A pose is six numbers — a rotation R and a translation t, together an element of the group SE(3). Chasles' theorem says any such motion, however tangled, is really one thing: a rotation about some fixed axis paired with a slide along it — a screw. Set a target pose, and watch its six numbers collapse into a single twist ξ; the exponential map exp(sξ) then unwinds that twist into the motion, tracing a clean helix along the screw axis.
x y z — the moving framescrew axis — rotate about, slide alongexp(sξ) — the geodesic helixnaïve lerp — straight-line t (off the manifold)A rigid motion lives in SE(3) = SO(3) ⋉ ℝ³, a curved six-dimensional manifold. Its tangent space at the identity — the Lie algebra 𝔰𝔢(3) — is the space of twists ξ = (v, ω): an angular part ω and a linear part v. The exponential map carries a twist to a motion, R = exp([ω]×) by Rodrigues' formula and t = V(ω)·v, where the left-Jacobian V couples rotation into translation — that coupling is exactly why a constant twist sweeps out a helix, not a straight line. Run it in reverse with the log map and any target pose reveals its unique screw: axis, angle, and pitch (Chasles). The straight-line comparison is what you get from lerping position and rotation separately; it leaves the geodesic and only rejoins it at the endpoints. This is the machinery behind pose-graph SLAM and IMU pre-integration, screw-based motion planning, and geodesic interpolation of trajectories — whenever a robot has to average, interpolate, or optimize over poses, it does it here, on the manifold, not in the six loose numbers.