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I don't want to change the geometry of the faces I want to connect to. I'm sure more seasoned 3D modellers have a simple way for this. Please enlighten!

In the tutorials for joining cup handles to the cup they line up so it's not an issue. Another tutorial even extruded the start and end of the handle and then joined them in the middle. I don't think either would work here.

Here is the cross section: enter image description here

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Do you mean you want to avoid face intersection? – Leon Cheung Mar 9 '14 at 6:57
up vote 2 down vote accepted

You can use Boolean modifier to join them together. If the handle is in the same mesh of the cup body, just simply separate it as a new object by P, then boolean the two object. You will get a final shape without intersection after apply it.

enter image description here

NOTE: When using Boolean modifier, the intersection part has to be closed shape.

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I get "Cannot execute boolean operation" -- no further information about the error is presented. – jacobsd Mar 9 '14 at 9:30
I think this may have what you meant by your note... but I had to add closing faces to both objects (where they were intersecting). @Leon Thank you – jacobsd Mar 9 '14 at 10:11
@jacobsd Hi, yep, that's the most common error message when experiencing with Boolean. I think you needn't add addtional face in your case. When separating, just don't miss this face. – Leon Cheung Mar 9 '14 at 10:24
I got it. I just have to clean up the mesh. Thanks again. – jacobsd Mar 9 '14 at 10:54

Set another vertex in the middle of the lower edge of the right strip (use knifetool [k] for example) and also add maybe a ring cut [CRTL]-[r] to the other part than join the vertecies with the corresponding ones of the other part of the mesh [Alt]-[m].

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How do I create the cuts exactly where the intersection happens? In other tools they might be called "pathfinding". If that helps. I don't mind some geometry clean up -- but I don't want to modify the geometry of the receiving faces. – jacobsd Mar 9 '14 at 7:12
I would use snapping, also you can slide the place where the cut happend aftewards. May I ask what is your reason for the "not wanting to modify"? A loopcut would go cleanly through the whole mesh, not leaving much traces. – user2567875 Mar 9 '14 at 7:59
I meant I didn't want to alter the receiving faces' angles. I don't care if I end up with more faces (as I expect that to be a requirement.) I don't want to have to "eyeball" the resulting geometry. – jacobsd Mar 9 '14 at 8:05

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