farcast wrote: ↑Sun Dec 31, 2023 6:20 amMost of my time spent on this graph was thinking about approximation functions, specifically for the inverse of the displacement of the train's stop point over time. For some reason, the formula for stop point is too hard for wolfram alpha to find an inverse of. I tried, but I can't figure it out either. I simplified it to (e^(-2x) + (a - 2)e^(-x) + ax) = y but couldn't get any further. 'a' is a variable that's based on train stats so I can't just replace it with a constant. I just decided to make my own approximation function.
mmmPI wrote: ↑Sun Dec 31, 2023 1:01 pmi will try to make it so that desmos can reuse the value it approximate when drawing the curve that is "Braking distance overtime + displacement overtime" or maybe get to understand why it cant. Maybe i could get wolfram to find the inverse of the braking distance over time and try to make the sum of those 2 inverse function , inverse braking distance and inverse displacement. But at this point i'm not yet understanding the reasonning behind the operation.
I am quite excited because i think i found a hack in Desmos while playing with the etiquettes for points made me happy , i haven't progressed much using it yet, but i feel it was worth sharing right away in case it can help saving some time. was investigating why it's so hard to combine the inverse of displacement overtime, and the inverse of braking distance overtime, to get the inverse of displacement of the stop point over time. It's just not possible to sum the 2 inverse functions, at least from a rigorous mathematical point of view ^^ , it feel like additions are very bad for inverse functions, x^x has an inverse, x^x +x doesn't ... so here's what i did :
Yep , i just cheated, i asked desmos to draw the function that wolfram alpha didn't find the inverse but not according to the X axis, but rather the Y axis. This is a visual representation of what the real inverse function if it existed looked like. I think it also show why it would be really hard to combine the 2 inverse function "according to x". But it is correct to combine the 2 functions if they are drawn based on the Y axis a simple sum is enough. Symmettry so powerful !
Then just drawing the function is not really helpful, because one need to be able to use that inverse to give it some input and know the output. So really now the question is " which input should i give to get [desired value] as output ?" Which is not how one is supposed to use a function but there is a way to make Desmos actually answer the question and for one user to be able to reuse that value.
That's the tilde ~ thing that enable in Desmos the regressions and it's being abused so that instead of trying to find coefficient of a polynomial or fitting curve to some datas it is attempting to find the parameter that would make the fake inverse function equal to the block length B. And it's doing it pretty accuratly.
This trick can also be used to find the maximum or the minimum of a function. If instead of B for block length, you write 1000000000000 , Desmos will try to find the parameter to give to the function so that the result is the closer possible to 1000000000000 , which in case the function is never reaching it, will just yield the parameter that yielded the highest output value from the function. ( thinking about the "max throughput" which has no analytical solution one could use this technique if you want to reuse the value of "max throughput".)
I'm not sure yet how this shall be used for more practical purposes, i'm still somewhat copying/following your reasonning and atm i'm a bit puzzled about the initial velocity, or rest speed. For now it's only based on an initial speed of 0 so that would model the stop and go traffic, or at least be a first step toward it. I think computing the braking time from the braking distance is also something i will have to add, and i will have a look at how you did it
Advanced alien : here you can use this spaceship it's almost ready to go back to your planet, just make sure to insert the standard galactic coordinate of your system.
in both case me =>
My technology hasn't reached that stage yet , but i'm taking this with me for studying purposes