6.21.9 Generating 2D Noise to Simulate 3D Noise

We have described a method for creating 2D noise functions. In the case of lattice noise, these 2D functions correspond to a 2D slice of the lattice. There are cases where we want to model a 3D noise function and where such a 2D function is inadequate. For example, to draw a vase that looks like it was carved from a solid block of marble, we cannot use a lattice 2D noise function.

However, we can create a 2D noise function that approximates the appearance of a true 3D noise function, using spot noise [100]. We take into account the object space coordinates of the geometry, and generate only spots that are close enough to the geometry to make a contribution to the 3D noise at those points. The difficulty is how to render the spot in such a way that at each fragment the value of the spot is determined by the object space distance from the center of the spot to that fragment. Depending on the complexity of the geometry, we may be able to make an acceptable approximation to the correct spot value by distorting the spot texture. One possible way to improve the approximation is to compensate for a nonuniform mapping of the noise texture to the geometry. Van Wijk describes how he does this by nonuniformly scaling a spot. Approximating the correct spot value is most important when generating the lower octaves, where the spots are largest and errors are most noticeable.