the Frenet-Serret frame
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Picture a tiny gyroscope riding along a curve, carrying its own set of three perpendicular axes that turn and tumble as the curve bends and twists. Rather than describe the curve from the outside using fixed room coordinates, the Frenet-Serret frame describes it from the point of view of a traveller moving along it — a personal, built-in coordinate system that follows the road. This 'frame on the move' is the central organizing idea of the local theory of space curves.
At each point of a regular space curve (parametrized by arc length, with nonzero curvature) the frame consists of three mutually perpendicular unit vectors: the unit tangent T = r'(s), pointing forward along the curve; the principal normal N = T'/|T'|, pointing toward the inside of the bend; and the binormal B = T x N, perpendicular to both and normal to the plane of bending. They form a right-handed orthonormal triple, an (x, y, z)-like axis system carried along the curve. The three planes they span have names: T and N span the osculating plane (the plane of bending), N and B span the normal plane (perpendicular to the curve), and T and B span the rectifying plane. As the point advances, the whole tripod rotates, and the Frenet-Serret formulas record exactly how T, N, B turn into one another, governed by just two numbers — the curvature and the torsion.
The power of the moving frame is that it turns geometry into bookkeeping you can integrate. Because the frame is attached to the curve itself, the rates at which its axes rotate (curvature kappa and torsion tau) are intrinsic features of the shape, independent of how you placed the curve in space. This is the engine behind the fundamental theorem of curves: prescribe kappa and tau as functions of arc length, and the Frenet-Serret formulas reconstruct the curve uniquely up to where you put it. Honest caveats: the standard Frenet frame requires the curvature to be nonzero (it breaks down on straight stretches and inflection points, where N is undefined), and on curves that have such points one uses alternatives like a Bishop (parallel-transport) frame to avoid the singularity.
On the helix r(t) = (cos t, sin t, t), the frame at the point (1, 0, 0) (t = 0) has T pointing forward-and-upward along the spiral, N = (-1, 0, 0) pointing horizontally inward toward the central axis, and B = T x N tilted off vertical. As you advance along the helix this tripod rotates steadily, with both a constant curvature and a constant torsion.
T, N, B: a right-handed tripod riding the curve, rotating by curvature and torsion.
The standard Frenet frame needs nonzero curvature: where kappa = 0 the principal normal N (and hence B) is undefined, so the frame is not defined along straight pieces or at inflection points. Curves with such points need a different frame, e.g. a Bishop frame.