A smooth curve is a curve which is a smooth function, where the word "curve" is interpreted in the analytic geometry context. In particular, a smooth curve is a continuous map from a one-dimensional space to an -dimensional space which on its domain has continuous derivatives up to a desired order.
Smooth Curve
See also
Closed Curve, Simple Curve, Smooth StructureThis entry contributed by Christopher Stover
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Stover, Christopher. "Smooth Curve." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/SmoothCurve.html