Abstract
A space curve might have generated by the motion of a point. The true geometric properties must be the properties that depend on the points of the curve and that do not depend on the manner in which these points are represented. Such properties must not change if a different representation of the points of the curve is chosen. They must be invariant under (1) transformations of the coordinate system in space, and (2) allowable parameter transformations of the curve. The importance of fundamental theorem of the theory of curves in Euclidean 3-spaceis that the implication that the curvature and torsion constitute a complete system of invariants of a curve. An invariant means a functional whose domain of definition includes curves and that is invariant under rigid motions of a curve, that is, whose value is the same for any two curves whose traces are congruent under a rigid motion.
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