What can be done about trimmed edges that are split into separate pieces?

Sometimes after trimming two surfaces against each other, the resulting edge is split into separate pieces, instead of forming a nice closed loop… This seems to cause issues later on with the FilletEdge command. See attached image. I have tried MergeEdge but this has no effect. Perhaps there is an issue with MergeEdge, or I am using it incorrectly?

All untrimmed NURBS surfaces have 4 sides (sometimes one is (or even 2 are) collapsed so that it looks like there are only 3 (or 1 in case of a sphere where 2 are collapsed) is visible). In your case the circular object is closed and 2-by-2 of its edges are joined to each other:

Now when another object splits one of those pairs - like in your example - the two edges are each split by that other object. In your picture you have selected a trim curve that is on one side of the surface. The other trim curve that you would like to be part of a nice closed loop is in fact on the complete other side of the surface (imagine unrolling that surface flat).

So in order to get what you need, you need to make sure that the natural edge of the surface is not in a position where it will be split by other objects. In the following picture you see the input curve of a revolved surface with the start-end position marked:

You could either make the start-end position of the original curve to be somewhere else and then revolve the curve again or you could use the SrfSeam command to move the seam of the revolved surface after it has been created - as done in this picture:

or put that seam totally on the other side (the inside) of that surface:

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Ah thanks! Such a simple solution. Will definitely pay more attention to seams in the future.