Grid Laplacian Node

The Grid Laplacian node computes the Laplacian of a scalar voxel grid. The Laplacian measures how a value at each voxel differs from the average of its neighbors—essentially, how much the field “curves” or deviates locally.

Mathematically, the Laplacian is defined as the divergence of the gradient of a scalar field. It is commonly used in physics and geometry for diffusion, smoothing, curvature analysis, and solving partial differential equations.

For a scalar field \(f(x, y, z)\), the Laplacian \(\nabla^2 f\) is given by:

\[\nabla^2 f = \nabla f = \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} + \frac{\partial^2 f}{\partial z^2}\]

The Laplacian is positive where the field has a local minimum (value smaller than its surroundings) and negative where it has a local maximum (value larger than its surroundings).

Inputs

Grid

The input scalar grid on which to compute the Laplacian. The grid must store scalar (float) values such as density, temperature, or a signed distance field.

Outputs

Laplacian

A float grid representing the Laplacian of the input field.

Each voxel contains the sum of the second derivatives of the input field along the X, Y, and Z axes. This can be used to detect local peaks and valleys or to drive smoothing and diffusion effects in procedural grids.