Lie Groups & Lie Algebras, Interactively
The mathematics of rotations and rigid motion, built from a compass needle up to IMU preintegration. Every idea comes with something you can drag, and every formula is checked numerically on the page.
A robot's heading, a camera's orientation and a drone's pose are not vectors: you cannot add two of them, average them with a mean, or step along a gradient by plain addition. Lie theory is the small, precise toolkit that fixes all three at once. You compose in the group, differentiate in the tangent space, and move between the two with Exp and Log. The four acts build that toolkit in order. Act I shows why it is needed and what a group is. Act II builds the algebra of rotations. Act III extends it to full rigid motion. Act IV does calculus with it: Jacobians, optimization, uncertainty and integration.
The prerequisites are matrix multiplication and a derivative. Linear Algebra, Part 20 and Calculus on manifolds are good warm-ups. For a single-page shortcut to just the parts the optimizer needs, the SO(3) primer covers it. Everything here is the machinery behind rotation estimation in Nonlinear Optimization and bundle adjustment in Multi-View Geometry. Keep the formula sheet open as you read.
The parts
Average two compass headings, add two sets of Euler angles, sum two rotation matrices - and watch each one fail in a way that points at the same missing idea.
The four group axioms tested live on SO(2), SO(3), SE(3) and friends, and the smooth, locally flat surface that turns a group into a Lie group.
The circle, its tangent line, exp as wrapping and log as unwrapping, and the plus and minus operators every later part generalises.
Differentiate R(t)R(t)ᵀ = I and a skew-symmetric matrix falls out: hat and vee, the three generators, and the Lie bracket as the cost of reordering two rotations.
Rodrigues' formula two ways, the Log map and its two numerical traps, and the ball of radius π that holds every rotation exactly once.
Drive a gimbal into lock, watch a quaternion take 720° to come home, and compare every way of writing a rotation down.
Homogeneous matrices, twists, and why the exponential of a constant velocity is a screw - a helix you can shape with two sliders.
One perturbation, applied on the left or on the right of a pose, and the matrix that converts between them - plus what each library means by a 'delta'.
Right and left Jacobians, Baker-Campbell-Hausdorff to first order, and a table of elementary derivatives you can check numerically with one click.
Gauss-Newton with a retraction: align two point clouds on SE(3), average noisy rotations, and close the loop on a small SE(2) pose graph.
Gaussians that live in the tangent space, the banana-shaped cloud of a robot driving with a shaky heading, and covariance propagated through composition.
Geodesics and slerp, integrating an angular velocity without drifting off the group, and the preintegrated gyro measurement that makes visual-inertial odometry fast.