invariants(const G& a, const G& b, double tol = 1e-9) {
  G e = traits<G>::Identity();
  return traits<G>::Equals(traits<G>::Compose(a, traits<G>::Inverse(a)), e, tol)
      && traits<G>::Equals(traits<G>::Between(a, b), traits<G>::Compose(traits<G>::Inverse(a), b), tol)
      && traits<G>::Equals(traits<G>::Compose(a, traits<G>::Between(a, b)), b, tol);
}
 
namespace internal {

template<class Class>
struct MultiplicativeGroupTraits {
  typedef group_tag structure_category;
  typedef multiplicative_group_tag group_flavor;
  static Class Identity() { return Class::Identity(); }
  static Class Compose(const Class &g, const Class & h) { return g * h;}
  static Class Between(const Class &g, const Class & h) { return g.inverse() * h;}
  static Class Inverse(const Class &g) { return g.inverse();}
};

template<class Class>
struct MultiplicativeGroup : MultiplicativeGroupTraits<Class>, Testable<Class> {};

template<class Class>
struct AdditiveGroupTraits {
  typedef group_tag structure_category;
  typedef additive_group_tag group_flavor;
  static Class Identity() { return Class::Identity(); }
  static Class Compose(const Class &g, const Class & h) { return g + h;}
  static Class Between(const Class &g, const Class & h) { return h - g;}
  static Class Inverse(const Class &g) { return -g;}
};

template<class Class>
struct AdditiveGroup : AdditiveGroupTraits<Class>, Testable<Class> {};
 
}  // namespace internal

template<typename G>
GTSAM_CONCEPT_REQUIRES(IsGroup<G>,G) //
compose_pow(const G& g, size_t n) {
  if (n == 0) return traits<G>::Identity();
  else if (n == 1) return g;
  else return traits<G>::Compose(compose_pow(g, n - 1), g);
}

template<typename G, typename H>
class DirectProduct: public std::pair<G, H> {
  GTSAM_CONCEPT_ASSERT(IsGroup<G>);
  GTSAM_CONCEPT_ASSERT(IsGroup<H>);
 
public:
  DirectProduct():std::pair<G,H>(traits<G>::Identity(),traits<H>::Identity()) {}
 
  // Construct from two subgroup elements
  DirectProduct(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
  // identity
  static DirectProduct Identity() { return DirectProduct(); }
 
  DirectProduct operator*(const DirectProduct& other) const {
    return DirectProduct(traits<G>::Compose(this->first, other.first),
        traits<H>::Compose(this->second, other.second));
  }
  DirectProduct inverse() const {
    return DirectProduct(this->first.inverse(), this->second.inverse());
  }
};
 
// Define any direct product group to be a model of the multiplicative Group concept
template<typename G, typename H>
struct traits<DirectProduct<G, H> > :
  internal::MultiplicativeGroupTraits<DirectProduct<G, H> > {};

template<typename G, typename H>
class DirectSum: public std::pair<G, H> {
  GTSAM_CONCEPT_ASSERT(IsGroup<G>);  // TODO(frank): check additive
  GTSAM_CONCEPT_ASSERT(IsGroup<H>);  // TODO(frank): check additive
 
  const G& g() const { return this->first; }
  const H& h() const { return this->second;}
 
public:
  DirectSum():std::pair<G,H>(traits<G>::Identity(),traits<H>::Identity()) {}
 
  // Construct from two subgroup elements
  DirectSum(const G& g, const H& h):std::pair<G,H>(g,h) {}
 
  // identity
  static DirectSum Identity() { return DirectSum(); }
 
  DirectSum operator+(const DirectSum& other) const {
    return DirectSum(g()+other.g(), h()+other.h());
  }
  DirectSum operator-(const DirectSum& other) const {
    return DirectSum(g()-other.g(), h()-other.h());
  }
  DirectSum operator-() const {
    return DirectSum(- g(), - h());
  }
};
 
// Define direct sums to be a model of the Additive Group concept
template<typename G, typename H>
struct traits<DirectSum<G, H> > :
  internal::AdditiveGroupTraits<DirectSum<G, H> > {};
 
}  // namespace gtsam

#define GTSAM_CONCEPT_GROUP_INST(T) template class gtsam::IsGroup<T>;
#define GTSAM_CONCEPT_GROUP_TYPE(T) typedef gtsam::IsGroup<T> _gtsam_IsGroup_##T;