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).

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

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

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