the fundamental theorem of curves
Imagine giving someone driving instructions for a winding mountain road using only two dials: one says how sharply to turn at each moment, the other says how much to tilt out of the current plane. The claim of the fundamental theorem of curves is astonishing in its tidiness: those two dials, read off as you go, contain the complete shape of the road. Nothing else is needed — the entire space curve is recoverable from just its curvature and its torsion.
Stated carefully: given any two smooth functions kappa(s) > 0 and tau(s) of a real variable s (arc length), with the curvature strictly positive, there EXISTS a regular space curve, parametrized by arc length, having exactly kappa as its curvature and tau as its torsion; and this curve is UNIQUE up to a rigid motion — that is, any two curves with the same kapp(s) and tau(s) can be carried onto each other by a rotation together with a translation (and possibly a reflection, depending on convention). The proof is a clean application of the Frenet-Serret formulas: those formulas form a linear system of differential equations for the frame (T, N, B); the existence-and-uniqueness theorem for differential equations guarantees a unique frame once you fix the frame at a starting point, and integrating T = r'(s) then reconstructs the curve. Curvature gives existence-of-bending, torsion gives existence-of-twist, and together they integrate up to the whole curve.
This theorem is the reason curvature and torsion are called a complete set of invariants for space curves: they capture everything about the shape and nothing about its position or orientation in space. It is the curve-theory analogue of the idea that an angle and side data can fix a triangle up to congruence. Two honest points. First, the 'up to rigid motion' is essential and not a flaw: the theorem deliberately ignores where you place the curve, because location is not part of the intrinsic shape — two identical springs on different tables are the same curve to this theorem. Second, the standard statement needs kappa > 0 everywhere; where the curvature vanishes, torsion is not even defined, so the clean version applies to curves that genuinely bend at every point, and extending it across kappa = 0 points needs extra care.
Set kappa = constant c > 0 and tau = constant d. Integrating the Frenet-Serret equations with these constants yields a circular helix; the radius and pitch are determined by c and d. Special cases: tau = 0 forces a circle (radius 1/c), and c = 0 would force a straight line. So 'constant curvature and constant torsion' is precisely the signature of a helix.
Prescribe kappa(s) and tau(s), integrate Frenet-Serret, and the curve comes out — unique up to rigid motion.
'Unique up to rigid motion' is the point, not a weakness: the theorem captures shape, deliberately discarding position and orientation. The standard form requires kappa > 0; where curvature vanishes, torsion is undefined and the clean statement does not directly apply.