/// @file
/// Direct product R X SO(2) - rotation and scaling in 2d.

#ifndef SOPHUS_RXSO2_HPP
#define SOPHUS_RXSO2_HPP

#include "so2.hpp"

namespace Sophus {
template <class Scalar_, int Options = 0>
class RxSO2;
using RxSO2d = RxSO2<double>;
using RxSO2f = RxSO2<float>;
}  // namespace Sophus

namespace Eigen {
namespace internal {

template <class Scalar_, int Options_>
struct traits<Sophus::RxSO2<Scalar_, Options_>> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using ComplexType = Sophus::Vector2<Scalar, Options>;
};

template <class Scalar_, int Options_>
struct traits<Map<Sophus::RxSO2<Scalar_>, Options_>>
    : traits<Sophus::RxSO2<Scalar_, Options_>> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using ComplexType = Map<Sophus::Vector2<Scalar>, Options>;
};

template <class Scalar_, int Options_>
struct traits<Map<Sophus::RxSO2<Scalar_> const, Options_>>
    : traits<Sophus::RxSO2<Scalar_, Options_> const> {
  static constexpr int Options = Options_;
  using Scalar = Scalar_;
  using ComplexType = Map<Sophus::Vector2<Scalar> const, Options>;
};
}  // namespace internal
}  // namespace Eigen

namespace Sophus {

/// RxSO2 base type - implements RxSO2 class but is storage agnostic
///
/// This class implements the group ``R+ x SO(2)``, the direct product of the
/// group of positive scalar 2x2 matrices (= isomorph to the positive
/// real numbers) and the two-dimensional special orthogonal group SO(2).
/// Geometrically, it is the group of rotation and scaling in two dimensions.
/// As a matrix groups, R+ x SO(2) consists of matrices of the form ``s * R``
/// where ``R`` is an orthogonal matrix with ``det(R) = 1`` and ``s > 0``
/// being a positive real number. In particular, it has the following form:
///
///     | s * cos(theta)  s * -sin(theta) |
///     | s * sin(theta)  s *  cos(theta) |
///
/// where ``theta`` being the rotation angle. Internally, it is represented by
/// the first column of the rotation matrix, or in other words by a non-zero
/// complex number.
///
/// R+ x SO(2) is not compact, but a commutative group. First it is not compact
/// since the scale factor is not bound. Second it is commutative since
/// ``sR(alpha, s1) * sR(beta, s2) = sR(beta, s2) * sR(alpha, s1)``,  simply
/// because ``alpha + beta = beta + alpha`` and ``s1 * s2 = s2 * s1`` with
/// ``alpha`` and ``beta`` being rotation angles and ``s1``, ``s2`` being scale
/// factors.
///
/// This class has the explicit class invariant that the scale ``s`` is not
/// too close to either zero or infinity. Strictly speaking, it must hold that:
///
///   ``complex().norm() >= Constants::epsilon()`` and
///   ``1. / complex().norm() >= Constants::epsilon()``.
///
/// In order to obey this condition, group multiplication is implemented with
/// saturation such that a product always has a scale which is equal or greater
/// this threshold.
template <class Derived>
class RxSO2Base {
 public:
  static constexpr int Options = Eigen::internal::traits<Derived>::Options;
  using Scalar = typename Eigen::internal::traits<Derived>::Scalar;
  using ComplexType = typename Eigen::internal::traits<Derived>::ComplexType;
  using ComplexTemporaryType = Sophus::Vector2<Scalar, Options>;

  /// Degrees of freedom of manifold, number of dimensions in tangent space
  /// (one for rotation and one for scaling).
  static int constexpr DoF = 2;
  /// Number of internal parameters used (complex number is a tuple).
  static int constexpr num_parameters = 2;
  /// Group transformations are 2x2 matrices.
  static int constexpr N = 2;
  using Transformation = Matrix<Scalar, N, N>;
  using Point = Vector2<Scalar>;
  using HomogeneousPoint = Vector3<Scalar>;
  using Line = ParametrizedLine2<Scalar>;
  using Hyperplane = Hyperplane2<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 RxSO2 operations.
  template <typename OtherDerived>
  using ReturnScalar = typename Eigen::ScalarBinaryOpTraits<
      Scalar, typename OtherDerived::Scalar>::ReturnType;

  template <typename OtherDerived>
  using RxSO2Product = RxSO2<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.
  ///
  /// For RxSO(2), it simply returns the identity matrix.
  ///
  SOPHUS_FUNC Adjoint Adj() const { return Adjoint::Identity(); }

