Discrete Plane Curves.

Let (γ0,,γn) be a regular polygon in the plane R2. Then there are unique real numbers 0,,n1>0 and unit vectors T0,,Tn1S1 such that the edge vectors of γ have the form

γj+1γj=jTj.

Furthermore, it never happens that Tj+1=Tj and therefore there are unique real numbers κ1,,κn1 such that

π<κj<πTj=eiκjTj1.

polygon-smallIt is easy to see that we have

Tj=eiαjT0

with

αj=jk=1κj.

 

So κ determines T uniquely up to a multiplicative constant a=T0 of norm one. Together with the edge lengths 0,,n1 the tangent directions T in turn determine γ up to an additive (translational) constant b=γ0:

γj=γ(0)+j1k=0Tj.

Thus for each collection of edge length 0,,n1 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 mZ such that

nk=1κ=αn=2πm.

m is called the tangent winding number of γ.

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