I'm working on a game that requires a seemingly random, but constrained and continuous wave function that controls the values of certain (think of stock market graphs).
I've created a wave function using a Factional Brownian Motion or Fractal Noise.
(See Nepeo's tutorial on FBM/Fractal Noise)
I then plug this function into a sigmoid which restricts the range of the function to (0,1), then translate and stretch this function to fit the range (-1,1).
y = 2(1/(1+e^(-1/8)a))-1
(Where a is the output of the fractal noise function)
The result is a beautifully random looking wave function that sits neatly between -1 and 1.
This is just a really cool piece of math and the concepts can be applied in many areas of game design. The ability to create apparent randomness from sine functions that follow a coherent pattern and smoothly transitions between values is incredibly useful.
I used Desmos to create the graphs and functions, you can see the actual graphs and tweak them yourself here: desmos.com/calculator/f7zznq1zeq
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Interesting. Aspects of that graph do have an uncanny similarity to long term market index graphs.