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The TNG Warp Speed Formula
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[QUOTE]Originally posted by machf: [QB] Well, I guess the "original" formula ("original" as in "shown on the first page of this thread") was arrived at by applying interpolation/extrapolation methods based on the available dataset (a certain number of pairs of values (W,v) ). So, I guess it can be done in pretty much the same way just by switching the axes around. Taking a closer look, it seems they started with the general approach v=W^(10/3) and looked for a function that would modify it in such a way that the results wouldn't be affected for W less than or equal to 9 but would give the asymptotic curve for 9 less than W less than 10, with v tending towards infinite as W tends towards 10... so basically, for W less than or equal to 9, you have a bunch of zeroes for, say, Y (where Y equates the product a(W)*b(W)*c(W) in that formula) and some more non-zero values for 9 less than W less than 10. I can see how this would be practical for calculating v as f(W), but for finding W as g(v) (where actually v=h(Y) and thus W=g(h(Y)) ) with several values of W for the same value of Y (that is, 0) it certainly wouldn't... So, let's just do it the other way around (compared to how it may have been originally done): start with W=v^(3/10) and find a function (or set of functions, if you like) such that W can be expressed as v^((3/10)*(1+F(v))) that passes through the known set of value pairs for (v,W)... the resulting values of W should approach 10 as v approaches infinity. Let me try to express it in simpler terms... the "original" formula is an approximation based on available data and the approach v=W^(10/3*(1+f(W))), solving f(W) based on data pairs not of (W,v) but rather (W,y) where y=(3/10)*ln(v)/ln(W)-1. So, the inverse formula should also be an approximation, based on the same available data and the approach W=v^(3/10*(1-g(v))), solving g(v) based on pairs not of (v,W) but rather (v,z) where z=1-(10/3)*ln(W)/ln(v). <s>(The problem with that, as I mentioned earlier, is that you'll end up with several instances of z=0 for different values of W, so maybe it's a better idea to choose a generic W=g(v) instead, or maybe, W=v^(3/10)*g(v) at most.)</s> You decide what kind of function you want g(v) to be: logarithmic, exponential, polynomial, sinusoidal, etc. (or even a combination thereof) and use the available data to find the proper coefficients (keep in mind that whatever function you choose to try to adjust the data to, it will have to have fewer coefficients than the amount of data pairs available). You can try different functions and see which one gives the best approximation. [/QB][/QUOTE]
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