# General solution for the Eikonal equation $| \nabla u|^2=1$

Does there exist a formula for the general solution of the Eikonal equation? $| \nabla u|^2=1$. I'm looking for something similar to "the general solution of $\dfrac{\partial u}{\partial x}(x,y)=0$ is $u=\varphi(y)$, for an arbitrary function $\varphi$". That is, the formula should include one arbitrary function.

Thank you

• for 1D it is not hard and 2D, you can try to use polar coordinate, for high dimensional cases, maybe you can apply the same method. I know some numerical method for it. – Yimin Apr 3 '13 at 15:06
• While more computationally focused, Peternell & Steiner's - A geometric idea to solve the eikonal equation [pdf], may be of interest. – GEL Apr 3 '13 at 15:53

No, but there are a few special cases. Take an arbitrary convex function such as $y=x^2$, and take as the domain of $u$ all the points with $y<x^2$. Then for each point of the domain, there is a unique closest point on the graph. Take $u$ to be the distance to that point. The simplest case is $y=0$, $u(x,y) = |y|$. The level sets of $u$ are curves parallel'' to the boundary, and even for $y=x^2$ you can't write a nice formula for $u$, though you can write an implicit one. Same idea works in any number of dimensions, take a convex set, and define $u(x)$ as the distance to the set, when $x$ is exterior to the set. This solves the eikonal equation because $|\nabla u|$ is the maximum rate of change of $u$, which occurs by moving $x$ directly away from that closest point, and when you do that move, $u$ changes by exactly the same distance that you moved $x$.