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From wikipedia, there's this animated GIF of a linkage.

Here's a frame from the animation:

Question: How does it calculate log? All I see there are trig functions and circles. log doesn't show up in geometric maths, does it?

My friend thinks it's just a (lucky) approximation.

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  • $\begingroup$ It does not need to do log functions since the scale can be calibrared to handle the difference. Anyway a fourbar function is really complex thing and can be made to do quite many things, luck has nothing to do with this. $\endgroup$
    – joojaa
    Commented Jan 3, 2020 at 5:26
  • $\begingroup$ @joojaa Both scales are linear. $\endgroup$
    – Greg Bell
    Commented Jan 3, 2020 at 6:37
  • $\begingroup$ @GregBell but not the same linear length... is the ratio of lengths about 2.7? $\endgroup$
    – Solar Mike
    Commented Jan 3, 2020 at 6:45
  • $\begingroup$ @SolarMike in this case 3.5 $\endgroup$
    – joojaa
    Commented Jan 3, 2020 at 7:10
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    $\begingroup$ It's not even a good approximation. Log 2 = 0.3010 but the diagram shows about 0.27. $\endgroup$
    – alephzero
    Commented Jan 3, 2020 at 10:26

2 Answers 2

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This is a classic linkage synthesis problem called "function generator". Before calculators, linkages like this were used to approximate mathematical functions. They died with the slide rule.
However, this principle of coordinated input and output links is still an important part of many linkage mechanisms. Look at your windshield wipers.
Any kinematics textbook covers this topic. Design of Machinery - Robert Norton, Theory of Machines and Mechanisms - Uicker and Pennock, Mechanism Design - Erdman and Sandor. A new CAD based method is recently published called pole and rotation angle constraints. See the book Planar Linkage Synthesis - Zimmerman www.prclinks.com

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Angles are related to trig functions, which are indeed related to exponentials, which in turn are related to logs. This means that logs and angles are distantly related (but in very interesting ways). That said, I do not know how to explain the workings of the mechanism, but note the following:

In the study of kinematics of mechanisms there is something called the dual-number formalism in which the rotary position of one end of a rotating element relative to its other end can be expressed in terms of dual number exponentials. Then the rotary position of the end of another element which is linked end-to-end with the first element can be expressed as the sum of exponentials- as if you were summing vectors by placing them end-to-end.

Professor An Tzu Yang at the University of California at Davis College of Engineering was a popularizer of this technique 50 years ago.

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