What Is This Function?

You may try finding X, Y coordinates of various points on the curve and then put those in "CurveExpert" a free software for fitting the points on the curve. There are many options like quadratic, spline, polynomial of different order for fitting. See which curve gives the best resemblance with this curve. You will get the equation for the curve fitting to the points.

GSuhas,

Wow, that is a beautiful application! CurveExpert is great. However, I still have not found a function that approximates the graph. I will keep playing with it, but if anyone can give me ideas I would appreciate it.

Can you allow a piecewise function that from x ~8-25 is a bit over 1/4 of a sine wave, from x ~75-92 likewise, and three linear segments?

It looks a lot like the graph for the change in frequency of a sound due to the doppler shift. You might try googling for that equation and then modify it to match your graph.

I would approach this as a combination of step functions. I was taught it easiest to deal with these using Laplace transforms. I'm not answering your question directly and perhaps not as helpful as the "CurveExpert" software, but I hope it points you in a helpful direction.

A segmented-piecewise approach is fine, but the function needs to be continuous across the range of inputs. Super-imposed continuous functions would be fine.

The Fourier Transform is the continuous function that can imitate your curve. Indeed it can approximate any curve over a given data set.

http://en.wikipedia.org/wiki/Fourier_transform

Motion control firmware have use similar for speed curves

0st - no motion

1st - S-curve acceleration

2nd - ramped deceleration

3rd - S-curve deceleration

but it will result in jerky motion

Google "Galil controllers" it might help

It would be difficult to imitate the sharp changes on either side of the ramp. (Sin to straight line)

for that an additional input is needed (such as timer. limit switch, position sensor, speed sensor, acceleration sensor , keyboard or whatever)

HP Basic (Rocy Mountain Basic) of the 70's to 80's had a nice feature called boolean-algebra. (True=1 , False=0). [Easy to duplicate in any programming system]

An exact formula could then be written to give the desired result.

What is the purpose or application ?

For the S curved parts you can use and adapt the pre-computer formula for Sin X

y = Sin x = x - x^3/3! + x^5/5! - x^7/7! + X^9/9! . . . . . . .

and the line

y = mx + c

Again, the fourier transform is your friend for this type of function.

you just need to get the higher frequency waves to obtain a good approximation.

programs are available for this.

they give yoy therefore a continuous function over the interval required.

Fourier transform is a good tool but I challenge you to scale and enter the graph and calculate and plot approximation superimposed on the original.

Hi patjdixon

What is the purpose or the application?