Two unit-speed plane curves which have the same curvature differ only by a Euclidean motion.
Fundamental Theorem of Plane Curves
See also
Fundamental Theorem of Space CurvesExplore with Wolfram|Alpha
References
Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 136-138, 1997.Referenced on Wolfram|Alpha
Fundamental Theorem of Plane CurvesCite this as:
Weisstein, Eric W. "Fundamental Theorem of Plane Curves." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FundamentalTheoremofPlaneCurves.html