Home > Mathematica, Mathematics > Walking in a straight line

Walking in a straight line

How to fold it inspired me to understand the Peaucellier’s linkage. Initially, I wrote a number of equations and rather foolishly (and lazily) asked Mathematica to solve them.  This was not successful but taught me:

  • about some of the algorithms used in solve
  • a limitation of using Solve:  this is more a failure to think about the problem deeply enough

I explored the Wolfram Demonstration Project and there was a project on the linkage. The source code was illuminating.  I started thinking about how this solution was derived.  Here is my approach. ( It never ceases to surprise me how often I took the wrong approach and how quickly a better approach is revealed. )

Linkage diagram

Consider the above diagram. Choosing suitable lengths for XA (=XB) and AC=BC=AD=BD one can solve for \alpha using \phi

I will present this using a number of images (the WordPress \LaTeX is laborious):

Definitions

Using position vectors to determine loci for A,B, D

Establishing the linkage at point D traces a straight line.

Animated gif of Peaucellier's linkage

I have also made an embedded CDF. The Wolfram Demonstration Project demonstration of Izidor Hafner referred to above is more instructive and shows another  linkage that produces a straight line.

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Categories: Mathematica, Mathematics

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