  /// Returns rotation angle.
  ///
  SOPHUS_FUNC Scalar angle() const { return SO2<Scalar>(complex()).log(); }

  /// Returns copy of instance casted to NewScalarType.
  ///
  template <class NewScalarType>
  SOPHUS_FUNC RxSO2<NewScalarType> cast() const {
    return RxSO2<NewScalarType>(complex().template cast<NewScalarType>());
  }

  /// This provides unsafe read/write access to internal data. RxSO(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 is
  /// not set close to zero.
  ///
  /// Note: The first parameter represents the real part, while the
  /// second parameter represent the imaginary part.
  ///
  SOPHUS_FUNC Scalar* data() { return complex_nonconst().data(); }

  /// Const version of data() above.
  ///
  SOPHUS_FUNC Scalar const* data() const { return complex().data(); }

  /// Returns group inverse.
  ///
  SOPHUS_FUNC RxSO2<Scalar> inverse() const {
    Scalar squared_scale = complex().squaredNorm();
    return RxSO2<Scalar>(complex().x() / squared_scale,
                         -complex().y() / squared_scale);
  }

  /// Logarithmic map
  ///
  /// Computes the logarithm, the inverse of the group exponential which maps
  /// element of the group (scaled rotation matrices) to elements of the tangent
  /// space (rotation-vector plus logarithm of scale factor).
  ///
  /// To be specific, this function computes ``vee(logmat(.))`` with
  /// ``logmat(.)`` being the matrix logarithm and ``vee(.)`` the vee-operator
  /// of RxSO2.
  ///
  SOPHUS_FUNC Tangent log() const {
    using std::log;
    Tangent theta_sigma;
    theta_sigma[1] = log(scale());
    theta_sigma[0] = SO2<Scalar>(complex()).log();
    return theta_sigma;
  }

  /// Returns 2x2 matrix representation of the instance.
  ///
  /// For RxSO2, the matrix representation is an scaled orthogonal matrix ``sR``
  /// with ``det(R)=s^2``, thus a scaled rotation matrix ``R``  with scale
  /// ``s``.
  ///
  SOPHUS_FUNC Transformation matrix() const {
    Transformation sR;
    // clang-format off
    sR << complex()[0], -complex()[1],
          complex()[1],  complex()[0];
    // clang-format on
    return sR;
  }

  /// Assignment-like operator from OtherDerived.
  ///
  template <class OtherDerived>
  SOPHUS_FUNC RxSO2Base<Derived>& operator=(
      RxSO2Base<OtherDerived> const& other) {
    complex_nonconst() = other.complex();
    return *this;
  }

  /// Group multiplication, which is rotation concatenation and scale
  /// multiplication.
  ///
  /// Note: This function performs saturation for products close to zero in
  /// order to ensure the class invariant.
  ///
  template <typename OtherDerived>
  SOPHUS_FUNC RxSO2Product<OtherDerived> operator*(
      RxSO2Base<OtherDerived> const& other) const {
    using ResultT = ReturnScalar<OtherDerived>;

    Scalar lhs_real = complex().x();
    Scalar lhs_imag = complex().y();
    typename OtherDerived::Scalar const& rhs_real = other.complex().x();
    typename OtherDerived::Scalar const& rhs_imag = other.complex().y();
    /// complex multiplication
    typename RxSO2Product<OtherDerived>::ComplexType result_complex(
        lhs_real * rhs_real - lhs_imag * rhs_imag,
        lhs_real * rhs_imag + lhs_imag * rhs_real);

    const ResultT squared_scale = result_complex.squaredNorm();

    if (squared_scale <
        Constants<ResultT>::epsilon() * Constants<ResultT>::epsilon()) {
      /// Saturation to ensure class invariant.
      result_complex.normalize();
      result_complex *= Constants<ResultT>::epsilonPlus();
    }
    if (squared_scale > Scalar(1.) / (Constants<ResultT>::epsilon() *
                                      Constants<ResultT>::epsilon())) {
      /// Saturation to ensure class invariant.
      result_complex.normalize();
      result_complex /= Constants<ResultT>::epsilonPlus();
    }
    return RxSO2Product<OtherDerived>(result_complex);
  }

