Wikipedia:Reference desk/Archives/Mathematics/2022 December 12

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December 12

Plotting a graph on paper in the good old times

I'm aware of things like slide rules for many basic operations, which were handy when you didn't have a computer and not even a pocket calculator available. But how did people plot graphs on paper back then when they cared about accuracy? I imagine that something like the sine function could be easily be plotted with simple gears. But what about more complex functions? Did they just chose some points of the function and did the best to connect them by hand? Is there a complex set of gears that would be helpful plotting functions? Bumptump (talk) 18:56, 12 December 2022 (UTC)

See Tide-predicting machine for an early example of an analog computing device that produced paper graph output. AndyTheGrump (talk) 19:01, 12 December 2022 (UTC)

A scientific draughtsperson would have a set of flat splines that could be used to draw a smooth curve connecting a given sequence of computed points on the graph of a smooth function. Alternatively, they could use French curves. For illustrations requiring high precision, a larger version could be made first, with corrections and adjustments made as needed, which was then transcribed to the desired scale using a pantograph, but often precision is not that important.  --Lambiam 19:55, 12 December 2022 (UTC)

If they really cared about accuracy they chose enough points. But normally that is unnecessary. For mathhematical graphs they'd find important points of the function like max and min and where it crossed the zero line and inflection points where the curve changes direction and then did a smooth curve like with a spline between them if they were reasonably close. NadVolum (talk) 00:04, 13 December 2022 (UTC)

The (clearly hand-drawn) illustration of the graph of the hyperbolic tangent function, figure 18 on p. 184 of the second edition (1937) of volume I of Courant's Differential and Integral Calculus,^[1] is also an illustration of such limited accuracy: the distance of the function values to the asymptotes shown for $x=\pm 2~$ is about twice that of the correct distance.  --Lambiam 09:15, 13 December 2022 (UTC)

If you have a look at Ship gun fire-control system they had enormous and very heavy analogue computers and it says the last one used in combat was in 1991. They would have no problem with plotting functions! NadVolum (talk) 13:22, 15 December 2022 (UTC)