SO(3) From Scratch

Starting point

A 3×3 matrix R rotates vectors — it never stretches, shears, or mirrors them. "Preserves lengths and angles" is exactly the statement (Ru)·(Rv) = u·v for every pair of vectors u, v. Expand the dot product and that condition collapses to one matrix equation:

That alone allows one more possibility than rotation: a reflection (flip one axis) also satisfies RᵀR = I. Reflections turn a right-handed coordinate frame into a left-handed one, which shows up as det(R) = -1 instead of +1. Ruling that out gives the actual definition used everywhere in robotics:

Build a rotation below with an axis and an angle (this is anticipating section 3 — for now just treat it as "some matrix generator") and watch both checks hold for every axis and every angle, while a plain "add 1 to a diagonal entry" matrix (a not-quite-rotation) fails them.

Go deeper → Groups, matrix groups and manifolds: the group axioms, the matrix-group zoo, and why SO(3) rather than O(3).

The building block

For a fixed unit vector k = (k₁, k₂, k₃), the map v ↦ k × v is linear in v — so it's some matrix times v. Write out the cross product component by component and that matrix falls out directly:

[k]ₓ (read "k hat" or "the skew of k") is skew-symmetric: flipping it over its diagonal negates it, [k]ₓᵀ = -[k]ₓ. That's not a coincidence — it's forced by k × v always being perpendicular to v, i.e. vᵀ(k×v) = 0 for every v, which is exactly what Aᵀ = -A means for a matrix. Pick a k below and see its skew matrix, and watch [k]ₓv come out perpendicular to v every time, no matter what v is.

Go deeper → The tangent space and so(3): hat and vee, angular velocity in two frames, and the Lie bracket.

From geometry to a formula

To rotate v by angle θ about unit axis k, split v into a piece parallel to k and a piece perpendicular to it:

The perpendicular part v⊥ and the vector k×v⊥ = k×v are perpendicular to each other and to k, and both have the same length — exactly the two "clock hand" directions of a circle in the plane through v⊥. Standard 2D rotation-by-θ in that plane, written back in 3D vectors, gives:

Substitute v∥ = v - v⊥, replace every cross product with [k]ₓ from section 2 (so k×v = [k]ₓv, and k×(k×v) = [k]ₓ²v = v⊥ - v, i.e. v⊥ = v + [k]ₓ²v), and everything collects into one matrix acting on v — Rodrigues' rotation formula:

Drag axis and angle below — the canvas shows v sweeping to Rv exactly along the circle this derivation describes, and the readout confirms it stays at constant radius from the axis.

Go deeper → Exp and Log on SO(3): the logarithm, its numerical traps near 0 and π, and the ball of radius π.

A second route to the same formula

The matrix exponential is defined the same way the scalar one is, as an infinite sum:

Set A = θ[k]ₓ for unit axis k. A short computation shows [k]ₓ³ = -[k]ₓ (a skew-symmetric matrix built from a unit vector cubes back to minus itself) — so every power beyond the square is just ±[k]ₓ or ±[k]ₓ², and the whole series sorts into two families:

Same formula as section 3, reached a completely different way. This is the function part 5 calls Exp(δ): feed it a 3-number vector δ (its direction is the axis k, its length is the angle θ), and it hands back a rotation matrix. Pick a truncation order below and watch the truncated series converge onto the exact matrix as you add more terms — by 3-4 terms it's already indistinguishable for a moderate angle.

Go deeper → Exp and Log on SO(3): the series derivation with a convergence chart for any angle.

The step that made part 5's update rule work

Adding two arbitrary rotation matrices entry-by-entry almost never gives another rotation — the sum generally fails RᵀR=I (try it: I + I = 2I, and (2I)ᵀ(2I) = 4I ≠ I). So "current estimate plus a small step" can't mean ordinary addition the way it does for a position in ℝ³. It has to mean compose — multiply on a small extra rotation:

Because Exp is built from an exact sin/cos formula, R·Exp(δ) is a genuine rotation matrix for any size of δ, not just small ones — there's no approximation being made by using it. The approximation people usually reach for is the opposite direction: for a small δ, the exponential series' higher terms are negligible, so Exp(δ) ≈ I + [δ]ₓ — which is exactly why an optimizer's tiny per-iteration step can be treated as approximately linear even though the underlying space is curved. That local flatness is what makes gradient descent, Gauss-Newton and Levenberg-Marquardt (all of which reason about a linear step) work at all on something curved like SO(3).

Drag δ below and compare the exact composed rotation to the linearized I+[δ]ₓ stand-in (renormalized back onto SO(3) so it's a fair comparison) — small δ, they agree; push it larger and the linear stand-in visibly drifts off, exactly the same shape of error part 5 showed for "naive" angle-adding.

Go deeper → SO(2): the whole story in one dimension: ⊕ and ⊖ where every formula fits on a line.

How the code actually computes a step

For a residual r(R) = measured - Rᵀn comparing a landmark direction rotated into the sensor frame against what was actually measured, part 5's Gauss-Newton needs ∂r/∂δ — but δ isn't a coordinate R is written in terms of, it's the tangent nudge from section 5. The fix: physically apply a tiny nudge in each of the 3 tangent directions, using the exact same R·Exp(δ) composition, and see how much r moved:

That's a literal transcription of gradAndHessNumeric in part 5's script: three tiny nudges (one per tangent axis), each composed on with Exp, each evaluated forward and backward, divided by 2h. Pick a landmark and a tangent axis below and watch this play out on the exact numbers — as the step size h shrinks from coarse to tiny, the estimate stabilizes onto essentially one value (until it's so tiny that floating-point rounding starts to dominate, visible on the far left of the slider).

Go deeper → Jacobians on Lie groups: the analytic right Jacobian that replaces these central differences, with a checkable table of elementary derivatives.

Where this rejoins part 5

With the numerical J from section 6 (stacked over all landmarks) and gradient g = Jᵀr, Gauss-Newton solves the exact same 3×3 linear system as every earlier part — (JᵀJ) δ = -g — for the tangent step δ. Levenberg-Marquardt damps it the same way too, adding λI before solving. The only thing that's new versus parts 1–4 is the last line:

That's the entire gap this page fills in: sections 1-2 built the vocabulary (what a rotation is, and the skew-symmetric matrix that turns a cross product into multiplication), sections 3-4 derived Exp two ways, section 5 explained why composing is mandatory and when the linear approximation is safe, and section 6 showed the Jacobian actually being computed with it. Everything downstream — the normal equations, the damping, the convergence loop — is unchanged from part 1.

Go deeper → Optimization on manifolds: Gauss–Newton on SO(3) and SE(3), rotation averaging and a pose graph.

Five quick questions

Recap