Signed distance function

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A disk (top) and its signed distance function (bottom, in red). The x-y plane in shown in blue.

A more complicated set (top) and its signed distance function (bottom, in red).

In mathematics and applications, the signed distance function of a set S in a metric space determines how close a given point x is to the boundary of S, with that function having positive values at points x inside S, it decreases in value as x approaches the boundary of S where the signed distance function is zero, and it takes negative values outside of S.

Formally, if (X, d) is a metric space, the signed distance function f is defined by

$f(x)= \begin{cases} d(x, \partial S) & \mbox{ if } x\in S \\ -d(x, \partial S)& \mbox{ if } x\not\in S \end{cases}$

where

$d(x, \partial S)=\inf_{y\in\partial S}d(x, y)$

(the ∂ symbol denotes the set boundary, while 'inf' is the infimum).

If S is a subset of the Euclidean space Rⁿ with piecewise smooth boundary, the signed distance function is differentiable almost everywhere, and its gradient satisfies the eikonal equation

$|\nabla f|=1.$