When using InsertKnot or Rebuild to a cut surface, the edge (tip) changes. Why?
Untitled.3dm (83.5 KB)
If you use the default accuracy, there is no problem. The problem document uses centimeters (accuracy 0.1). Little accuracy?
How to correctly determine the necessary accuracy?
If you use millimeters and reduce tolerance, the problem simply decreases, but it remains 
InsertKnot should not change the shape. InsertControlPoint will usually change the shape.
Rebuild will usually change a shape.
Which object(s) are you modifying? Can you post a .3dm file with the objects before and after modification and specify exactly which commands were used?
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Can you post a .3dm file with the before and after surfaces as well as the trim curve? If you put the surfaces on different layers it will be easier to see the differences.
I think I can duplicate what you are seeing. The differences I see are on the order of 1.e-13 which is the usual precision of Rhino math and not significant.
I understand, but sometimes a weak accuracy gives a complete error
Use Layers
Untitled.3dm (48.1 KB)
The difference is .000003 which is much smaller than your absolute tolerance of 0.1.
In joining works. However, in order to avoid possible problems, it is worth using the accuracy of 0.01. 
By the way, smoothing with a radius of 0.1 gives an error.
However, the deviation of the edge is easily seen in the eye, strange 
The edge, by the way, is corrected at joining. Why does this happen?
Hello - generally keep the units tolerance in the .01 > .000001 range to keep Rhino happiest.
-Pascal
I agree, although 0.0001 is usually enough. 
Plus, there still it is necessary not to forget about the size of the model in the document.
Only if you zoom in.
Can you somehow see the real size?
Configure your screen using the Zoom1to1Calibrate command, then run the Zoom command > click on the 1to1 command line option. This should show you the size of the object if it were a physical one.
Something I do not understand, can you show an example of use?