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eikons

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Possible Dictionary Clues
Plural form of eikon.

Eikons 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.

Eikons 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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