DERIV-round-shape — A circle or a capsule as a rectangle rounded by half its smaller side

Used by: F10 as built in E01, then moved to E14 (research note 0010); the capsule stays a rectangle rounded by half its smaller side. Sources: DERIV-local-radius-clamp (setback and factor per edge), DERIV-fillet-setback (s = r · tan(|τ|/2)), REF-MATHWORLD-STADIUM (a rectangle capped by two half circles), REF-SVG2-SHAPES §10.2 (as a contrast: SVG caps a rect's rx at half its width and ry at half its height separately, which gives elliptical corners, not a capsule).

Notation

Symbol	Definition
w, h	width and height of the round shape's box, integers ≥ 0
m	the smaller side, min(w, h)
r	the radius given to each of the four corners

Step 1 — Every corner rounded by half the smaller side

The box is a rectangle: four right angles, |τ| = π/2, so each setback is s = r · tan(π/4) = r (DERIV-fillet-setback). With r = m/2:

• on a side of length m, the two setbacks add up to m: the factor of that side is 1 (DERIV-local-radius-clamp step 2), the two arcs meet in its middle and the side is a half circle;
• on a side of length M = max(w, h) ≥ m, they add up to m ≤ M: the factor is 1 too, and a straight part of length M − m remains.

No radius is reduced: the drawing is exactly the rectangle capped by two half circles of radius m/2, a stadium (REF-MATHWORLD-STADIUM); when w = h the straight parts have no length and the four quarter arcs make the circle of diameter w (as F01: a square of side c with radius c/2). In floating point tan(π/4) is 0.9999999999999999: the arcs stop about 1e-16 · m short of the middle (1.4e-14 for m = 100), a piece without length dropped from the contour while it stays below EPSILON (hasLength), that is for sides up to a few million units, as for any rectangle at its maximal radius (Q20).

Step 2 — Converted to a free shape

A free shape stores integer radii (ADR-0029, ADR-0033). The conversion is the rectangle tool's free shape for the box (the unit square stretched to w × h, ADR-0032, valid at a size of 0), each of its 4 corners getting ⌈m/2⌉. For an even m it is m/2, step 1. For an odd m it is (m + 1)/2: on a side of length m the setbacks add up to m + 1 > m, the factor is m/(m + 1), and the effective radius is (m + 1)/2 · m/(m + 1) = m/2 (DERIV-local-radius-clamp step 3): the same drawing. On the longer side, m + 1 ≤ M unless w = h, where both sides give the same factor.

Checks (test cases)

Case	Computation	Expected
100 × 100	r = 50	the circle of diameter 100: four quarter arcs of radius 50
120 × 80	r = 40; top side 120: straight part 40	a capsule: half circles of radius 40, straight sides from x 40 to 80
80 × 120	r = 40; left side 120: straight part 40	a standing capsule
81 × 81 converted	radii 41; factor 81/82	effective radius 40.5: the same circle
120 × 80 converted	radii 40	the same capsule
0 × 50	r = 0	a vertical line, as a rectangle of width 0 (Q15)
0 × 50 converted	the unit square stretched to 0 × 50, radii 0	a valid free shape, the same line