  /// Group action on 2-points.
  ///
  /// This function rotates a 2 dimensional point ``p`` by the SO2 element
  /// ``bar_R_foo`` (= rotation matrix) and scales it by the scale factor ``s``:
  ///
  ///   ``p_bar = s * (bar_R_foo * p_foo)``.
  ///
  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 matrix() * p;
  }

  /// 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 auto rsp = *this * p.template head<2>();
    return HomogeneousPointProduct<HPointDerived>(rsp(0), rsp(1), p(2));
  }

  /// Group action on lines.
  ///
  /// This function rotates a parameterized line ``l(t) = o + t * d`` by the SO2
  /// element and scales it by the scale factor
  ///
  /// Origin ``o`` is rotated and scaled
  /// Direction ``d`` is rotated (preserving it's norm)
  ///
  SOPHUS_FUNC Line operator*(Line const& l) const {
    return Line((*this) * l.origin(), (*this) * l.direction() / scale());
  }

  /// Group action on hyper-planes.
  ///
  /// This function rotates a hyper-plane ``n.x + d = 0`` by the SO2
  /// element and scales offset by the scale factor
  ///
  /// Normal vector ``n`` is rotated
  /// Offset ``d`` is scaled
  ///
  /// Note that in 2d-case hyper-planes are just another parametrization of
  /// lines
  ///
  SOPHUS_FUNC Hyperplane operator*(Hyperplane const& p) const {
    const auto this_scale = scale();
    return Hyperplane((*this) * p.normal() / this_scale,
                      this_scale * p.offset());
  }

  /// In-place group multiplication. This method is only valid if the return
  /// type of the multiplication is compatible with this SO2's Scalar type.
  ///
  /// Note: This function performs saturation for products close to zero in
  /// order to ensure the class invariant.
  ///
  template <typename OtherDerived,
            typename = typename std::enable_if<
                std::is_same<Scalar, ReturnScalar<OtherDerived>>::value>::type>
  SOPHUS_FUNC RxSO2Base<Derived>& operator*=(
      RxSO2Base<OtherDerived> const& other) {
    *static_cast<Derived*>(this) = *this * other;
    return *this;
  }

  /// Returns derivative of  this * RxSO2::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;
    J << -complex().y(), complex().x(), complex().x(), complex().y();
    return J;
  }

  /// Returns internal parameters of RxSO(2).
  ///
  /// It returns (c[0], c[1]), with c being the  complex number.
  ///
  SOPHUS_FUNC Sophus::Vector<Scalar, num_parameters> params() const {
    return complex();
  }

  /// Sets non-zero complex
  ///
  /// Precondition: ``z`` must not be close to either zero or infinity.
  SOPHUS_FUNC void setComplex(Vector2<Scalar> const& z) {
    SOPHUS_ENSURE(z.squaredNorm() > Constants<Scalar>::epsilon() *
                                        Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon.");
    SOPHUS_ENSURE(z.squaredNorm() < Scalar(1.) / (Constants<Scalar>::epsilon() *
                                                  Constants<Scalar>::epsilon()),
                  "Inverse scale factor must be greate-equal epsilon.");
    static_cast<Derived*>(this)->complex_nonconst() = z;
  }

  /// Accessor of complex.
  ///
  SOPHUS_FUNC ComplexType const& complex() const {
    return static_cast<Derived const*>(this)->complex();
  }

  /// Returns rotation matrix.
  ///
  SOPHUS_FUNC Transformation rotationMatrix() const {
    ComplexTemporaryType norm_quad = complex();
    norm_quad.normalize();
    return SO2<Scalar>(norm_quad).matrix();
  }

  /// Returns scale.
  ///
  SOPHUS_FUNC
  Scalar scale() const { return complex().norm(); }

  /// Setter of rotation angle, leaves scale as is.
  ///
  SOPHUS_FUNC void setAngle(Scalar const& theta) { setSO2(SO2<Scalar>(theta)); }

