((new)): Rigid3d Tutorial

SE3d T_ba = T_ab.inverse();

In robotics, computer vision, and 3D graphics, the ability to represent rotations and translations in 3D space is fundamental. The Rigid3D object (often found in libraries like Sophus , Eigen , geometry_msgs , or tf2 ) is the industry-standard way to do this. Unlike a 4x4 homogeneous matrix, Rigid3D separates rotation (SO(3)) and translation, offering better numerical stability and mathematical clarity. rigid3d tutorial

# Rotation: 90 deg around Z r = R.from_euler('z', 90, degrees=True) t = np.array([1.0, 0.0, 0.0]) T = np.eye(4) T[:3,:3] = r.as_matrix() T[:3, 3] = t 4. Applying the Transformation Transform a 3D point ( p = (0, 1, 0) ) from frame A to frame B. SE3d T_ba = T_ab

Rigid3D typically stores these as a unit quaternion (for rotation) and a 3-vector (for translation). For this tutorial, we'll use the Sophus::SE3d (C++) or scipy.spatial.transform.Rotation (Python). If you're working with ROS, tf2::Transform is analogous. C++ (Eigen + Sophus) #include <sophus/se3.hpp> #include <Eigen/Core> #include <iostream> using Sophus::SE3d; using Eigen::Vector3d; using Eigen::Quaterniond; Python (NumPy + SciPy or transforms3d) import numpy as np from scipy.spatial.transform import Rotation as R # Or use `from transforms3d.quaternions import quat2mat` 3. Creating a Rigid3D Object Let’s create a transformation that represents: rotate 90° about Z-axis, then translate by (1, 0, 0). # Rotation: 90 deg around Z r = R

SE3d T_ab = SE3d(q_ab, t_ab); SE3d T_bc = SE3d(q_bc, t_bc); SE3d T_ac = T_ab * T_bc;

Vector3d p_a(0.0, 1.0, 0.0); Vector3d p_b = T_ab * p_a; std::cout << "p_b: " << p_b.transpose() << std::endl; // Expected: after 90° Z rot: (0,1,0) -> (-1,0,0) then + translation (1,0,0) -> (0,0,0)

[ p_B = R_AB \cdot p_A + t_AB ]