DERIV-free-shape-stretch — Stretching a free shape by its width and height, without loss

Used by: ADR-0032, F09; ADR-0033, F08 (fractional points and size). Sources: REF-SVG2-COORDS §8.2 (the viewBox transform with align = none: one scale per axis, then a translation), REF-CSS-TRANSFORMS §3 (a transform about an origin). fitInBox (DERIV-regular-polygon-fit) shares the scale, with one factor for both axes, but not the translation: it places the box by its anchor, while this map keeps the top-left fixed.

Notation

Symbol	Definition
p₀ … pₙ₋₁	the points of the free shape, as stored, in contour order (Q11)
o = (oₓ, o_y)	top-left of their bounding box: the smallest x and the smallest y
nₓ, n_y	width and height of that box: largest minus smallest coordinate on each axis
w, h	width and height asked for the shape, integers ≥ 0
p'ᵢ	the point drawn

Step 1 — A scale about a fixed point

SVG maps a box onto another with one scale per axis when align is none: scale-x = target width / box width, scale-y likewise, then a translation (REF-SVG2-COORDS §8.2). CSS builds a transform about an origin by translating by the origin, applying the transform, and translating back (REF-CSS-TRANSFORMS §3). With the origin o:

p' = o + (sx · (pₓ − oₓ), sy · (p_y − o_y))

Step 2 — The factors that give the asked size, the top-left fixed

sx = w / nₓ, sy = h / n_y when nₓ, n_y > 0. Computed as p'ₓ = oₓ + w · (pₓ − oₓ) / nₓ, multiplying first: with integers the product is exact below 2⁵³, and one correctly rounded division gives w exactly at the far edge ((29 / 7) · 7 is 29.000000000000004, (29 · 7) / 7 is 29). Likewise for y. The point o maps to itself, so the top-left of the box stays fixed (Product Owner, F09).

The drawn box has width sx · nₓ = w and height sy · n_y = h: the point at the smallest x maps to oₓ, the one at the largest to oₓ + sx · nₓ = oₓ + w, and a scale with sx ≥ 0 keeps the order of the coordinates. Likewise for y. At its own size (w = nₓ) the formula gives back each point mathematically, but not always bit for bit in floating point (o + w·(p − o)/w may differ from p by an ulp): the code therefore does not compute that axis and draws it as stored, bit for bit, which keeps a converted polygon exactly where it was (ADR-0033); otherwise, with fractional inputs, a drawn point may differ from the exact one by an ulp (about 1e-16 of the size), far below the 5 decimals written (Q10). The far edge is exact while w · nₓ stays below 2⁵³; beyond (sizes near the largest safe integer, Q20), it can be off by a unit — accepted, as for the fit of F02.

Degenerate box: when nₓ = 0 (every point on one vertical line), no factor gives a width (w / 0 is infinite). Such a shape is refused: the stored points must span a box of positive width and height (ADR-0032). The rectangle tool never gives one (step 4).

Step 3 — Without loss

The stored points never change: the drawn points are a function of the stored points and of w, h only. So w = 0 draws every point at oₓ, and w back to its first value draws exactly the first points: nothing was lost (Product Owner, F09: stretching must never be destructive).

Step 4 — The rectangle tool

The rectangle tool stores the unit square (0, 0), (1, 0), (1, 1), (0, 1), top-left first, clockwise (Q11): o = (0, 0), nₓ = n_y = 1, so sx = w, sy = h, and the points drawn are (0, 0), (w, 0), (w, h), (0, h) — the rectangle of F01. The products 1 · w, 0 · w are exact in floating point: the drawing is the same as before, to the last bit.

Checks (test cases)

Case	Computation	Expected points drawn
Rectangle tool, 100 × 50	sx = 100, sy = 50	(0, 0), (100, 0), (100, 50), (0, 50)
Rectangle tool, 0 × 50, then 100 × 50	sx = 0, then 100	all x at 0, then the line above
Points (1, 0), (2, 2), (0, 2) to 10 × 4	o = (0, 0), n = 2 × 2, sx = 5, sy = 2	(5, 0), (10, 4), (0, 4)
Points (3, 3), (5, 3), (5, 7) to 4 × 8	o = (3, 3), n = 2 × 4, sx = 2, sy = 2	(3, 3), (7, 3), (7, 11): top-left kept
Points (0, 0), (7, 0), (7, 1) to 29 × 1	29 · 7 / 7, not 29 / 7 · 7	(0, 0), (29, 0), (29, 1): the far edge exact
Points (0, 0), (3, 0), (3, 2), (0, 2) to 100 × 50	sx = 100 / 3, sy = 25	(0, 0), (100, 0), (100, 50), (0, 50): a rectangle stays one