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Ok, so lets say I have a mesh like the one on the left, and I want to make a hole in it like the one on the right, as illustrated here:

enter image description here

but, using the methods I can think of, you end up with inner vertices that are out of alignment with the outer edge, thus producing a non-flat surface as you can kind of see here:

enter image description here

also, as I just eyeballed it, the shape and curves of the hole are not totally symmetrical at each corner, so, what is the best way to cut a hole like this, with symmetrical measurements, in a surface that is at an angle, and keeping everything in perfect alignment with the surface itself without creating any n-gons and having clean topology?

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    $\begingroup$ Hit i in your keyboard to use the inset tool $\endgroup$ – Sebastián Mestre May 14 '17 at 15:23
  • $\begingroup$ Knife Project was made for cases like that. I suggest you read up on that $\endgroup$ – J. Brosch May 17 '17 at 12:44
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Maybe you should use boolean modyfier.

  1. Create an object that has the shape of the hole you want to create. enter image description here
  2. Add boolean modyfier to the object you want to cut a hole into.
  3. Make sure that the operation type was set to difference. Choose object that you want to use as holecutter.

enter image description here

  1. Slide the holecutter into the place where you want hole to be. enter image description here

  2. Apply the modyfier and remove the holecutter. enter image description here Hope you understood my bad english, but if this post helped then let me know by votng this answer up. Cheers! :)

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