Mathematical centring is correct almost always. Three cases are exceptions. They're worth knowing well, because the temptation is to invent a fourth and start nudging everything by feel, and that destroys the grid that took an hour to build.
Why does a circle look smaller than a square?
The first case is round shapes against square ones. A circle whose diameter equals a square's side has an area of π/4 = 0.785398 of the square, which is 21.46% less. Set them side by side at the same nominal size and the circle looks smaller, because the eye compares areas and not bounding boxes. To make the areas equal you'd need a circle 12.8379% wider than the square, and that overshoot looks like an error in the other direction. In practice people settle between 2% and 3%. A 2% overshoot leaves the circle 18.29% smaller in area than the square, and a 3% overshoot leaves it 16.68% smaller. The eye is satisfied long before the areas match. That tells you the correction is about how prominent the circle looks, and the geometry is only where you start.
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Why do capitals sit too low in a box?
The second case is capitals inside a box, because a font's box isn't symmetric about its own centre. Take a typical sans face with an ascender at 0.75 em above the baseline and a descender at 0.21 em below. The box is 0.96 em tall, and its centre is 0.27 em above the baseline. A line of capitals has no descenders, so its visual centre is at about half the cap height, or roughly 0.35 em above the baseline. Centre that line mathematically inside a shape and the capitals end up 0.08 em too low. At 18 point that's 1.44 points, at 24 point it's 1.92 points, and at 32 point it's 2.56 points. That last one is 0.0356 inches, and you'll see it clearly in a button or a badge.
Which other cases need an optical correction?
The third case is pointed shapes. A right-pointing triangle 0.6 inches wide has its bounding-box centre 0.3 inches from the flat edge and its centroid at 0.2 inches, because the mass is concentrated at the base. Centre a play triangle inside a circle by bounding box and it looks visibly left of centre. The correction is 0.1 inches, which is 7.2 points, or one sixth of the shape's width. The same logic applies to any asymmetric mass: an arrow, a chevron, a speech bubble with a tail.
A fourth case is a close cousin, and it's about hanging text instead of centring it. If it isn't hung outside the measure, an opening quotation mark at 24 point overhangs its text block by roughly 0.28 em, or 0.0933 inches. Picture a quote slide where the text block's left edge is meant to line up with the chart below it. That 0.0933 inches is just the kind of break the alignment score catches and the author can't see.
When is an optical correction legitimate?
The rule for all four is short. An optical correction is legitimate when you can name which case you're in and state the correction as a number. "It looked better" isn't a case, and a slide where three elements have been nudged by feel has a grid nobody can maintain.