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Concept
Signed distance function
Summary
In mathematics and its applications, the signed distance function (or oriented distance function) is the orthogonal distance of a given point x to the boundary of a set Ω in a metric space, with the sign determined by whether or not x is in the interior of Ω. The function has positive values at points x inside Ω, it decreases in value as x approaches the boundary of Ω where the signed distance function is zero, and it takes negative values outside of Ω. However, the alternative convention is also sometimes taken instead (i.e., negative inside Ω and positive outside).
If Ω is a subset of a metric space X with metric d, then the signed distance function f is defined by
where denotes the boundary of . For any ,
where inf denotes the infimum.
If Ω is a subset of the Euclidean space Rn with piecewise smooth boundary, then the signed distance function is differentiable almost everywhere, and its gradient satisfies the eikonal equation
If the boundary of Ω is Ck for k ≥ 2 (see Differentiability classes) then d is Ck on points sufficiently close to the boundary of Ω. In particular, on the boundary f satisfies
where N is the inward normal vector field. The signed distance function is thus a differentiable extension of the normal vector field. In particular, the Hessian of the signed distance function on the boundary of Ω gives the Weingarten map.
If, further, Γ is a region sufficiently close to the boundary of Ω that f is twice continuously differentiable on it, then there is an explicit formula involving the Weingarten map Wx for the Jacobian of changing variables in terms of the signed distance function and nearest boundary point. Specifically, if T(∂Ω, μ) is the set of points within distance μ of the boundary of Ω (i.e. the tubular neighbourhood of radius μ), and g is an absolutely integrable function on Γ, then
where det denotes the determinant and dSu indicates that we are taking the surface integral.
Algorithms for calculating the signed distance function include the efficient fast marching method, fast sweeping method and the more general level-set method.
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