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eikonia

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There are 7 letters in EIKONIA ( A1E1I1K5N1O1 )

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Eikonia might refer to
The eikonal equation (from German Eikonal, which is from Greek εἰκών, image) is a non-linear partial differential equation encountered in problems of wave propagation, when the wave equation is approximated using the WKB theory. It is derivable from Maxwell's equations of electromagnetics, and provides a link between physical (wave) optics and geometric (ray) optics.
* The eikonal equation is of the form*
*
*
*
* |
*
* ∇
* u
* (
* x
* )
*
* |
*
* =
* 1
*
* /
*
* f
* (
* x
* )
* ,
*
* x
* ∈
* Ω
*
*
* {\displaystyle |\nabla u(x)|=1/f(x),\ x\in \Omega }
* subject to
*
*
*
* u
*
*
* |
*
*
* ∂
* Ω
*
*
* =
* 0
*
*
* {\displaystyle u|_{\partial \Omega }=0}
* , where
*
*
*
* Ω
*
*
* {\displaystyle \Omega }
* is an open set in
*
*
*
*
*
* R
*
*
* n
*
*
*
*
* {\displaystyle \mathbb {R} ^{n}}
* with well-behaved boundary,
*
*
*
* f
* (
* x
* )
*
*
* {\displaystyle f(x)}
* is a function with positive values,
*
*
*
* ∇
*
*
* {\displaystyle \nabla }
* denotes the gradient and
*
*
*
*
* |
*
* ⋅
*
* |
*
*
*
* {\displaystyle |\cdot |}
* is the Euclidean norm. Here, the right-hand side
*
*
*
* f
* (
* x
* )
*
*
* {\displaystyle f(x)}
* is typically supplied as known input. Physically, the solution
*
*
*
* u
* (
* x
* )
*
*
* {\displaystyle u(x)}
* is the shortest time needed to travel from the boundary
*
*
*
* ∂
* Ω
*
*
* {\displaystyle \partial \Omega }
* to
*
*
*
* x
*
*
* {\displaystyle x}
* inside
*
*
*
* Ω
* ,
*
*
* {\displaystyle \Omega ,}
* with
*
*
*
* f
* (
* x
* )
*
*
* {\displaystyle f(x)}
* being the speed at
*
*
*
* x
*
*
* {\displaystyle x}
* .
* In the special case when
*
*
*
* f
* =
* 1
*
*
* {\displaystyle f=1}
* , the solution gives the signed distance from
*
*
*
* ∂
* Ω
*
*
* {\displaystyle \partial \Omega }
* .
* One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method.
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