An eikonal equation (from Greek εἰκών, image) is a non-linear first-order partial differential equation that is encountered in problems of wave propagation.

The classical eikonal equation in geometric optics is a differential equation of the form

where ${\displaystyle x}$ lies in an open subset of ${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle n(x)}$ is a positive function, ${\displaystyle \nabla }$ denotes the gradient, and ${\displaystyle |\cdot |}$ is the Euclidean norm. The function ${\displaystyle n}$ is given and one seeks solutions ${\displaystyle u}$. In the context of geometric optics, the function ${\displaystyle n}$ is the refractive index of the medium.

More generally, an eikonal equation is an equation of the form

where ${\displaystyle H}$ is a function of ${\displaystyle 2n}$ variables. Here the function ${\displaystyle H}$ is given, and ${\displaystyle u}$ is the solution. If ${\displaystyle H(x,y)=|y|-n(x)}$, then equation (2) becomes (1).

Eikonal equations naturally arise in the WKB method and the study of Maxwell's equations. Eikonal equations provide a link between physical (wave) optics and geometric (ray) optics.

One fast computational algorithm to approximate the solution to the eikonal equation is the fast marching method.

The term "eikonal" was first used in the context of geometric optics by Heinrich Bruns. However, the actual equation appears earlier in the seminal work of William Rowan Hamilton on geometric optics.