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# non linear ratios

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For values x: (0, xMax), where xMax is user defined, the values are evenly spaced and linear. The y values must start at one and should tend to zero over the range ofd x values. Before the application of a modifier function to x, the equation describing how x and y relate to each other is: y = 1 - x/xMax. This isn't very interesting or realistic for my purpose. Two others I'm using are
INVERSE:
y = 1/(x+1)  // need to make it x+1 because when n/0 = error & when x = 0  y = 1
// y only tends to 0 as x-> infinity
// also initial fall of is too rapid

COSINE:
linear = x/xMax;             // good core shape but as the range of x-values
y = (cos(linear * PI) + 1)/2 // increases it becomes quite flat though it
// can be improved by re-applying f'n to result


I'm looking for a bell-shaped curved like the right-hand-side of a normal distribution curve, or a signal function (though these are y: (0, 1) so technically it'd be a -ve signal, which I'm not sure exists). However, other than the cosine one above and doing some very protracted stuff with sinh and ln I haven't had much joy. I am numerically able but looking through mathematical stuff on the web all of a sudden I am very aware that I have become completely ignorant of the proper syntax and nomenclature so please make suggestions readily digestible to the layman! If I haven't been clear please ask and I'll try to specify. Thank you in advance for you help.

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Perhaps it's better if you explain what you need this for, or how you intend to use it?

edit: that is, since you're looking for a bell curve, but not using it, is there any particular reason?

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Alternately, rather than trying to find a function that produces the shape you want, you could always just use a lookup table (interpolated linearly or with splines) -- that way you can draw in the exact response you want.

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Write y = A + Bx + Cx^2 + Dx^3
so
dy/dx = B + 2Cx + 3Dx^2

when x = 0, you know y = 1 and dy/dx = 0

when x = xmax, you know y = 0 and dy/dx = 0

Gives you 4 equations, though two are trivial (A = 1 and B = 0). Solve the remaining two, and you have a simple polynomial that fits the boundary conditions, and smoothly goes from 1 to 0 over the range.