# Fundamental Theorem of Space Curves

This is not a question I need to work out but I'm trying to understand this theorem.

My lecture notes state: 'This theorem states the existence of solutions to the Frenet - Serret Equations that, apart from the possibility of a rigid motion, are uniquely determined by any choice of smooth functions k > 0, and  torsion(s).'

WolframAlpha states: 'If two single-valued continuous functions kappa(s) (curvature) and tau(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for orientation and position in space (i.e., up to a Euclidean motion), where s is the arc length, kappa is the curvature, and tau is the torsion. '

I want to go by Wolframs explanation since it's simpler but does anyone have any examples or diagrams that can help me understand this further? I'm having trouble seeing the importance or practical uses of this theorem.

## Answers and Replies

Fundamental theorem of curves is of great importance. She says that the curvature and torsion of a curve does not depend on the Cartesian benchmark defined curve. So, it says that the curvature and torsion are some intrinsic and objectively parameters of the curve.

Are you still interested in this theorem?