Signed distance function

In mathematics and its applications, the signed distance function (or oriented distance function) of a set Ω in a metric space determines the distance of a given point x from the boundary of Ω, with the sign determined by whether x is in 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). In mathematics and its applications, the signed distance function (or oriented distance function) of a set Ω in a metric space determines the distance of a given point x from the boundary of Ω, with the sign determined by whether x is in 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 ∂ Ω {displaystyle partial Omega } denotes the boundary of Ω {displaystyle Omega } . For any x ∈ X {displaystyle xin X} , 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 indicates the determinant and dSu indicates that we are taking the surface integral.