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| [[Image:Signed distance1.png|right|thumb|A disk (top) and its signed distance function (bottom, in red). The ''x''-''y'' plane is shown in blue.]]
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| [[Image:Signed distance2.png|right|thumb|A more complicated set (top) and its signed distance function (bottom, in red).]]
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| In [[mathematics]] and applications, the '''signed distance function''' of a set ''Ω'' in a [[metric space]], also called the '''oriented distance function''', determines the distance of a given point ''x'' from the [[boundary (topology)|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''.
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| ==Definition==
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| If (''X'', ''d'') is a metric space, the ''signed distance function'' ''f'' is defined by
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| :<math>f(x)=
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| \begin{cases}
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| d(x, \Omega^c) & \mbox{ if } x\in\Omega \\
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| -d(x, \Omega)& \mbox{ if } x\in\Omega^c
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| \end{cases}
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| </math>
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| where
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| : <math> d(x, \Omega)=\inf_{y\in \Omega}d(x, y)</math>
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| and 'inf' denotes the [[infimum]].
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| [[Algorithm]]s for calculating the signed distance function include the efficient [[fast marching method]] and the more general but slower [[level set method]].
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| Signed distance functions are applied for example in [[computer vision]].
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| ==Properties in Euclidean space==
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| If ''Ω'' is a subset of the [[Euclidean space]] '''R'''<sup>''n''</sup> with [[piecewise]] [[smooth function|smooth]] boundary, then the signed distance function is differentiable [[almost everywhere]], and its [[gradient]] satisfies the [[eikonal equation]]
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| : <math>|\nabla f|=1.</math> | |
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| If the boundary of ''Ω'' is ''C''<sup>''k''</sup> for ''k''≥2 (see [[differentiability classes]]) then ''d'' is ''C''<sup>''k''</sup> on points sufficiently close to the boundary of ''Ω''.{{sfn|Gilbarg|1983|loc=Lemma 14.16}} In particular, '''''on''''' the boundary ''f'' satisfies
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| :<math>\nabla f(x) = N(x),</math>
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| 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]].
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| 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''<sub>''x''</sub> 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 function on ''Γ'', then
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| :<math>\int_{\Gamma} g(x)\,dx = \int_{\partial\Omega}\int_{-\mu}^\mu g(u+\lambda N(u))\, \det(I-\lambda W_u) \,d\lambda \,dS_u,</math>
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| where det indicates the [[determinant]] and ''dS''<sub>''u''</sub> indicates that we are taking the [[surface integral]].{{sfn|Gilbarg|1983|loc=Equation (14.98)}}
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| ==See also==
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| * [[Level set method]]
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| * [[Eikonal equation]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{cite book | author=Stanley J. Osher and Ronald P. Fedkiw | title=Level Set Methods and Dynamic Implicit Surfaces | publisher=Springer | year=2002}}
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| *{{cite book | author1=Gilbarg, D. | author2=Trudinger, N. S. | year=1983 | edition=2nd | title=Elliptic Partial Differential Equations of Second Order | publisher=Springer-Verlag | volume=224 | series=Grundlehren der mathematischen Wissenschaften }} (or the Appendix of the 1977 1st ed.)
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| {{mathapplied-stub}}
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| [[Category:Applied mathematics]]
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