Let (γ0,…,γn) be a regular polygon in the plane R2. Then there are unique real numbers ℓ0,…,ℓn−1>0 and unit vectors T0,…,Tn−1∈S1 such that the edge vectors of γ have the form
γj+1−γj=ℓjTj.
Furthermore, it never happens that Tj+1=−Tj and therefore there are unique real numbers κ1,…,κn−1 such that
−π<κj<πTj=eiκjTj−1.
It is easy to see that we have
Tj=eiαjT0
with
αj=j∑k=1κj.
So κ determines T uniquely up to a multiplicative constant a=T0 of norm one. Together with the edge lengths ℓ0,…,ℓn−1 the tangent directions T in turn determine γ up to an additive (translational) constant b=γ0:
γj=γ(0)+j−1∑k=0Tj.
Thus for each collection of edge length ℓ0,…,ℓn−1 and each curvature function κ there exists a discrete curve γ with curvature κ. Every other such curve ˜γ differs from γ only by a euclidean motion:
˜γ=aγ+b
with |a|=1.
For a closed discrete curve we must have Tn=T0 which means that there has to be an integer m∈Z such that
n∑k=1κ=αn=2πm.
m is called the tangent winding number of γ.