I know this is an old question, but here is one of the clearest explanations regarding interpolation between the fixed points that make up 1d Perlin noise http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf
One of the most important things you need to know, useful throughout programming, is the interpolation function ...
http://paulbourke.net/miscellaneous/interpolation/
Once you have random points with smoothstep interpolation, you have a kind of smooth 1d noise function.
see smoothstep on the wiki. a lot on the topic through Google. https://en.wikipedia.org/wiki/Smoothstep
obviously the hyperlink is unstable, here it is again:
Simplex, demystified noise
Ken Perlin introduced "simplex noise", replacing it with the classic noise algorithm. The classic “Perlin noise” received an award at the academy and became the ubiquitous procedural primitive for computer graphics for many years, but in retrospect it has many limitations. Ken Perlin himself designed simplex noise specifically to overcome these limitations, and he spent a lot of good on it. Therefore, it is better than its original algorithm.
Some of the most important benefits:
• Simplex noise has lower computational complexity and requires fewer multiplications.
• Simplex noise scales to higher sizes (4D, 5D and higher) with much lower computational costs, the complexity lies in the size instead of classical noise.
• Simplex noise has no noticeable directional artifacts.
• Simplex noise has a well-defined and continuous gradient everywhere that can be calculated quite cheaply.
• Simplex noise is easy to use in hardware.
Unfortunately, even now in early 2005, very few people seem to understand simplex noise and almost no one uses it, so I wrote this. I will try to explain the algorithm a little more thoroughly than Ken Perlin managed to do in my commentary on the course from Siggraph 2001 and 2002, and I hope it is clear that this is not so difficult to understand as it seems at first glance. From what Ive found out what confuses people the most is the impenetrable nature of Ken Perlin's reference implementation in Java. It presents a very compact and uninhibited code to demonstrate the principle, but this code is clearly not intended to be read as a textbook. After several attempts, I abandoned the code and instead read its article, which was much clearer.
Not a Crystal Clear, however, since it represents the algorithm mainly in words and code fragments. I would appreciate some graphs and numbers and some useful equations, and this is what I am trying to provide here to make it easier for others to understand the grandeur and beauty of simplex noise. I will also first explain things in one and two dimensions to simplify the explanation with the help of graphs and images, and then move on to three and four dimensions. Classic noise To explain simplex noise, it is helpful to have a good understanding of classic Perlin noise. I saw a lot of bad and misguided explanations in this area, so to make sure that you have the necessary foundation, I will first introduce Perlin's classic noise.
Perlin noise is the so-called gradient noise, which means that you set a pseudo-random gradient at regular intervals in space and interpolate a smooth function between these points. To generate Perlin noises in one dimension, you associate a pseudo-random gradient (or slope) for the noise function with each integer coordinate and sets the value of the function at each integer coordinate to zero.
For a given point somewhere between two integer points, the value is interpolated between two values, namely the values that would be the result if the closest linear slopes from left and right were extrapolated to the specified point. This interpolation is a smooth step.