  /// Setter of complex using rotation matrix ``R``, leaves scale as is.
  ///
  /// Precondition: ``R`` must be orthogonal with determinant of one.
  ///
  SOPHUS_FUNC void setRotationMatrix(Transformation const& R) {
    setSO2(SO2<Scalar>(R));
  }

  /// Sets scale and leaves rotation as is.
  ///
  SOPHUS_FUNC void setScale(Scalar const& scale) {
    using std::sqrt;
    complex_nonconst().normalize();
    complex_nonconst() *= scale;
  }

  /// Setter of complex number using scaled rotation matrix ``sR``.
  ///
  /// Precondition: The 2x2 matrix must be "scaled orthogonal"
  ///               and have a positive determinant.
  ///
  SOPHUS_FUNC void setScaledRotationMatrix(Transformation const& sR) {
    SOPHUS_ENSURE(isScaledOrthogonalAndPositive(sR),
                  "sR must be scaled orthogonal:\n {}", sR);
    complex_nonconst() = sR.col(0);
  }

  /// Setter of SO(2) rotations, leaves scale as is.
  ///
  SOPHUS_FUNC void setSO2(SO2<Scalar> const& so2) {
    using std::sqrt;
    Scalar saved_scale = scale();
    complex_nonconst() = so2.unit_complex();
    complex_nonconst() *= saved_scale;
  }

  SOPHUS_FUNC SO2<Scalar> so2() const { return SO2<Scalar>(complex()); }

 private:
  /// Mutator of complex is private to ensure class invariant.
  ///
  SOPHUS_FUNC ComplexType& complex_nonconst() {
    return static_cast<Derived*>(this)->complex_nonconst();
  }
};

/// RxSO2 using storage; derived from RxSO2Base.
template <class Scalar_, int Options>
class RxSO2 : public RxSO2Base<RxSO2<Scalar_, Options>> {
 public:
  using Base = RxSO2Base<RxSO2<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;
  using ComplexMember = Eigen::Matrix<Scalar, 2, 1, Options>;

  /// ``Base`` is friend so complex_nonconst can be accessed from ``Base``.
  friend class RxSO2Base<RxSO2<Scalar_, Options>>;

  using Base::operator=;

  /// Define copy-assignment operator explicitly. The definition of
  /// implicit copy assignment operator is deprecated in presence of a
  /// user-declared copy constructor (-Wdeprecated-copy in clang >= 13).
  SOPHUS_FUNC RxSO2& operator=(RxSO2 const& other) = default;

  static int constexpr DoF = Base::DoF;
  static int constexpr num_parameters = Base::num_parameters;

  EIGEN_MAKE_ALIGNED_OPERATOR_NEW

  /// Default constructor initializes complex number to identity rotation and
  /// scale to 1.
  ///
  SOPHUS_FUNC RxSO2() : complex_(Scalar(1), Scalar(0)) {}

  /// Copy constructor
  ///
  SOPHUS_FUNC RxSO2(RxSO2 const& other) = default;

  /// Copy-like constructor from OtherDerived.
  ///
  template <class OtherDerived>
  SOPHUS_FUNC RxSO2(RxSO2Base<OtherDerived> const& other)
      : complex_(other.complex()) {}

  /// Constructor from scaled rotation matrix
  ///
  /// Precondition: rotation matrix need to be scaled orthogonal with
  /// determinant of ``s^2``.
  ///
  SOPHUS_FUNC explicit RxSO2(Transformation const& sR) {
    this->setScaledRotationMatrix(sR);
  }

  /// Constructor from scale factor and rotation matrix ``R``.
  ///
  /// Precondition: Rotation matrix ``R`` must to be orthogonal with determinant
  ///               of 1 and ``scale`` must not to be close to either zero or
  ///               infinity.
  ///
  SOPHUS_FUNC RxSO2(Scalar const& scale, Transformation const& R)
      : RxSO2((scale * SO2<Scalar>(R).unit_complex()).eval()) {}

