text: #ifndef SOPHUS_SIM_DETAILS_HPP
  #define SOPHUS_SIM_DETAILS_HPP
  
  #include "types.hpp"
  
  namespace Sophus {
  namespace details {
  
  template <class Scalar, int N>
  Matrix<Scalar, N, N> calcW(Matrix<Scalar, N, N> const& Omega,
                             Scalar const theta, Scalar const sigma) {
    using std::abs;
    using std::exp;
    using std::sin;
    using std::cos;
    static Matrix<Scalar, N, N> const I = Matrix<Scalar, N, N>::Identity();
    static Scalar const one(1);
    static Scalar const half(0.5);
    Matrix<Scalar, N, N> const Omega2 = Omega * Omega;
    Scalar const scale = exp(sigma);
    Scalar A, B, C;
    if (abs(sigma) < Constants<Scalar>::epsilon()) {
      C = one;
      if (abs(theta) < Constants<Scalar>::epsilon()) {
        A = half;
        B = Scalar(1. / 6.);
      } else {
        Scalar theta_sq = theta * theta;
        A = (one - cos(theta)) / theta_sq;
        B = (theta - sin(theta)) / (theta_sq * theta);
      }
    } else {
      C = (scale - one) / sigma;
      if (abs(theta) < Constants<Scalar>::epsilon()) {
        Scalar sigma_sq = sigma * sigma;
        A = ((sigma - one) * scale + one) / sigma_sq;
        B = ((half * sigma * sigma - sigma + one) * scale) / (sigma_sq * sigma);
      } else {
        Scalar theta_sq = theta * theta;
        Scalar a = scale * sin(theta);
        Scalar b = scale * cos(theta);
        Scalar c = theta_sq + sigma * sigma;
        A = (a * sigma + (one - b) * theta) / (theta * c);
        B = (C - ((b - one) * sigma + a * theta) / (c)) * one / (theta_sq);
      }
    }
    return A * Omega + B * Omega2 + C * I;
  }
  
  template <class Scalar, int N>
  Matrix<Scalar, N, N> calcWInv(Matrix<Scalar, N, N> const& Omega,
                                Scalar const theta, Scalar const sigma,
                                Scalar const scale) {
    using std::abs;
    using std::sin;
    using std::cos;
    static Matrix<Scalar, N, N> const I = Matrix<Scalar, N, N>::Identity();
    static Scalar const half(0.5);
    static Scalar const one(1);
    static Scalar const two(2);
    Matrix<Scalar, N, N> const Omega2 = Omega * Omega;
    Scalar const scale_sq = scale * scale;
    Scalar const theta_sq = theta * theta;
    Scalar const sin_theta = sin(theta);
    Scalar const cos_theta = cos(theta);
  
    Scalar a, b, c;
    if (abs(sigma * sigma) < Constants<Scalar>::epsilon()) {
      c = one - half * sigma;
      a = -half;
      if (abs(theta_sq) < Constants<Scalar>::epsilon()) {
        b = Scalar(1. / 12.);
      } else {
        b = (theta * sin_theta + two * cos_theta - two) /
            (two * theta_sq * (cos_theta - one));
      }
    } else {
      Scalar const scale_cu = scale_sq * scale;
      c = sigma / (scale - one);
      if (abs(theta_sq) < Constants<Scalar>::epsilon()) {
        a = (-sigma * scale + scale - one) / ((scale - one) * (scale - one));
        b = (scale_sq * sigma - two * scale_sq + scale * sigma + two * scale) /
            (two * scale_cu - Scalar(6) * scale_sq + Scalar(6) * scale - two);
      } else {
        Scalar const s_sin_theta = scale * sin_theta;
        Scalar const s_cos_theta = scale * cos_theta;
        a = (theta * s_cos_theta - theta - sigma * s_sin_theta) /
            (theta * (scale_sq - two * s_cos_theta + one));
        b = -scale *
            (theta * s_sin_theta - theta * sin_theta + sigma * s_cos_theta -
             scale * sigma + sigma * cos_theta - sigma) /
            (theta_sq * (scale_cu - two * scale * s_cos_theta - scale_sq +
                         two * s_cos_theta + scale - one));
      }
    }
    return a * Omega + b * Omega2 + c * I;
  }
  
  }  // details
  }  // Sophus
  
  #endif
  