The diagram on the left goes step by step
through the construction of 2D noise.
The point being evaluated is shown in red.
Try clicking on the diagram.
First you will see a linear gradient
rising from one of the four surrounding grid points.
This is the pseudo-random gradient the algorithm
has chosen at that point.

Click again, and you will see the drop-off filter centered
at that grid point.
This filter, which makes the influence of this grid point
gradually drop to zero at a unit distance,
is a product of two
cubic drop-off curves, one in x and the other in y.
The shape of each cubic is computed by 3t^{2}-2t^{3}.

Click again, and
you will see the the linear gradient multiplied by
the weighting filter. This will be the contribution from
that grid point to the final result.

Click three more times.
You will see, in turn, the contributions from each of the other three surrounding grid points.

Finally, click one more time, and you will see the final result -
the sum of the contributions from the four surrounding grid points.