  /// Constructor from scale factor and SO2
  ///
  /// Precondition: ``scale`` must not be close to either zero or infinity.
  ///
  SOPHUS_FUNC RxSO2(Scalar const& scale, SO2<Scalar> const& so2)
      : RxSO2((scale * so2.unit_complex()).eval()) {}

  /// Constructor from complex number.
  ///
  /// Precondition: complex number must not be close to either zero or infinity
  ///
  SOPHUS_FUNC explicit RxSO2(Vector2<Scalar> const& z) : complex_(z) {
    SOPHUS_ENSURE(complex_.squaredNorm() >= Constants<Scalar>::epsilon() *
                                                Constants<Scalar>::epsilon(),
                  "Scale factor must be greater-equal epsilon: {} vs {}",
                  complex_.squaredNorm(),
                  Constants<Scalar>::epsilon() * Constants<Scalar>::epsilon());
    SOPHUS_ENSURE(
        complex_.squaredNorm() <= Scalar(1.) / (Constants<Scalar>::epsilon() *
                                                Constants<Scalar>::epsilon()),
        "Inverse scale factor must be greater-equal epsilon: % vs %",
        Scalar(1.) / complex_.squaredNorm(),
        Constants<Scalar>::epsilon() * Constants<Scalar>::epsilon());
  }

  /// Constructor from complex number.
  ///
  /// Precondition: complex number must not be close to either zero or inifnity.
  ///
  SOPHUS_FUNC explicit RxSO2(Scalar const& real, Scalar const& imag)
      : RxSO2(Vector2<Scalar>(real, imag)) {}

  /// Accessor of complex.
  ///
  SOPHUS_FUNC ComplexMember const& complex() const { return complex_; }

  /// Returns derivative of exp(x) wrt. ``x``
  ///
  SOPHUS_FUNC static Sophus::Matrix<Scalar, num_parameters, DoF> Dx_exp_x(
      Tangent const& a) {
    using std::cos;
    using std::exp;
    using std::sin;
    Scalar const theta = a[0];
    Scalar const sigma = a[1];

    Sophus::Matrix<Scalar, num_parameters, DoF> J;
    J << -sin(theta), cos(theta), cos(theta), sin(theta);
    return J * exp(sigma);
  }

  /// 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;
    static Scalar const i(1.);
    static Scalar const o(0.);
    J << o, i, i, o;
    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 (= rotation angle
  /// plus logarithm of scale) and returns the corresponding element of the
  /// group RxSO2.
  ///
  /// To be more specific, this function computes ``expmat(hat(theta))``
  /// with ``expmat(.)`` being the matrix exponential and ``hat(.)`` being the
  /// hat()-operator of RSO2.
  ///
  SOPHUS_FUNC static RxSO2<Scalar> exp(Tangent const& a) {
    using std::exp;
    using std::max;
    using std::min;

    Scalar const theta = a[0];
    Scalar const sigma = a[1];
    Scalar s = exp(sigma);
    // Ensuring proper scale
    s = max(s, Constants<Scalar>::epsilonPlus());
    s = min(s, Scalar(1.) / Constants<Scalar>::epsilonPlus());
    Vector2<Scalar> z = SO2<Scalar>::exp(theta).unit_complex();
    z *= s;
    return RxSO2<Scalar>(z);
  }

  /// Returns the ith infinitesimal generators of ``R+ x SO(2)``.
  ///
  /// The infinitesimal generators of RxSO2 are:
  ///
  /// ```
  ///         |  0 -1 |
  ///   G_0 = |  1  0 |
  ///
  ///         |  1  0 |
  ///   G_1 = |  0  1 |
  /// ```
  ///
  /// Precondition: ``i`` must be 0, or 1.
  ///
  SOPHUS_FUNC static Transformation generator(int i) {
    SOPHUS_ENSURE(i >= 0 && i <= 1, "i should be 0 or 1.");
    Tangent e;
    e.setZero();
    e[i] = Scalar(1);
    return hat(e);
  }

