Noise - Graphics Shaders - page 214

slope of the function at each point is set to be zero or
to a natural slope (more on this later). These properties
are directly interpolated to generate a piecewise cubic
or quintic function across the entire domain. In gradient
noise , the function value is set at each fixed point, and
the pseudo-random values at that fixed point are used
as the gradient (slope) of the curve through the point.
Gradient noise with the fixed values set to zero is the
type of noise originally defined for the Photorealistic
RenderMan package [37]. It is also popular to set both
the value and the gradient at the fixed points with
pseudo-random numbers, the value+gradient noise .
Noise that is based on pseudorandom number (PRN)
gradients is often called Perlin noise [34], [35]. For both
kinds of noise, these values are used to determine the
equation of a piecewise cubic noise function between
the two points, as described later in this chapter.
The noisegraph application is a tool that lets you
test out these different noise types and parameters for
yourself. It produces line graphs of 1D noise functions
and pseudocolor regions for 2D noise. The application
is driven by a menu with choices for the kind of noise,
the number of octaves, and the kind of pseudocolor
display; it also lets you make other choices for options
discussed in this chapter.
All three types of continuous noise functions are
illustrated by noisegraph displays in Figure 10.1. As you
can tell from these graphs, value noise can, by chance,
have regions of similar values, while gradient-only
noise always passes through zero at regular points.
This affects the kind of control you would have with
each. This distinction forms a kind of dual relationship between the functions
and is the key to how they are defined. We will see how it affects the actual
function expressions when we consider those a litle later in the chapter.

Figure 10.1. A one-dimensional value-
only noise function (top), a gradient-only
function (middle), and value+gradient
noise function (botom).

Cubic and Quintic Interpolation

We can use several different options as we create the piecewise interpolation
of the initial pseudo-random values that is needed to create the overall noise
function. Two common ones are to use a cubic interpolation or to use a quin-