Editing Perlin noise (section)

===Dot product===
[[File:PerlinNoiseDotProducts.svg|alt=|none|thumb|256x256px|The dot product of each point with its nearest grid node gradient value. The dot product with the other three nodes in the cell is not shown.]]

For working out the value of any candidate point, first find the unique grid cell in which the point lies. Then, identify the {{math|2<sup>''n''</sup>}} corners of that cell and their associated gradient vectors. Next, for each corner, calculate an offset vector. An offset vector is a displacement vector from that corner to the candidate point.

For each corner, we take the [[dot product]] between its gradient vector and the offset vector to the candidate point. This dot product will be zero if the candidate point is exactly at the grid corner.

For a point in a two-dimensional grid, this will require the computation of four offset vectors and dot products, while in three dimensions it will require eight offset vectors and eight dot products. In general, the algorithm has {{math|''O''(2<sup>''n''</sup>)}} [[Big O notation|complexity]] in {{mvar|n}} dimensions.