In mathematics and its applications, the signed 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.
Definition
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 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 nearestboundary point. Specifically, if T is the set of points within distance μ of the boundary of Ω, 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.
Signed distance functions are applied, for example, in real-time rendering and computer vision. They have also been used in a method to render smooth fonts at large sizes using GPU acceleration. Valve's method computed signed distance fields in raster space in order to avoid the computational complexity of solving the problem in the vector space. More recently piece-wise approximation solutions have been proposed, but even this way the computation can be too slow for real-time rendering, and it has to be assisted by grid-based discretization techniques to approximate the distance to points that are too far away. In 2020 fossgame engine Godot 4.0 received SDF based real-time global illumination or SDFGI, that became a compromise between more realistic voxel-based GI and baked GI. Its core advantage is infinite space it can be applied to, that allows developers to use it for open-world games.
