DERIV-arc-bounding-box — The straight box of a contour of segments and fillet arcs

Used by: F04 (the straight box of a turned shape, EN-015). Sources: REF-SVG2-COORDS §8.10 (the bounding box is the tightest axis-aligned rectangle around the geometric shape, as getBBox returns it), REF-OPENSTAX-UNIT-CIRCLE (the point of the unit circle at angle t is (cos t, sin t)), REF-MATHWORLD-PERP-DOT (a⊥·b = |a| |b| sin θ).

Notation

Symbol	Definition
C, r	center and radius of a fillet arc
S, E	its start and end; s = S − C, e = E − C
σ	+1 when the arc turns in the positive direction of angles (clockwise on screen), −1 otherwise
u × v	the perp dot product uₓ v_y − u_y vₓ
d	one of the four axis directions (1, 0), (0, 1), (−1, 0), (0, −1)

Step 1 — A segment

A segment lies between its ends: its box is the box of its two ends. A contour's box is the box of its pieces' boxes, so only the arcs need more.

Step 2 — Where an arc reaches its extreme x and y

A point of the arc is C + r (cos t, sin t) (REF-OPENSTAX-UNIT-CIRCLE, scaled by r, moved to C). Along the unit circle cos t goes from 1 to −1 as t goes from 0 to π, and back from π to 2π: it only turns at multiples of π, where the point is C ± r (1, 0). Likewise sin t only turns at π/2 + kπ, the points C ± r (0, 1). So on an arc, the extreme x and y are at its ends or at the axis points C + r d the arc passes through.

Step 3 — Whether the arc passes through an axis point

A fillet sweeps the turning angle of its corner, more than 0 and less than π (DERIV-fillet-arc); an arc without length (a corner that keeps no fillet, dropped by hasLength) has s = e and is counted by its ends only. The perp dot product has the sign of sin θ, θ the angle from one vector to the other (REF-MATHWORLD-PERP-DOT). For an arc of less than π, the direction d lies on it when it is reached from s and reaches e in the arc's direction:

σ · (s × d) ≥ 0 and σ · (d × e) ≥ 0

Both angles are then in [0, π] in the arc's direction, and their sum is the sweep, so d lies between s and e. The signs are computed in the same frame as σ (y down), so no screen convention enters.

Step 4 — The box

The box of a contour is the box of: the ends of every piece, and the axis points C + r d of every arc that passes through them (step 3). It is tight: every point counted is on the drawn outline, and no point of the outline lies outside (step 2), which is the box getBBox returns (REF-SVG2-COORDS §8.10). Strokes are not in it.

Checks (test cases)

Case	Points counted	Expected box
Square 100 × 100, sharp, turned 45° about (50, 50)	the four corners	x and y from −20.71068 to 120.71068
Square 100 × 100, radius 50 (a circle), any angle	the axis points of the four arcs	x and y from 0 to 100
Square 100 × 100, radius 10, straight	arc ends only: each quarter arc ends on an axis	x and y from 0 to 100
Square 100 × 100, radius 10, turned 45°	the arcs crossing the axes	x and y from 50 − (40√2 + 10) = −16.56854 to 116.56854