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  "body": "\n## Definition\n\nLet $\\Gamma(t) \\subset \\R^d$, $t \\ge 0$, be a family of closed hypersurfaces that bound open sets $\\Omega(t)$.\nLet $V(x, t)$ be a given speed of $\\Gamma(t)$ in its outward normal direction.\nThe *level-set method* represents and evolves $\\Gamma(t)$ as follows.\n\n1. Representation: choose a continuous function $\\phi \\colon \\R^d \\times [0, \\infty) \\to \\R$ with $\\Omega(t) = \\set{x : \\phi(x, t) \u003c 0}$ and $\\Gamma(t) = \\set{x : \\phi(x, t) = 0}$, the zero [[level set|0B2RZ0778N]] of $\\phi(\\cdot, t)$.\n2. Evolution: with $V$ extended near $\\Gamma(t)$, $\\phi$ solves the level-set equation\n$$\\phi_t + V \\abs{\\nabla \\phi} = 0.$$\nThis is a [[Hamilton–Jacobi equation|5TRXQKP7DY]], understood in the [[viscosity sense|H1ERJ3T75T]] where $\\phi$ is not smooth.\n3. Discretization: the equation is solved on a fixed grid, usually with upwind finite differences. The interface $\\Gamma(t)$ is read off as the zero set of the discrete $\\phi$.\n\nIf the interface is carried by a velocity field $v(x, t)$, the evolution equation is $\\phi_t + v \\cdot \\nabla \\phi = 0$, which is the case $V = v \\cdot n$.\n\n## Remarks\n\nWhere $\\nabla \\phi \\ne 0$, the outward unit normal is $n = \\nabla \\phi / \\abs{\\nabla \\phi}$, with $\\nabla$ the [[gradient|0ABK9V9Y64]].\nThe [[divergence|0MWV6ZTK6S]] $\\kappa = \\nabla \\cdot n$ is the sum of the principal curvatures of $\\Gamma$, a multiple of its [[mean curvature|PA78NFEPCF]], and it is used for motion by curvature with $V = -\\kappa$.\nOne usually keeps $\\phi$ close to a signed distance function, with $\\abs{\\nabla \\phi} = 1$, by periodic reinitialization.\n\nExample: in $\\R^2$ let $V = 1$ and $\\phi(x, t) = \\abs{x} - (r_0 + t)$ with $r_0 \u003e 0$.\nFor $x \\ne 0$ one has $\\phi_t = -1$ and $\\abs{\\nabla \\phi} = 1$, so $\\phi_t + V \\abs{\\nabla \\phi} = 0$, and $\\Gamma(t)$ is the circle of radius $r_0 + t$.\n\nNon-example: $\\phi(x, t) = \\abs{x} - r_0$ represents a circle but fails clause 2 for $V = 1$, since $\\phi_t + \\abs{\\nabla \\phi} = 1 \\ne 0$.\nIts zero set does not move.\n",
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