Improve clarity

include/manif/impl/sgal3/SGal3Tangent_base.h

@@ -230,7 +230,15 @@ SGal3TangentBase<_Derived>::ljac() const {
  * 
  * Tangent space is tau = [rho ; nu ; theta ; t] in R^10
  * 
- * Matrix blocks D, E, L, M, N, N1, N2 are called different here. We specify the correspondences in the comments.
+ * Matrix blocks D, E, L, M, N, N1, N2 are referred to in the comments and correspond to eqs. in Kelly's paper:
+ * 
+ * D: (18) = Jl_SO3(theta)
+ * E: (19)
+ * L: (32)
+ * M: (33) = Q(nu,theta)
+ * N: (34) = N1 - N2
+ * N1: (35) = Q(rho,theta)
+ * N2: (36)
  */
 
 
@@ -245,53 +253,48 @@ SGal3TangentBase<_Derived>::ljac() const {
  Jl.template block<3, 3>(3, 3) = D; // Block D
  Jl.template block<3, 3>(6, 6) = D; // Block D
 
- // Compute block E
+ // Block E
  Eigen::Matrix<Scalar,3,3> E;
  fillE(E, so3); // Block E
  Jl.template block<3,1>(0,9) = E * lin2(); // Block E * nu
- // fillE(Jl.template block<3, 3>(6, 6), so3); // Block E
- // Jl.template block<3, 1>(0, 9) = Jl.template block<3, 3>(6, 6) * lin2(); // Block E * nu
 
+ // Skew matrix W = theta^
+ Eigen::Matrix<Scalar,3,3> W = so3.hat();
+ // Skew matrix V = nu^
+ Eigen::Matrix<Scalar,3,3> V = skew(lin2());
+
+ // Angles and trigonometry
  const Scalar theta_sq = so3.coeffs().squaredNorm();
- const Scalar theta = sqrt(theta_sq); // rotation angle
+ const Scalar theta = sqrt(theta_sq); // rotation angle
+ const Scalar theta_cu = theta * theta_sq;
 
  const Scalar sin_t = sin(theta);
  const Scalar cos_t = cos(theta);
- const Scalar theta_cu = theta * theta_sq;
 
- // Compute skew matrix W = theta^
- // Jl.template block<3, 3>(6, 6) = so3.hat(); // Use as temporary storage with simpler ref
- // ConstRef33 W = Jl.template block<3, 3>(6, 6);
- Eigen::Matrix<Scalar,3,3> W = so3.hat();
- // Compute skew matrix V = nu^
- // Jl.template bottomRightCorner<3, 3>() = skew(lin2()); // Use as temporary storage with simpler ref
- // ConstRef33 V = Jl.template bottomRightCorner<3, 3>();
- Eigen::Matrix<Scalar,3,3> V = skew(lin2());
-
- // Compute block L
+ // Block L
  Scalar cA, cB;
  // small angle approx.
- if (theta_sq > Constants<Scalar>::eps) {
+ if (theta_cu > Constants<Scalar>::eps) {
  cA = (sin_t - theta * cos_t) / theta_sq;
  cB = (theta_sq + Scalar(2) * (Scalar(1) - theta * sin_t - cos_t)) / (Scalar(2) * theta_sq);
  } else {
  cA = Scalar(1./3.) * theta - Scalar(1./30.) * theta_cu;
  cB = Scalar(1./8.) * theta_sq;
  }
- Jl.template block<3, 3>(0, 3).noalias() = -t() * ( // Block L * t
+ Jl.template block<3, 3>(0, 3).noalias() = -t() * ( // Block - L * t
  I33(Scalar(0.5)) + cA * W + cB * W * W // Block L
  );
 
- // Compute block M = Q(nu,theta)
+ // Block M = Q(nu,theta)
  SE3Tangent<Scalar>::fillQ(Jl.template block<3, 3>(3, 6), coeffs().template segment<6>(3)); // Block M = Q(nu,theta)
 
 
- // Compute block N1, part of N
+ // Block N1, part of N. N1 = Q(rho,theta)
  Eigen::Matrix<Scalar, 6, 1> rho_theta;
- rho_theta << coeffs().template head<3>(), coeffs().template segment<3>(6);
+ rho_theta << lin(), ang();
  SE3Tangent<Scalar>::fillQ(Jl.template block<3, 3>(0, 6), rho_theta); // block N1 = Q(rho,theta)
 
- // Compute block N2, part of N
+ // Block N2, part of N
  Scalar cC, cD, cE, cF;
  if (theta_cu > Constants<Scalar>::eps) {
  std::cout << "large" << std::endl;
@@ -325,7 +328,7 @@ SGal3TangentBase<_Derived>::ljac() const {
  // Block 1
  Jl(9, 9) = Scalar(1);
 
- // Complete with blocks of zeros 
+ // Blocks of zeros 
  Jl.template bottomLeftCorner<7, 3>().setZero();
  Jl.template block<4, 3>(6, 3).setZero();
  Jl.template block<1, 3>(9, 6).setZero();