Signed Distance Function - Properties in Euclidean Space

Properties in Euclidean Space

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 W_x for the Jacobian of changing variables in terms of the signed distance function and nearest boundary point. Specifically, if Γ is the set of points within distance μ of the boundary of Ω, and g is an absolutely integrable funciton on Γ, then

where det indicates the determinant and dS_u indicates that we are taking the surface integral.