An upright label clears a line by its support width and a vertex by its nearest corner, not by its centre's distance

Fact

For an axis-aligned label box w×h centred at c, the gap to a line through the origin with unit normal n is |n·c| − ((w/2)|nₓ| + (h/2)|n_y|). The subtracted term — the box's support along the normal — changes with the line's angle. The gap to a point, such as a vertex with an arc drawn around it, is the distance to the box's nearest corner or edge, up to √(w²+h²)/2 less than |c| on a diagonal.

Why it matters

A label put "d from the vertex along the bisector" with d judged from its centre looks right when the bisector is horizontal or vertical and overlaps on the diagonals, so hand-checked layouts pass and random geometry does not. The needed distance also grows as 1/sin(half-angle): a narrow angle pushes its label past the arm ends and out of a figure that was fitted without it.

How to apply

• On a bisector of half-angle α, place the centre at d = max over both sides of (support(n_side) + gap) / sin α.
• Then step d outward until the nearest-point distance to the vertex clears the arc radius plus a gap.
• Size the box from the tallest glyph actually rendered (a large "?" has a taller line box than digits); measure it with getBBox once instead of guessing.
• Include the label's box when fitting the figure to its viewport.
• Assert it over thousands of random figures: segment-vs-box (Liang–Barsky) for every line, nearest-point distance for arcs, box overlap between labels.

Related: [[svg-paint-order-hides-content]].