lio-cpp/lio/include/sophus/se2.hpp

/// @file
/// Special Euclidean group SE(2) - rotation and translation in 2d.

#ifndef SOPHUS_SE2_HPP
#define SOPHUS_SE2_HPP

#include "so2.hpp"

namespace Sophus {
template <class Scalar_, int Options = 0>
class SE2;
using SE2d = SE2<double>;
using SE2f = SE2<float>;
}  // namespace Sophus

namespace Eigen {
namespace internal {

template <class Scalar_, int Options>
struct traits<Sophus::SE2<Scalar_, Options>> {
  using Scalar = Scalar_;
  using TranslationType = Sophus::Vector2<Scalar, Options>;
  using SO2Type = Sophus::SO2<Scalar, Options>;
};

template <class Scalar_, int Options>
struct traits<Map<Sophus::SE2<Scalar_>, Options>>
    : traits<Sophus::SE2<Scalar_, Options>> {
  using Scalar = Scalar_;
  using TranslationType = Map<Sophus::Vector2<Scalar>, Options>;
  using SO2Type = Map<Sophus::SO2<Scalar>, Options>;
};

template <class Scalar_, int Options>
struct traits<Map<Sophus::SE2<Scalar_> const, Options>>
    : traits<Sophus::SE2<Scalar_, Options> const> {
  using Scalar = Scalar_;
  using TranslationType = Map<Sophus::Vector2<Scalar> const, Options>;
  using SO2Type = Map<Sophus::SO2<Scalar> const, Options>;
};
}  // namespace internal
}  // namespace Eigen

namespace Sophus {

/// SE2 base type - implements SE2 class but is storage agnostic.
///
/// SE(2) is the group of rotations  and translation in 2d. It is the
/// semi-direct product of SO(2) and the 2d Euclidean vector space.  The class
/// is represented using a composition of SO2Group  for rotation and a 2-vector
/// for translation.
///
/// SE(2) is neither compact, nor a commutative group.
///
/// See SO2Group for more details of the rotation representation in 2d.
///
template <class Derived>
class SE2Base {
 public:
  using Scalar = typename Eigen::internal::traits<Derived>::Scalar;
  using TranslationType =
      typename Eigen::internal::traits<Derived>::TranslationType;
  using SO2Type = typename Eigen::internal::traits<Derived>::SO2Type;

  /// Degrees of freedom of manifold, number of dimensions in tangent space
  /// (two for translation, three for rotation).
  static int constexpr DoF = 3;
  /// Number of internal parameters used (tuple for complex, two for
  /// translation).
  static int constexpr num_parameters = 4;
  /// Group transformations are 3x3 matrices.
  static int constexpr N = 3;
  using Transformation = Matrix<Scalar, N, N>;
  using Point = Vector2<Scalar>;
  using HomogeneousPoint = Vector3<Scalar>;
  using Line = ParametrizedLine2<Scalar>;
  using Tangent = Vector<Scalar, DoF>;
  using Adjoint = Matrix<Scalar, DoF, DoF>;

  /// For binary operations the return type is determined with the
  /// ScalarBinaryOpTraits feature of Eigen. This allows mixing concrete and Map
  /// types, as well as other compatible scalar types such as Ceres::Jet and
  /// double scalars with SE2 operations.
  template <typename OtherDerived>
  using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<
      Scalar, typename OtherDerived::Scalar>::ReturnType;

  template <typename OtherDerived>
  using SE2Product = SE2<ReturnScalar<OtherDerived>>;

  template <typename PointDerived>
  using PointProduct = Vector2<ReturnScalar<PointDerived>>;

  template <typename HPointDerived>
  using HomogeneousPointProduct = Vector3<ReturnScalar<HPointDerived>>;

