Hi geekmidget, it's from the automatic rebuild mechanism that shows replacement edit points on circles, arcs, and ellipses so that you will get a smooth result if you manipulate the points.

Exact circles are formed in NURBS by a sequence of arcs. The actual internal control points of your 300 degree arc are like this:



The problem with this is that if you manipulate one of those points the result will have a sharp corner in it like this:





So to avoid that when you turn on control points on an arc or circle you see the points of an automatically rebuilt curve that does not have interior corner points and so will deform smoothly.

Note that the location of the corner points corresponds to the spans that you noticed have 4 control points instead of 3:



This "auto rebuild" mechanism will use the actual control points if there are only 3 points in the segment. After trimming there are more than 3 points in the curve in those spots where the segments are glued together. That's what causes the different auto rebuild behavior.

- Michael