Triangle dissimilarity - part 2

We have now 4 different triangle dissimilarity measures defined. It is time to test with (suitably sensible) random triangles how they compare with each other.

Vertex dissimilarity and overlap ratio are rather well related with small random variation (some noise added to the first triangle). Groth and PT both have their own character. No doubt the shape space is approx. 3 dimensional (d= 2.8 with some estimates not covered here). This intuition comes from the following facts:

• three edge lengths
• if one edge is fixed (with the length as one degree of freedom, and orientation being neutral), the free vertex has 2 degrees of freedom. 
• Groth definition of a vertex angle and ratio of two adjacent angles (and scale as the third degree of freedom)
• and so on.. 

The vertex dissimilarity is the fastest to compute and seems to give good results in many applications, but it requires

• scaling with the perimeter length to become scale invariant (S), (this is cheap)
• sparse rotation to become orientation invariant (O) (and this is rather expensive),
• centering by the triangle mean to be location invariant (L) (this is cheap).