  /// Adjoint transformation
  ///
  /// This function return the adjoint transformation ``Ad`` of the group
  /// element ``A`` such that for all ``x`` it holds that
  /// ``hat(Ad_A * x) = A * hat(x) A^{-1}``. See hat-operator below.
  ///
  SOPHUS_FUNC Adjoint Adj() const {
    Matrix<Scalar, 2, 2> const& R = so2().matrix();
    Transformation res;
    res.setIdentity();
    res.template topLeftCorner<2, 2>() = R;
    res(0, 2) = translation()[1];
    res(1, 2) = -translation()[0];
    return res;
  }

  /// Returns copy of instance casted to NewScalarType.
  ///
  template <class NewScalarType>
  SOPHUS_FUNC SE2<NewScalarType> cast() const {
    return SE2<NewScalarType>(so2().template cast<NewScalarType>(),
                              translation().template cast<NewScalarType>());
  }

  /// Returns derivative of  this * exp(x)  wrt x at x=0.
  ///
  SOPHUS_FUNC Matrix<Scalar, num_parameters, DoF> Dx_this_mul_exp_x_at_0()
      const {
    Matrix<Scalar, num_parameters, DoF> J;
    Sophus::Vector2<Scalar> const c = unit_complex();
    Scalar o(0);
    J(0, 0) = o;
    J(0, 1) = o;
    J(0, 2) = -c[1];
    J(1, 0) = o;
    J(1, 1) = o;
    J(1, 2) = c[0];
    J(2, 0) = c[0];
    J(2, 1) = -c[1];
    J(2, 2) = o;
    J(3, 0) = c[1];
    J(3, 1) = c[0];
    J(3, 2) = o;
    return J;
  }

  /// Returns group inverse.
  ///
  SOPHUS_FUNC SE2<Scalar> inverse() const {
    SO2<Scalar> const invR = so2().inverse();
    return SE2<Scalar>(invR, invR * (translation() * Scalar(-1)));
  }

  /// Logarithmic map
  ///
  /// Computes the logarithm, the inverse of the group exponential which maps
  /// element of the group (rigid body transformations) to elements of the
  /// tangent space (twist).
  ///
  /// To be specific, this function computes ``vee(logmat(.))`` with
  /// ``logmat(.)`` being the matrix logarithm and ``vee(.)`` the vee-operator
  /// of SE(2).
  ///
  SOPHUS_FUNC Tangent log() const {
    using std::abs;

    Tangent upsilon_theta;
    Scalar theta = so2().log();
    upsilon_theta[2] = theta;
    Scalar halftheta = Scalar(0.5) * theta;
    Scalar halftheta_by_tan_of_halftheta;

    Vector2<Scalar> z = so2().unit_complex();
    Scalar real_minus_one = z.x() - Scalar(1.);
    if (abs(real_minus_one) < Constants<Scalar>::epsilon()) {
      halftheta_by_tan_of_halftheta =
          Scalar(1.) - Scalar(1. / 12) * theta * theta;
    } else {
      halftheta_by_tan_of_halftheta = -(halftheta * z.y()) / (real_minus_one);
    }
    Matrix<Scalar, 2, 2> V_inv;
    V_inv << halftheta_by_tan_of_halftheta, halftheta, -halftheta,
        halftheta_by_tan_of_halftheta;
    upsilon_theta.template head<2>() = V_inv * translation();
    return upsilon_theta;
  }

  /// Normalize SO2 element
  ///
  /// It re-normalizes the SO2 element.
  ///
  SOPHUS_FUNC void normalize() { so2().normalize(); }

  /// Returns 3x3 matrix representation of the instance.
  ///
  /// It has the following form:
  ///
  ///   | R t |
  ///   | o 1 |
  ///
  /// where ``R`` is a 2x2 rotation matrix, ``t`` a translation 2-vector and
  /// ``o`` a 2-column vector of zeros.
  ///
  SOPHUS_FUNC Transformation matrix() const {
    Transformation homogenious_matrix;
    homogenious_matrix.template topLeftCorner<2, 3>() = matrix2x3();
    homogenious_matrix.row(2) =
        Matrix<Scalar, 1, 3>(Scalar(0), Scalar(0), Scalar(1));
    return homogenious_matrix;
  }