  /// hat-operator
  ///
  /// It takes in the 2-vector representation ``a`` (= rotation angle plus
  /// logarithm of scale) and  returns the corresponding matrix representation
  /// of Lie algebra element.
  ///
  /// Formally, the hat()-operator of RxSO2 is defined as
  ///
  ///   ``hat(.): R^2 -> R^{2x2},  hat(a) = sum_i a_i * G_i``  (for i=0,1,2)
  ///
  /// with ``G_i`` being the ith infinitesimal generator of RxSO2.
  ///
  /// The corresponding inverse is the vee()-operator, see below.
  ///
  SOPHUS_FUNC static Transformation hat(Tangent const& a) {
    Transformation A;
    // clang-format off
    A << a(1), -a(0),
         a(0),  a(1);
    // clang-format on
    return A;
  }

  /// Lie bracket
  ///
  /// It computes the Lie bracket of RxSO(2). To be more specific, it computes
  ///
  ///   ``[omega_1, omega_2]_rxso2 := 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 RxSO2.
  ///
  SOPHUS_FUNC static Tangent lieBracket(Tangent const&, Tangent const&) {
    Vector2<Scalar> res;
    res.setZero();
    return res;
  }

  /// Draw uniform sample from RxSO(2) manifold.
  ///
  /// The scale factor is drawn uniformly in log2-space from [-1, 1],
  /// hence the scale is in [0.5, 2)].
  ///
  template <class UniformRandomBitGenerator>
  static RxSO2 sampleUniform(UniformRandomBitGenerator& generator) {
    std::uniform_real_distribution<Scalar> uniform(Scalar(-1), Scalar(1));
    using std::exp2;
    return RxSO2(exp2(uniform(generator)),
                 SO2<Scalar>::sampleUniform(generator));
  }

  /// vee-operator
  ///
  /// It takes the 2x2-matrix representation ``Omega`` and maps it to the
  /// corresponding vector representation of Lie algebra.
  ///
  /// This is the inverse of the hat()-operator, see above.
  ///
  /// Precondition: ``Omega`` must have the following structure:
  ///
  ///                |  d -x |
  ///                |  x  d |
  ///
  SOPHUS_FUNC static Tangent vee(Transformation const& Omega) {
    using std::abs;
    return Tangent(Omega(1, 0), Omega(0, 0));
  }

 protected:
  SOPHUS_FUNC ComplexMember& complex_nonconst() { return complex_; }

  ComplexMember complex_;
};

}  // namespace Sophus

namespace Eigen {

/// Specialization of Eigen::Map for ``RxSO2``; derived from  RxSO2Base.
///
/// Allows us to wrap RxSO2 objects around POD array (e.g. external z style
/// complex).
template <class Scalar_, int Options>
class Map<Sophus::RxSO2<Scalar_>, Options>
    : public Sophus::RxSO2Base<Map<Sophus::RxSO2<Scalar_>, Options>> {
  using Base = Sophus::RxSO2Base<Map<Sophus::RxSO2<Scalar_>, Options>>;

 public:
  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;

  /// ``Base`` is friend so complex_nonconst can be accessed from ``Base``.
  friend class Sophus::RxSO2Base<Map<Sophus::RxSO2<Scalar_>, Options>>;

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

  SOPHUS_FUNC Map(Scalar* coeffs) : complex_(coeffs) {}

  /// Accessor of complex.
  ///
  SOPHUS_FUNC
  Map<Sophus::Vector2<Scalar>, Options> const& complex() const {
    return complex_;
  }

 protected:
  SOPHUS_FUNC Map<Sophus::Vector2<Scalar>, Options>& complex_nonconst() {
    return complex_;
  }

  Map<Sophus::Vector2<Scalar>, Options> complex_;
};

/// Specialization of Eigen::Map for ``RxSO2 const``; derived from  RxSO2Base.
///
/// Allows us to wrap RxSO2 objects around POD array (e.g. external z style
/// complex).
template <class Scalar_, int Options>
class Map<Sophus::RxSO2<Scalar_> const, Options>
    : public Sophus::RxSO2Base<Map<Sophus::RxSO2<Scalar_> const, Options>> {
 public:
  using Base = Sophus::RxSO2Base<Map<Sophus::RxSO2<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) : complex_(coeffs) {}

  /// Accessor of complex.
  ///
  SOPHUS_FUNC
  Map<Sophus::Vector2<Scalar> const, Options> const& complex() const {
    return complex_;
  }

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

#endif  /// SOPHUS_RXSO2_HPP
