Main Theorem of Curve Theory
Given n functions
with
then there exists a unique (up to transformations using the Euclidean group) Cn+1-curve γ which is regular of order n and has the following properties
where the set
is the Frenet frame for the curve.
By additionally providing a start t0 in I, a starting point p0 in Rn and an initial positive orthonormal Frenet frame {e1, ..., en-1} with
we can eliminate the Euclidean transformations and get unique curve γ.
Read more about this topic: Differential Geometry Of Curves
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