6 Image Analysis of Curvature Using Classical Mechanics, the Elastica 65 Fig. 6.3 Examples of various Elastica shapes 6.2 Methodology The method for measuring curvature is to fit an Elastica curve to set of points along a deformed shape using a least squares regression. Once the best fitting Elastica curve is found, the magnitude and location of the maximum curvature of the shape can be computed directly. Levien [1] showed that the equations governing the inflectional Elastica can be parameterized as: x(s) =s −2E (am(s,m),m) y(s) =−2 √mcn(s,m) for 0 ≤s ≤4K(m) where m is the elliptic modulus and s is the arclength along the curve. The functions E, K, am and cn are elliptic integrals and Jacobi elliptic functions defined as: E(ϕ,m) = ϕ 0 1−msin 2 (t)dt K(m) = π/2 0 1 1−msin 2 (t) dt am(u,m) =the value of ϕ suchthat u = ϕ 0 1 1−msin 2 (t) dt cn(u,m) =cos(am(u,m)) Curvature can be defined as: κ (s,m) =2 √mcn(s,m)
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