I guess you're trying to projection map onto irregular shapes.
Try working a bit backwards,
create a polygon that has about the required number of edge points, (don't worry, you can add or remove points later ay time by a double click)
then align it in Edit points mode in the Output transformation to your object.
Now go back to the Input selection, right click on the slice and select Match output shape.
This should get your output distortions flat again, and you can move the Input slice to the position you'd like.
Have a look at this link for a quick explanation why polygons are triangles: https://www.quora.com/Why-do-we-use-tri ... of-squares