First of all, you snip the selvedge at the point where you need to ..
Then, you pull the two sides apart very gently until you have a prominent thread from the weft which runs perpendicular to the selvedge edge.
Pull this thread until it gathers the fabric into a pucker - like when you do a gathering stitch - continue until the whole width of fabric is puckered, like this ... then, pull the fabric straight again.
You'll be left with a visible line in the fabric where your thread has broken the weave of the fabric. If you hold it up to the light, you can see that the thread is missing.
|held up to the light the line becomes much clearer|
Then all you do is cut along the line and voila! You have fabric cut on the grain, with no waste. All nice and tidy.
I haven't tried this with cotton so can't tell you if it works or not, and you do get different effects with different silks. The above example is a standard dupion, and below is some heavy spotty silk.. It also works an absolute treat with fine floaty silks which are nigh impossible to cut straight by eye!