Similarity in shapes I have been thinking which countries in the world are similar in shape, then I had this problem. 

For example we have country maps as seen in the figure and we need to determine which country is similar in shape to the country A. Intiutively we can say that the 5.th country is more and less similar, but it is difficult to determine the second most similar country. 
The sizes of the maps do not matter, we have option to scale. I have been thinking to solve the problem like least squares method but we have to determine the functions of the maps.
Is there any branch of mathematics study these things?
 A: This would fall under the heading "shape registration" or "shape matching", and is a common task in computer vision - for instance, the mail sorting machines used by the post office use shape recognition to read the addresses on letters and packages.  
You might take a look in to image registration for starters, and possibly this paper about shape similarity measures.  The bibliography in that paper would also be a good place to look.
In general, the problem isn't as easy as one might think, since like you say we have to consider mappings of the shapes in addition to just their boundaries.  It is always possible to measure the "distance" between two shapes using the Hausdorff distance, but that measurement only will only help for your problem once the shapes have been "registered", i.e. shifted and scaled so they match as closely as possible.
One naive numerical method might be to use the Hausdorff distance to register the shapes as follows: 


*

*Approximate the center of mass of the shapes, and translate one so their centers match up

*Minimize the Hausdorff distance between the shapes by rotating and scaling

*The minimized Hausdorff distance gives a measurement of the distance between the two registered shapes


Here's another link I've just found.  This is fun stuff to play with, enjoy!
