Matching

Comparison of polygonal arcs

The comparison of (arbitrary, not necessarily maximal convex) arcs a₁, a₂ is derived in the following steps:

Step 1
The arcs are transposed into tangent space representation.
Step 2
The resulting curves are scaled to the same length. Let l(a₁) > l(a₂), then the scaling factor sf is simply given by
sf := l(a₁) / l(a₂)
Step 3
The tangent space curve of a₂ is shifted vertically, equalizing the weighted average of the angle values of a₁ and a₂.
For the original arc representation this means a rotation of a₂ to fit better.
Step 4
The L₁ difference is taken. L(a₁,a₂) := || a₁ - a₂ || _L1, hence L is the area between the two tangent space curves.
Step 5
To gain the arc measure, L is multiplied by two factors:
- first: the scaling cost, given by sf
- second: the length af the longer arc, in our case l(a₁).
  This factor is assumed to give the meaning of this arc to the whole curve.

Hence the arc measure D(a₁,a₂) is given by:

D(a₁,a₂)= L(a₁,a₂) * sf * l(a₁)
= L(a₁,a₂) * l²(a₁) / l(a₂)