  /// Returns the significant first two rows of the matrix above.
  ///
  SOPHUS_FUNC Matrix<Scalar, 2, 3> matrix2x3() const {
    Matrix<Scalar, 2, 3> matrix;
    matrix.template topLeftCorner<2, 2>() = rotationMatrix();
    matrix.col(2) = translation();
    return matrix;
  }

  /// Assignment operator.
  ///
  SOPHUS_FUNC SE2Base& operator=(SE2Base const& other) = default;

  /// Assignment-like operator from OtherDerived.
  ///
  template <class OtherDerived>
  SOPHUS_FUNC SE2Base<Derived>& operator=(SE2Base<OtherDerived> const& other) {
    so2() = other.so2();
    translation() = other.translation();
    return *this;
  }

  /// Group multiplication, which is rotation concatenation.
  ///
  template <typename OtherDerived>
  SOPHUS_FUNC SE2Product<OtherDerived> operator*(
      SE2Base<OtherDerived> const& other) const {
    return SE2Product<OtherDerived>(
        so2() * other.so2(), translation() + so2() * other.translation());
  }

  /// Group action on 2-points.
  ///
  /// This function rotates and translates a two dimensional point ``p`` by the
  /// SE(2) element ``bar_T_foo = (bar_R_foo, t_bar)`` (= rigid body
  /// transformation):
  ///
  ///   ``p_bar = bar_R_foo * p_foo + t_bar``.
  ///
  template <typename PointDerived,
            typename = typename std::enable_if<
                IsFixedSizeVector<PointDerived, 2>::value>::type>
  SOPHUS_FUNC PointProduct<PointDerived> operator*(
      Eigen::MatrixBase<PointDerived> const& p) const {
    return so2() * p + translation();
  }

  /// Group action on homogeneous 2-points. See above for more details.
  ///
  template <typename HPointDerived,
            typename = typename std::enable_if<
                IsFixedSizeVector<HPointDerived, 3>::value>::type>
  SOPHUS_FUNC HomogeneousPointProduct<HPointDerived> operator*(
      Eigen::MatrixBase<HPointDerived> const& p) const {
    const PointProduct<HPointDerived> tp =
        so2() * p.template head<2>() + p(2) * translation();
    return HomogeneousPointProduct<HPointDerived>(tp(0), tp(1), p(2));
  }

  /// Group action on lines.
  ///
  /// This function rotates and translates a parametrized line
  /// ``l(t) = o + t * d`` by the SE(2) element:
  ///
  /// Origin ``o`` is rotated and translated using SE(2) action
  /// Direction ``d`` is rotated using SO(2) action
  ///
  SOPHUS_FUNC Line operator*(Line const& l) const {
    return Line((*this) * l.origin(), so2() * l.direction());
  }

  /// In-place group multiplication. This method is only valid if the return
  /// type of the multiplication is compatible with this SO2's Scalar type.
  ///
  template <typename OtherDerived,
            typename = typename std::enable_if<
                std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type>
  SOPHUS_FUNC SE2Base<Derived>& operator*=(SE2Base<OtherDerived> const& other) {
    *static_cast<Derived*>(this) = *this * other;
    return *this;
  }

  /// Returns internal parameters of SE(2).
  ///
  /// It returns (c[0], c[1], t[0], t[1]),
  /// with c being the unit complex number, t the translation 3-vector.
  ///
  SOPHUS_FUNC Sophus::Vector<Scalar, num_parameters> params() const {
    Sophus::Vector<Scalar, num_parameters> p;
    p << so2().params(), translation();
    return p;
  }

  /// Returns rotation matrix.
  ///
  SOPHUS_FUNC Matrix<Scalar, 2, 2> rotationMatrix() const {
    return so2().matrix();
  }

