Partial polygon matching
There is quite a bit of work on this important problem. Some of the most
insightful work is by Helmut Alt and collaborators. He wrote a survey in 2009:


  Helmut Alt. "The computational geometry of comparing shapes." Efficient Algorithms. Springer Berlin Heidelberg, 2009. 235-248.
  (Springer link.)



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Image from Helmut Alt &amp; Leonidas J. Guibas. 
"Discrete geometric shapes: Matching, interpolation, and approximation." Handbook of computational geometry 1 (1999): 121-153.).