Here is a plot of the cut angle you get versus the fullness of hit. For example, a fullness of 0.5 -- an exact half-ball hit -- gives a cut angle of 30 degrees. (This ignores throw which can be several degrees depending on speed, spin, cut angle and ball conditions, but that is mostly a separate issue.)
A couple of other fullnesses to note are 3/4 full which gives an angle slightly under 15 degrees and 1/4 full which produces about a 48-degree cut.
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A larger version of this is available at http://www.sfbilliards.com/fract.pdf as a PDF.
That is interesting Bob, as I do move from thick to thin at 1/2 ball, and thin to very thin at 1/4 ball.