  /// Takes in complex number, and normalizes it.
  ///
  /// Precondition: The complex number must not be close to zero.
  ///
  SOPHUS_FUNC void setComplex(Sophus::Vector2<Scalar> const& complex) {
    return so2().setComplex(complex);
  }

  /// Sets ``so3`` using ``rotation_matrix``.
  ///
  /// Precondition: ``R`` must be orthogonal and ``det(R)=1``.
  ///
  SOPHUS_FUNC void setRotationMatrix(Matrix<Scalar, 2, 2> const& R) {
    SOPHUS_ENSURE(isOrthogonal(R), "R is not orthogonal:\n %", R);
    SOPHUS_ENSURE(R.determinant() > Scalar(0), "det(R) is not positive: %",
                  R.determinant());
    typename SO2Type::ComplexTemporaryType const complex(
        Scalar(0.5) * (R(0, 0) + R(1, 1)), Scalar(0.5) * (R(1, 0) - R(0, 1)));
    so2().setComplex(complex);
  }

  /// Mutator of SO3 group.
  ///
  SOPHUS_FUNC
  SO2Type& so2() { return static_cast<Derived*>(this)->so2(); }

  /// Accessor of SO3 group.
  ///
  SOPHUS_FUNC
  SO2Type const& so2() const {
    return static_cast<Derived const*>(this)->so2();
  }

  /// Mutator of translation vector.
  ///
  SOPHUS_FUNC
  TranslationType& translation() {
    return static_cast<Derived*>(this)->translation();
  }

  /// Accessor of translation vector
  ///
  SOPHUS_FUNC
  TranslationType const& translation() const {
    return static_cast<Derived const*>(this)->translation();
  }

  /// Accessor of unit complex number.
  ///
  SOPHUS_FUNC
  typename Eigen::internal::traits<Derived>::SO2Type::ComplexT const&
  unit_complex() const {
    return so2().unit_complex();
  }
};

/// SE2 using default storage; derived from SE2Base.
template <class Scalar_, int Options>
class SE2 : public SE2Base<SE2<Scalar_, Options>> {
 public:
  using Base = SE2Base<SE2<Scalar_, Options>>;
  static int constexpr DoF = Base::DoF;
  static int constexpr num_parameters = Base::num_parameters;

  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;
  using SO2Member = SO2<Scalar, Options>;
  using TranslationMember = Vector2<Scalar, Options>;

  EIGEN_MAKE_ALIGNED_OPERATOR_NEW

  /// Default constructor initializes rigid body motion to the identity.
  ///
  SOPHUS_FUNC SE2();

  /// Copy constructor
  ///
  SOPHUS_FUNC SE2(SE2 const& other) = default;

  /// Copy-like constructor from OtherDerived
  ///
  template <class OtherDerived>
  SOPHUS_FUNC SE2(SE2Base<OtherDerived> const& other)
      : so2_(other.so2()), translation_(other.translation()) {
    static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
                  "must be same Scalar type");
  }

  /// Constructor from SO3 and translation vector
  ///
  template <class OtherDerived, class D>
  SOPHUS_FUNC SE2(SO2Base<OtherDerived> const& so2,
                  Eigen::MatrixBase<D> const& translation)
      : so2_(so2), translation_(translation) {
    static_assert(std::is_same<typename OtherDerived::Scalar, Scalar>::value,
                  "must be same Scalar type");
    static_assert(std::is_same<typename D::Scalar, Scalar>::value,
                  "must be same Scalar type");
  }

  /// Constructor from rotation matrix and translation vector
  ///
  /// Precondition: Rotation matrix needs to be orthogonal with determinant
  /// of 1.
  ///
  SOPHUS_FUNC
  SE2(typename SO2<Scalar>::Transformation const& rotation_matrix,
      Point const& translation)
      : so2_(rotation_matrix), translation_(translation) {}

