In response to our latest Make It Last Build Series project, the Drawbot, Bill Ola Rasmussen posted a link to this piece he wrote on the math involved in V-plotter design. Thanks, Bill!
What is the optimal configuration of control lines for an area to be plotted? Obviously, we can’t have a drawing area above the control lines — our friend gravity sees to that. But, can we do better than hand waving and “somewhere below the control lines” for the plot area? Yes: we think up some constraints and model them with math and code (two more friends!):
Tension: We can imagine that the control lines have to be under some tension in order to be effective. For the purposes of this article we say that both string tensions must be in the range [ m/2, m*1.5 ], where m is the mass of the plotter head. Lines can neither be too slack nor too heavily loaded. The effect of this constraint is to prevent any line from being too close to horizontal or too close to vertical.
Resolution: There is a change in resolution when we map a change of length in one or both of the strings into X and Y coordinates. I.e. Coordinate system conversion causes a non-uniform step resolution. We say that, for each control line, a one unit change causes at most a 1.4 unit change in the X,Y coordinate system. We limit plotting to the area of reasonable resolution. Here our definition of reasonable is a 40% change.To implement these requirements, we need to know how to calculate them. The next few sections talk math and have pretty pictures.
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