Clouds, like smoke, have an amorphous structure without well defined surfaces and boundaries. In recent times, computationally intensive physical modeling techniques have given way to simplified mathematical models which are both computationally tractable and aesthetically pleasing [34,28].

The main idea behind these techniques involves generating a
realistic 2D or 3D texture function using a fractal or spectral based
function. Gardner suggests a Fourier-like sum of sine waves with phase shifts

with the relationships

Care must be taken using this technique to choose values to avoid a regular pattern in the texture. Alternatively, texture generation techniques described in Section 6.20.2 can be used.

A stochastic method, based on work by Fournier and Miller [32,66],
uses a midpoint displacement technique called Diamond-Square for generating
a set of random values on a uniform grid. These generated values
are interpreted as opacity values and correspond to the cloud density at a
given point. The algorithm is iterative and during each iteration
two steps are executed. The first, the *diamond* step takes four corners
of a square and produces a new value at the center of the square by averaging
the values at the four corners and adding a random number in the range .
The second step, the *square* step, consists of taking the corners of
the four diamonds that were generated in the diamond step (they share the
center point of the diamond step) and generating a new center value for each
diamond by averaging its four corners and adding a random number in the
range . During the square step, attention must be paid to diamonds
at the edges of the grid as they will wrap around to the opposite side of the
grid. During each iteration the number of squares processed
is increased by a factor of four. To produce smooth variations in
the generated values, the range of the random value added during the
generation of center points is reduced by some fraction for each iteration.

Seed values for the first few iterations of the algorithm may be used to control the overall shape of the cloud.

Any of these techniques can be used to produce a 2D texture
which can be used to render a *cloud layer*. A cloud layer
is simulated by drawing a large textured polygon in the sky
at a fixed altitude. A luminance cloud texture is used to blend
a white constant texture environment color into a blue sky polygon.

Some of the dynamic aspects of clouds can be simulated by vary parameters over time. Cloud development can be simulated by scaling and biasing the luminance values in the texture. Drifting can be simulated by moving the texture pattern across the sky, i.e., transforming the texture coordinates. Ground fog can be simulated by drawing the thin cloud layer between the viewer and ground rather than the viewer and the sky.

Gardner also suggests using ellipsoids to simulate 3D cloud structures. The texture data is generated using a 3-dimensional extension of the Fourier synthesis method outlined above and the textures are applied with increasing transparency near the boundary of the ellipsoid. These 3D textures can also be combined with the volume rendering techniques described in Section 16.2 to produce 3D cloud images. In order to improve the performance of the rendering, the full volume rendering algorithm need not be used. In particular, the cloud may be assumed to be elliptical and opaque at the center. Therefore, the interior of the cloud can be drawn as a polygonal shell and the outer edges of the cloud using the volume rendering techniques.