Fundamental Theorem for Curves

Differential Geometry · Axiom Academy

LESSON Fundamental Theorem for Curves How curvature and torsion completely determine a space curve Fundamental Theorem for Space Curves Given two smooth functions κ(s) > 0 and τ(s) defined on an interval [0, L], there exists a unique regular curve α(s) (up to rigid motion) with arc-length parametrization such that: The phrase "up to rigid motion" means the curve is unique except for its position and orientation in space. You can translate or rotate it, but its shape is completely determined. 2. Curvature and Torsion as Complete Invariants What does "complete invariants" mean? An invariant is a property that doesn't change under certain transformations (like rotation or translation). Complete invariants contain all the geometric information about an object. Curvature κ(s): Measures how much the curve bends at each point (deviation from a straight line) Torsion τ(s): Measures how much the curve twists out of its osculating plane Together, these two functions capture everything about the curve's shape. Two curves with the same κ(s) and τ(s) must be congruent (identical except for position/orientation). 3. Proof Idea: Existence and Uniqueness The proof relies on the Frenet-Serret equations: These are a system of first-order ODEs that govern how the Frenet frame T, N, B evolves along the curve: N'(s) = -κ(s) T(s) + τ(s) B(s) Existence: Given κ(s) and τ(s), solve the Frenet-Serret ODEs with initial conditions T(0), N(0), B(0) to get the frame along the curve

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