  /// Constructor from rotation angle and translation vector.
  ///
  SOPHUS_FUNC SE2(Scalar const& theta, Point const& translation)
      : so2_(theta), translation_(translation) {}

  /// Constructor from complex number and translation vector
  ///
  /// Precondition: ``complex`` must not be close to zero.
  SOPHUS_FUNC SE2(Vector2<Scalar> const& complex, Point const& translation)
      : so2_(complex), translation_(translation) {}

  /// Constructor from 3x3 matrix
  ///
  /// Precondition: Rotation matrix needs to be orthogonal with determinant
  /// of 1. The last row must be ``(0, 0, 1)``.
  ///
  SOPHUS_FUNC explicit SE2(Transformation const& T)
      : so2_(T.template topLeftCorner<2, 2>().eval()),
        translation_(T.template block<2, 1>(0, 2)) {}

  /// This provides unsafe read/write access to internal data. SO(2) is
  /// represented by a complex number (two parameters). When using direct write
  /// access, the user needs to take care of that the complex number stays
  /// normalized.
  ///
  SOPHUS_FUNC Scalar* data() {
    // so2_ and translation_ are layed out sequentially with no padding
    return so2_.data();
  }

  /// Const version of data() above.
  ///
  SOPHUS_FUNC Scalar const* data() const {
    /// so2_ and translation_ are layed out sequentially with no padding
    return so2_.data();
  }

  /// Accessor of SO3
  ///
  SOPHUS_FUNC SO2Member& so2() { return so2_; }

  /// Mutator of SO3
  ///
  SOPHUS_FUNC SO2Member const& so2() const { return so2_; }

  /// Mutator of translation vector
  ///
  SOPHUS_FUNC TranslationMember& translation() { return translation_; }

  /// Accessor of translation vector
  ///
  SOPHUS_FUNC TranslationMember const& translation() const {
    return translation_;
  }

  /// Returns derivative of exp(x) wrt. x.
  ///
  SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF> Dx_exp_x(
      Tangent const& upsilon_theta) {
    using std::abs;
    using std::cos;
    using std::pow;
    using std::sin;
    Sophus::Matrix<Scalar, num_parameters, DoF> J;
    Sophus::Vector<Scalar, 2> upsilon = upsilon_theta.template head<2>();
    Scalar theta = upsilon_theta[2];

    if (abs(theta) < Constants<Scalar>::epsilon()) {
      Scalar const o(0);
      Scalar const i(1);

      // clang-format off
      J << o, o, o, o, o, i, i, o, -Scalar(0.5) * upsilon[1], o, i,
          Scalar(0.5) * upsilon[0];
      // clang-format on
      return J;
    }

    Scalar const c0 = sin(theta);
    Scalar const c1 = cos(theta);
    Scalar const c2 = 1.0 / theta;
    Scalar const c3 = c0 * c2;
    Scalar const c4 = -c1 + Scalar(1);
    Scalar const c5 = c2 * c4;
    Scalar const c6 = c1 * c2;
    Scalar const c7 = pow(theta, -2);
    Scalar const c8 = c0 * c7;
    Scalar const c9 = c4 * c7;

    Scalar const o = Scalar(0);
    J(0, 0) = o;
    J(0, 1) = o;
    J(0, 2) = -c0;
    J(1, 0) = o;
    J(1, 1) = o;
    J(1, 2) = c1;
    J(2, 0) = c3;
    J(2, 1) = -c5;
    J(2, 2) =
        -c3 * upsilon[1] + c6 * upsilon[0] - c8 * upsilon[0] + c9 * upsilon[1];
    J(3, 0) = c5;
    J(3, 1) = c3;
    J(3, 2) =
        c3 * upsilon[0] + c6 * upsilon[1] - c8 * upsilon[1] - c9 * upsilon[0];
    return J;
  }

  /// Returns derivative of exp(x) wrt. x_i at x=0.
  ///
  SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF>
  Dx_exp_x_at_0() {
    Sophus::Matrix<Scalar, num_parameters, DoF> J;
    Scalar const o(0);
    Scalar const i(1);

