Signed distance function

If Ω is a subset of a metric space, X, with metric, d, then the signed distance function, f, is defined by

${\displaystyle f(x)={\begin{cases}d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega \\-d(x,\partial \Omega )&{\mbox{ if }}x\in \Omega ^{c}\end{cases}}}$

where ${\displaystyle \partial \Omega }$ denotes the boundary of ${\displaystyle \Omega }$. For any ${\displaystyle x\in X}$,

${\displaystyle d(x,\partial \Omega ):=\inf _{y\in \partial \Omega }d(x,y)}$

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

${\displaystyle |\nabla f|=1.}$

If the boundary of Ω is C^k for k≥2 (see differentiability classes) then d is C^k on points sufficiently close to the boundary of Ω.[3] In particular, on the boundary f satisfies

${\displaystyle \nabla f(x)=N(x),}$

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 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

${\displaystyle \int _{T(\partial \Omega ,\mu )}g(x)\,dx=\int _{\partial \Omega }\int _{-\mu }^{\mu }g(u+\lambda N(u))\,\det(I-\lambda W_{u})\,d\lambda \,dS_{u},}$

where det indicates the determinant and dS_u indicates that we are taking the surface integral.[4]

Algorithms for calculating the signed distance function include the efficient fast marching method, fast sweeping method[5] and the more general level-set method.

Signed distance functions are applied, for example, in real-time rendering[6] and computer vision.[7][8]

They have also been used in a method (advanced by Valve) to render smooth fonts at large sizes (or alternatively at high DPI) using GPU acceleration.[9] Valve's method computed signed distance fields in raster space in order to avoid the computational complexity of solving the problem in the (continuous) vector space. More recently piece-wise approximation solutions have been proposed (which for example approximate a Bézier with arc splines), 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 (and cull from the computation) the distance to points that are too far away.[10]

• Distance function
• Level-set method
• Eikonal equation
• Parallel (aka offset) curve

• Stanley J. Osher and Ronald P. Fedkiw (2003). Level Set Methods and Dynamic Implicit Surfaces. Springer. ISBN 9780387227467.
• Gilbarg, D.; Trudinger, N. S. (1983). Elliptic Partial Differential Equations of Second Order. Grundlehren der mathematischen Wissenschaften. 224 (2nd ed.). Springer-Verlag. (or the Appendix of the 1977 1st ed.)