Editing Perlin noise (section)

===Interpolation===
[[File:PerlinNoiseInterpolated.svg|alt=|none|thumb|256x256px|The final interpolated result]]
The final step is interpolation between the {{math|2<sup>''n''</sup>}} dot products. Interpolation is performed using a function that has zero first [[derivative]] (and possibly also second derivative) at the {{math|2<sup>''n''</sup>}} grid nodes. Therefore, at points close to the grid nodes, the output will approximate the dot product of the gradient vector of the node and the offset vector to the node. This means that the noise function will pass through 0 at every node, giving Perlin noise its characteristic look.

If {{math|''n'' {{=}} 1}}, an example of a function that interpolates between value {{math|''a''<sub>0</sub>}} at grid node 0 and value {{math|''a''<sub>1</sub>}} at grid node 1 is

:<math>f(x) = a_0 + \operatorname{smoothstep}(x)\cdot(a_1-a_0) \quad \text{for }0\leq x\leq 1</math>

where the [[smoothstep]] function was used.

Noise functions for use in computer graphics typically produce values in the range {{closed-closed|–1.0, 1.0}} and can be scaled accordingly.