    // clang-format off
    J << o, o, o, o, o, i, i, o, o, o, i, o;
    // clang-format on
    return J;
  }

  /// Returns derivative of exp(x).matrix() wrt. ``x_i at x=0``.
  ///
  SOPHUS_FUNC static Transformation Dxi_exp_x_matrix_at_0(int i) {
    return generator(i);
  }

  /// Group exponential
  ///
  /// This functions takes in an element of tangent space (= twist ``a``) and
  /// returns the corresponding element of the group SE(2).
  ///
  /// The first two components of ``a`` represent the translational part
  /// ``upsilon`` in the tangent space of SE(2), while the last three components
  /// of ``a`` represents the rotation vector ``omega``.
  /// To be more specific, this function computes ``expmat(hat(a))`` with
  /// ``expmat(.)`` being the matrix exponential and ``hat(.)`` the hat-operator
  /// of SE(2), see below.
  ///
  SOPHUS_FUNC static SE2<Scalar> exp(Tangent const& a) {
    Scalar theta = a[2];
    SO2<Scalar> so2 = SO2<Scalar>::exp(theta);
    Scalar sin_theta_by_theta;
    Scalar one_minus_cos_theta_by_theta;
    using std::abs;

    if (abs(theta) < Constants<Scalar>::epsilon()) {
      Scalar theta_sq = theta * theta;
      sin_theta_by_theta = Scalar(1.) - Scalar(1. / 6.) * theta_sq;
      one_minus_cos_theta_by_theta =
          Scalar(0.5) * theta - Scalar(1. / 24.) * theta * theta_sq;
    } else {
      sin_theta_by_theta = so2.unit_complex().y() / theta;
      one_minus_cos_theta_by_theta =
          (Scalar(1.) - so2.unit_complex().x()) / theta;
    }
    Vector2<Scalar> trans(
        sin_theta_by_theta * a[0] - one_minus_cos_theta_by_theta * a[1],
        one_minus_cos_theta_by_theta * a[0] + sin_theta_by_theta * a[1]);
    return SE2<Scalar>(so2, trans);
  }

  /// Returns closest SE3 given arbitrary 4x4 matrix.
  ///
  template <class S = Scalar>
  static SOPHUS_FUNC enable_if_t<std::is_floating_point<S>::value, SE2>
  fitToSE2(Matrix3<Scalar> const& T) {
    return SE2(SO2<Scalar>::fitToSO2(T.template block<2, 2>(0, 0)),
               T.template block<2, 1>(0, 2));
  }

  /// Returns the ith infinitesimal generators of SE(2).
  ///
  /// The infinitesimal generators of SE(2) are:
  ///
  /// ```
  ///         |  0  0  1 |
  ///   G_0 = |  0  0  0 |
  ///         |  0  0  0 |
  ///
  ///         |  0  0  0 |
  ///   G_1 = |  0  0  1 |
  ///         |  0  0  0 |
  ///
  ///         |  0 -1  0 |
  ///   G_2 = |  1  0  0 |
  ///         |  0  0  0 |
  /// ```
  ///
  /// Precondition: ``i`` must be in 0, 1 or 2.
  ///
  SOPHUS_FUNC static Transformation generator(int i) {
    SOPHUS_ENSURE(i >= 0 || i <= 2, "i should be in range [0,2].");
    Tangent e;
    e.setZero();
    e[i] = Scalar(1);
    return hat(e);
  }

