The same point, rounded two different ways, shows how rounding can affect grid-fitting at different point sizes. The first example uses the round-to-half-grid method, and the second uses the round-to-double-grid method:
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Two point sizes using rounded to the half-grid (top) and double grid (bottom).
In this example, the round-to-half-grid method always gets the control point at the crotch on the center of the pixel, but it doesn’t always get it in the middle of the stroke in-between. For a stroke that is one pixel wide, the exact middle is on the pixel center, while the center of a stroke that is two pixels wide falls between the two pixels. Conversely, round-to-double-grid always gets the control point at the crotch in the exact middle, but the pixel pattern might not be optimal.