The least obvious part of all of this is how to construct that metric tensor G from a given vector, but it turns out it's super straight forward. The attached example shows how to do so for a 3D metric tensor. In our paper we outline specific bounds for the sizing of this metric.

Just to elucidate the idea a bit more, on the left we have the original perlin noise algorithm, and on the right we have the steerable version, with the changes underlined in red. We simply add a second set of weights, and incorporate our metric into the original dot product step

New at #SIGGRAPH2025:

Can we make Perlin Noise stretch along some underlying vector field? Well it turns out it's possible with two simple additions to the original method! No need for advection or convolutions.

Find the paper and implementations here:
github.com/jakericedesi...

Distances to LP Voronoi Edges :)

shadertoy.com/view/MXVyz1

Distances to the boundary of voronoi in L4.

shadertoy.com/view/lfKfWV

Hopefully it doesn't crash webgl on your device! :)