Plane Curve Classification Theorem

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Let $I = \closedint a b \subseteq \R$ be a closed real interval.

Let $\gamma, \tilde \gamma: I \to \R^2$ be smooth unit-speed curves on a plane.

Let $N$ and $\tilde N$ be their normal vector fields.

Let $\map {\kappa_N} t$ and $\map {\kappa_{\tilde N}} t$ be curvatures at $\map \gamma t$ and $\map {\tilde \gamma} t$ respectively.

Then $\gamma$ and $\tilde \gamma$ are congruent by a direction-preserving congruence iff:

$\forall t \in I : \map {\kappa_N} t = \map {\kappa_{\tilde N}} t$