the Frenet-Serret formulas
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We have a moving tripod riding along a space curve — the unit tangent T, principal normal N, and binormal B. As the bug advances, this tripod rotates, and a natural question is: exactly how? The Frenet-Serret formulas are the precise answer. They are three equations telling you the rate of change of each frame vector, expressed back in terms of the frame itself, with all the geometry packaged into just two numbers, the curvature kappa and the torsion tau.
Working with arc length s (unit speed), the formulas are: T'(s) = kappa N (the tangent turns toward the normal at rate kappa — pure bending); N'(s) = -kappa T + tau B (the normal turns back toward minus the tangent and forward toward the binormal); and B'(s) = -tau N (the binormal turns toward minus the normal at rate tau — pure twisting). Read them as a single statement: the entire frame rotates rigidly, and the angular velocity of that rotation is the Darboux vector omega = tau T + kappa B, so that T' = omega x T, N' = omega x N, B' = omega x B. The beautiful structure is that the matrix relating (T', N', B') to (T, N, B) is antisymmetric — only two independent entries, kappa and tau — which is the algebraic shadow of the fact that an orthonormal frame can only rotate, never stretch.
These three formulas are the workhorse of the local theory of curves. They convert the geometry of a space curve into a system of linear differential equations driven by kappa(s) and tau(s); solving that system, given a starting point and starting frame, reconstructs the curve. This is precisely the mechanism behind the fundamental theorem of curves: prescribe kappa and tau, integrate the Frenet-Serret equations, and out comes a unique curve up to rigid motion. The standing honest caveat is the same as for the frame itself: the formulas in this form assume the curvature is nonzero so that N and B are defined; on segments where kappa = 0 the standard frame breaks down and one must patch with an alternative frame.
For the unit circle in the plane, kappa = 1 and tau = 0, so the formulas reduce to T' = N and N' = -T, with B constant. These are exactly the equations of uniform rotation — differentiate the rotating pair (T, N) and they cycle into each other — which is why the frame on a circle simply spins at a steady rate and the circle stays planar.
T' = kappa N, N' = -kappa T + tau B, B' = -tau N: rotation driven by curvature and torsion.
The compact 'antisymmetric matrix' form encodes that an orthonormal frame can only rotate, not deform. The formulas assume kappa is nonzero so that N and B exist; where curvature vanishes the standard frame and these equations break down.