# Function with specific characteristics

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Hi all,

I'm working on something horror-themed, possibly for a portfolio. I have a visual effect which can be applied a certain amount, let's say a floating point number from 0 to 100 for the sake of argument. In keeping with the horror theme I want the effect to progress in a somewhat jerky unpredictable manner with pauses. Any assistance with choosing an appropriate function would be appreciated. Some desired characteristics are below:

• Monotonic
• Some pseudo-random component
• Has pauses (plateaux)
• Has some sense of momentum (e.g. derivative has some pattern, not just spikes)

Some approaches that I considered which don't quite make the grade:

• A random walk with another function as the envelope (no pattern to derivative, timing too predictable)
• Some monotonic function plus noise (timing too predictable, may not be monotonic)

Maybe I'll have to go piecewise, but I hope not.

Any thoughts would be appreciated!

JT

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Just off the top of my head: you might consider the shape of tann(t) for t from some negative number tstart to some positive number tend, and n = positive odd integers. You would likely have to apply it piecewise but it's monotonic, the slope flattens at the origin, and the overall form can vary with n. Select a random small value greater than the slope of tann(0). The "pause" would be between d/dt(tann(-t))==slope and d/dt(tann(+t))==slope. You should be able to get a large set of variations by varying values for tstart, tend, n, and the slope for each piece.

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Take some 1D noise function with a range that includes 0, then take the maximum of that function and 0. Now compute the integral of this function. I believe that should fit the bill.

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You might want to consider multi-octave noise. What you're describing seems to be a random walk with high and low frequency (pause) components, which can easily be obtained by mixing noise functions from different octaves.

This particular article talks about multi-octave perlin noise in a single dimension:

http://freespace.virgin.net/hugo.elias/models/m_perlin.htm

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Thanks all, some options for me to try out. I suspect the tan option won't be right, the parts that tend to infinity would need to be worked around, plus I'm not sure I want that extreme a slope. Alvaro, I suspect that you and frying_pan are talking about similar things. Using perlin noise as the noise function may give nice results.

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Just in case you want to go the multi-octave route, I'd suggest looking at simplex noise, which is essentially the same concept but implemented more efficiently (using a dimensionally minimal set of gradients). There are really nice c++ implementations that are plug and play here. The lowest dimension noise you can get with that code is 2D, but you can just ignore one dimension.

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If I recall correctly, Perlin noise suffers badly in higher-dimension forms, I think 1D with a few octaves will be fine. ;) Plus I already have the code (somewhere).

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Yeah, I was going to suggest something similar. All of the trigonometric functions have patterns which can be pretty spiky.

And yes, doing it piecewise would be the way to go. If you want it to flow then you have to check the limits of the slopes at or near the ends.

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