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I apologize ahead of time for not knowing the right terminology to ask this question well...

I am trying to calculate a table of values using the "fastest" ramp/rate/curve based on specified limits over a period (time).

For example, assuming I have a starting value of 3000 and a target of 2000 over 10 days, so the LINEAR data would look like so: (from python pandas testing I am playing with)

rt = { "y": [3000, np.nan, np.nan, np.nan, np.nan, np.nan, np.nan, np.nan, np.nan, 2000]}
test = pd.DataFrame(rt)
d2 = test.interpolate('linear')
d2

0  3000.00
1  2888.88
2  2777.77
3  2666.66
4  2555.55
5  2444.44
6  2333.33
7  2222.22
8  2111.11
9  2000.00

I am wanting to calculate the curve representing the fastest rate of change with limits, for example where maxDailyChange = 350 such that no daily value change is > maxDailyChange

I do not know what to call this, I am sure there is a name, terminology...

In a perfect scenario, or perhaps another option, the curve would be smoothed, such that the
rate of change was relative to the remaining distance to target ? (again, my lack of understanding makes it hard to describe well)

I am using python pandas and numpy to play around with the problem, but any help just understanding the problem would be of great help !

Cheers and thank you for any insight !

BBALEY
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  • Welcome. So you want the 'fastest' curve between two values, where the speed is limited, but the time is fixed? Then what do you mean by 'fastest'? Is the curve where you increase at the fastest allowable rate and then maintain position 'faster' than other curves? – crubow Feb 01 '24 at 00:14

1 Answers1

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I think I have found a suitable method for what I was looking for. I remembered using a Savitsky-Golay filter previously for smoothing interpolated missing values, and it turns out this function provides the desired result (or at least close enough).

Given the following data:

data = { "linear": [3000, 2888.88, 2777.77, 2666.66, 2555.55, 2444.44, 2333.33, 2222.22, 2111.11, 2000] },
         "linearsgol": [3040.454545, 2845.30303, 2668.712121, 2510.681818, 2371.212121, 2250.30303, 2147.954545, 2064.166667, 1998.939394, 1952.272727] },
         "fastest": [3000, 2800, 2600, 2400, 2200, 2000, 2000, 2000, 2000, 2000] },
         "fastestsgol": [3045.454545, 2778.787879, 2550, 2359.090909, 2206.060606, 2090.909091, 2013.636364, 1974.242424, 1972.727273, 2009.090909]
        }

Where "fastest" is a simply a vector using the fastest possible rate, until the target is reached (target=2000),

and the maximum rate of change is 200,

applying the Savitsky-Golay filter to the "fastest" curve gives I think what I am looking for - which I think is a "Brachistochrone curve" ??

Plot of results

BBALEY
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  • The brachistochrone curve is similar, but the idea is you're constraining a falling object to a path (like a wire) subject only to gravity, so literally, you can call it a brachistochrone curve but you'll cause confusion because this name is associated with the gravity problem. It has a fun history you should check out. – crubow Feb 01 '24 at 20:52