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Suppose β = β(s) is a curve parametrized by arc-length. If β lies on a sphere...

Suppose β = β(s) is a curve parametrized by arc-length. If β lies on a sphere of center c and radius r, prove β −c = −ρn−ρ0σb where ρ = 1/κ and σ = 1/τ.

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the proof is too much tricky and uses too much concepts so if you stuck somewhere in the proof you can ask in comments

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