  /// hat-operator
  ///
  /// It takes in the 3-vector representation (= twist) and returns the
  /// corresponding matrix representation of Lie algebra element.
  ///
  /// Formally, the hat()-operator of SE(3) is defined as
  ///
  ///   ``hat(.): R^3 -> R^{3x33},  hat(a) = sum_i a_i * G_i``  (for i=0,1,2)
  ///
  /// with ``G_i`` being the ith infinitesimal generator of SE(2).
  ///
  /// The corresponding inverse is the vee()-operator, see below.
  ///
  SOPHUS_FUNC static Transformation hat(Tangent const& a) {
    Transformation Omega;
    Omega.setZero();
    Omega.template topLeftCorner<2, 2>() = SO2<Scalar>::hat(a[2]);
    Omega.col(2).template head<2>() = a.template head<2>();
    return Omega;
  }

  /// Lie bracket
  ///
  /// It computes the Lie bracket of SE(2). To be more specific, it computes
  ///
  ///   ``[omega_1, omega_2]_se2 := vee([hat(omega_1), hat(omega_2)])``
  ///
  /// with ``[A,B] := AB-BA`` being the matrix commutator, ``hat(.)`` the
  /// hat()-operator and ``vee(.)`` the vee()-operator of SE(2).
  ///
  SOPHUS_FUNC static Tangent lieBracket(Tangent const& a, Tangent const& b) {
    Vector2<Scalar> upsilon1 = a.template head<2>();
    Vector2<Scalar> upsilon2 = b.template head<2>();
    Scalar theta1 = a[2];
    Scalar theta2 = b[2];

    return Tangent(-theta1 * upsilon2[1] + theta2 * upsilon1[1],
                   theta1 * upsilon2[0] - theta2 * upsilon1[0], Scalar(0));
  }

  /// Construct pure rotation.
  ///
  static SOPHUS_FUNC SE2 rot(Scalar const& x) {
    return SE2(SO2<Scalar>(x), Sophus::Vector2<Scalar>::Zero());
  }

  /// Draw uniform sample from SE(2) manifold.
  ///
  /// Translations are drawn component-wise from the range [-1, 1].
  ///
  template <class UniformRandomBitGenerator>
  static SE2 sampleUniform(UniformRandomBitGenerator& generator) {
    std::uniform_real_distribution<Scalar> uniform(Scalar(-1), Scalar(1));
    return SE2(SO2<Scalar>::sampleUniform(generator),
               Vector2<Scalar>(uniform(generator), uniform(generator)));
  }

  /// Construct a translation only SE(2) instance.
  ///
  template <class T0, class T1>
  static SOPHUS_FUNC SE2 trans(T0 const& x, T1 const& y) {
    return SE2(SO2<Scalar>(), Vector2<Scalar>(x, y));
  }

  static SOPHUS_FUNC SE2 trans(Vector2<Scalar> const& xy) {
    return SE2(SO2<Scalar>(), xy);
  }

  /// Construct x-axis translation.
  ///
  static SOPHUS_FUNC SE2 transX(Scalar const& x) {
    return SE2::trans(x, Scalar(0));
  }

  /// Construct y-axis translation.
  ///
  static SOPHUS_FUNC SE2 transY(Scalar const& y) {
    return SE2::trans(Scalar(0), y);
  }

  /// vee-operator
  ///
  /// It takes the 3x3-matrix representation ``Omega`` and maps it to the
  /// corresponding 3-vector representation of Lie algebra.
  ///
  /// This is the inverse of the hat()-operator, see above.
  ///
  /// Precondition: ``Omega`` must have the following structure:
  ///
  ///                |  0 -d  a |
  ///                |  d  0  b |
  ///                |  0  0  0 |
  ///
  SOPHUS_FUNC static Tangent vee(Transformation const& Omega) {
    SOPHUS_ENSURE(
        Omega.row(2).template lpNorm<1>() < Constants<Scalar>::epsilon(),
        "Omega: \n%", Omega);
    Tangent upsilon_omega;
    upsilon_omega.template head<2>() = Omega.col(2).template head<2>();
    upsilon_omega[2] = SO2<Scalar>::vee(Omega.template topLeftCorner<2, 2>());
    return upsilon_omega;
  }

 protected:
  SO2Member so2_;
  TranslationMember translation_;
};

