In this document (http://webstaff.itn.liu.se/~stegu/simplexnoise/simplexnoise.pdf) about Perlin (and Simplex) Noise you can find an explanation about 1D Perlin Noise (at the top).
For a given point x somewhere between two integer points, the value is interpolated between two values, namely the values that would have been the result if the closest linear slopes from the left and from the right had been extrapolated to the point in question.
Am I right to say that the yellow dots are the extrapolations to the point in question?
This interpolation is not linear with distance, because that would not satisfy the constraint that the derivative of the noise function should be continuous also at the integer points. Instead, a blending function is used.
Am I right to say that if the interpolation was linear, it would have been the blue dot? Instead of using linear interpolation, Perlin uses a third or fifth degree polynomial.
I thought the interpolation would be:
y = f(1 - x) * x + f(x) * x, where f is the blending function. Unfortunately, this isn't equal with the black noise curve...