# The Algorithm

The algorithm is slightly different depending on whether the rightmost reflex vertex is on the upper or lower chain of the polygon. I present the case where it is on the lower chain, the oher case is symmetrical.

Until the polygon is convex:

• Find a righmost reflex joint;
In this example, it's point 1, shown in red.

• Relabel the joints counterclockwise along the polygon's perimeter so that this rightmost reflex joint is j1, and all straight joints are ignored.

• Find the index p such that jp is the first joint not on the right of the ray ( j0, 1 )
In this case, it is the joint 4.

• Perform the move described below until either the joints j0,j1, and jp-1 become colinear, or one of the moving joints straightens. In this instance, we rotate point 1 counterclockwise about point 8, maintaining the lengths of all the links. Only the angles at four points are allowed to change: joints 8, 1, 3 and 4.

• If the joints j0, j1, and jp-1 become colinear first, decrement p and continue the move with the new jp and jp-1.
In the example, the joints 8, 1 and 3 become colinear. So joint 3 becomes the new jp.

• If a joint straightens, the iteration is complete, go on to find the new righmost reflex joint.
In the example shown here, this will happen after we update p and continue the move: the joint 1 straightens. Now one reflex vertex has been straightened.

### The Move

Here j1 is the rightmost reflex vertex, and jp is the first joint not to the right of the ray ( j0, j1 ).

First we fix the position of all the joints jp+1 through jn-1 (which aren't shown here) as well as j0 and jp. Then we fix all the joint angles, except for the four joints we will be moving: j0, j1, jp-1 and jp.

This has for effect of fixing the distances || j0 jp || and || j1 jp-1 || (shown as dotted lines).

Now simply rotate j1 clockwise about j0, the motion of the other two moving joints is uniquely defined since all the link lengths are to be maintained.

Notice that we completely ignore the joints between j1 and jp-1 for the move. We can do this because because we have fixed all the angles on that chain, so it can just be considered an edge between j1 and jp-1 for the purpose of executing the move.