template <class Scalar, int Options>
SE2<Scalar, Options>::SE2() : translation_(TranslationMember::Zero()) {
  static_assert(std::is_standard_layout<SE2>::value,
                "Assume standard layout for the use of offsetof check below.");
  static_assert(
      offsetof(SE2, so2_) + sizeof(Scalar) * SO2<Scalar>::num_parameters ==
          offsetof(SE2, translation_),
      "This class assumes packed storage and hence will only work "
      "correctly depending on the compiler (options) - in "
      "particular when using [this->data(), this-data() + "
      "num_parameters] to access the raw data in a contiguous fashion.");
}

}  // namespace Sophus

namespace Eigen {

/// Specialization of Eigen::Map for ``SE2``; derived from SE2Base.
///
/// Allows us to wrap SE2 objects around POD array.
template <class Scalar_, int Options>
class Map<Sophus::SE2<Scalar_>, Options>
    : public Sophus::SE2Base<Map<Sophus::SE2<Scalar_>, Options>> {
 public:
  using Base = Sophus::SE2Base<Map<Sophus::SE2<Scalar_>, Options>>;
  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;

  // LCOV_EXCL_START
  SOPHUS_INHERIT_ASSIGNMENT_OPERATORS(Map);
  // LCOV_EXCL_STOP

  using Base::operator*=;
  using Base::operator*;

  SOPHUS_FUNC
  Map(Scalar* coeffs)
      : so2_(coeffs),
        translation_(coeffs + Sophus::SO2<Scalar>::num_parameters) {}

  /// Mutator of SO3
  ///
  SOPHUS_FUNC Map<Sophus::SO2<Scalar>, Options>& so2() { return so2_; }

  /// Accessor of SO3
  ///
  SOPHUS_FUNC Map<Sophus::SO2<Scalar>, Options> const& so2() const {
    return so2_;
  }

  /// Mutator of translation vector
  ///
  SOPHUS_FUNC Map<Sophus::Vector2<Scalar>, Options>& translation() {
    return translation_;
  }

  /// Accessor of translation vector
  ///
  SOPHUS_FUNC Map<Sophus::Vector2<Scalar>, Options> const& translation() const {
    return translation_;
  }

 protected:
  Map<Sophus::SO2<Scalar>, Options> so2_;
  Map<Sophus::Vector2<Scalar>, Options> translation_;
};

/// Specialization of Eigen::Map for ``SE2 const``; derived from SE2Base.
///
/// Allows us to wrap SE2 objects around POD array.
template <class Scalar_, int Options>
class Map<Sophus::SE2<Scalar_> const, Options>
    : public Sophus::SE2Base<Map<Sophus::SE2<Scalar_> const, Options>> {
 public:
  using Base = Sophus::SE2Base<Map<Sophus::SE2<Scalar_> const, Options>>;
  using Scalar = Scalar_;
  using Transformation = typename Base::Transformation;
  using Point = typename Base::Point;
  using HomogeneousPoint = typename Base::HomogeneousPoint;
  using Tangent = typename Base::Tangent;
  using Adjoint = typename Base::Adjoint;

  using Base::operator*=;
  using Base::operator*;

  SOPHUS_FUNC Map(Scalar const* coeffs)
      : so2_(coeffs),
        translation_(coeffs + Sophus::SO2<Scalar>::num_parameters) {}

  /// Accessor of SO3
  ///
  SOPHUS_FUNC Map<Sophus::SO2<Scalar> const, Options> const& so2() const {
    return so2_;
  }

  /// Accessor of translation vector
  ///
  SOPHUS_FUNC Map<Sophus::Vector2<Scalar> const, Options> const& translation()
      const {
    return translation_;
  }

 protected:
  Map<Sophus::SO2<Scalar> const, Options> const so2_;
  Map<Sophus::Vector2<Scalar> const, Options> const translation_;
};
}  // namespace Eigen

#endif