Interpolation

A AbstractMeshField stores values at grid nodes and nothing in between. An InterpolatedField equips it with piecewise-polynomial interpolation, turning it into a continuous, callable function you can evaluate — and differentiate — anywhere in the domain. It is also the engine underneath reinitialization, NewtonSDF, and the ImplicitIntegration quadrature, though most users meet it through the off-grid queries below.

Building an interpolant

Wrap a AbstractMeshField in an InterpolatedField, choosing the polynomial order. The result is callable and evaluates the local interpolant at the query point — coordinates may be passed as separate scalars or as a single point-like object such as an SVector:

using LevelSetMethods
star = MeshField(CartesianGrid((-2.0, -2.0), (2.0, 2.0), (64, 64))) do x   # see the geometry page
    r, θ = hypot(x...), atan(x[2], x[1])
    return r - (1 + 0.25 * cos(5θ))
end
ϕ = InterpolatedField(star, 3)
ϕ(0.5, 0.5)   # evaluate at an arbitrary point inside the grid

-0.11610956979486423

The construction is identical in any dimension.

Derivatives

The interpolant differentiates its local polynomial patch exactly (via ForwardDiff), so LevelSetMethods.gradient and LevelSetMethods.hessian return analytic derivatives rather than further finite-difference approximations. When you need the value alongside its derivatives, the fused value_and_gradient and value_gradient_hessian compute them together in a single pass:

using StaticArrays
val, grad = LevelSetMethods.value_and_gradient(ϕ, SVector(0.1, 0.2))

(-0.9597097997417927, [3.6051486271923157, -0.8027543450694594])