Continuous rigid-motion symmetry of a simple closed space curve
= Continuous rigid-motion symmetry of a simple closed space curve
A simple closed curve in $\mathbb R^3$ preserved by a nonconstant smooth family of proper rigid motions is a circle. Differentiating the normalized family at the identity gives a nonzero <infinitesimal proper Euclidean motion> tangent to the curve. Compactness rules out its line and nonclosed helical orbits, leaving a circular rotation orbit.