# Roulette Cover

- Measures the shape a roulette walls off inside its own disc, the question the spirograph page asks, and prints only.
- Every curve is a wall. One flood starts at the raster's edge, the fluid poured from outside; one starts at the centre pixel, the fluid poured at the centre of the track. The shape is the rest of the disc, the walls and their pockets included, and `covered` is its share of the disc.
- The disc is the track's centre, the radius `rho + max d` no curve leaves and the radius `min abs(rho - d)` no curve enters, `rho` the centre circle's radius and `d` a seat's distance from the wheel's centre: the inner radius is the least over the seats and not the outermost seat's own, since seats on either side of `rho` each keep their own inner radius. A trace of 200001 points per curve meets both radii on all 48 cases, worst gap `3.55e-15`.
- The instrument is `mrlynum::spirograph::cover` on rasters of side 256, 512, 1024 and 2048. A raster shape carries a boundary error of the order of the polylines' length times the pixel side, so every printed digit is the Richardson limit of the ladder under that law, with a bar from the previous rung; the wall's own share of the disc is printed beside it, since the wall is counted inside the shape and is most of what the limit removes.
- The self-check is `winding`, the mean signed winding number of the disc's pixel centres, read by scanline against the polylines. Its expectation is the Green's theorem closed form `areas`, `pi b rho (rho -+ d^2/r)` summed over the distinct curves and divided by the disc's area: the two share no code. It checks the polylines and the raster against Green's theorem and never the floods, which it cannot see; the floods are guarded instead by the sample spacing of at most half a pixel, which makes the wall eight-connected and a four-connected flood unable to cross it.
- The candidates tested against the extrapolated cover are the single simple curve's form `rho (rho -+ d^2/r) / (rho + d)^2`, the full annulus `4 rho d / (rho + d)^2`, which is `4 d (a - b) b / (a - b + b d)^2` inside, and the sum of the enclosed areas over the disc's area.
- The cases are one pencil at every wheel ratio `a/b` in lowest terms with `1 <= b <= 4` and `b < a <= 9`, inside and outside; two and four quarter-turn copies of one seat at `7/3` and `5/2` inside; and the carpet's fills and corners at `7/3` and `5/2` inside and `5/8` outside, 48 in all, every one at reach `0.9` wheel radii.
- Every table runs on every case, and the report's lines are computed from the tables they summarise, never asserted.

## RUN

- `CARGO_BUILD_JOBS=4 cargo run --release -p roulette-cover`
- About two and a half seconds; prints only, writes nothing. Peak memory is the 2048 raster, about 30 MB.

## WITNESSES

- The disc: on all 48 cases the traced maximum and minimum radius meet `rho + max d` and `min abs(rho - d)` with worst gap `3.55e-15`, including the five inside ratios whose seat sits past `rho`, `3/2, 4/3, 5/3, 5/4, 7/4`, where the inner radius is `d - rho` and not `rho - d`: one pencil at `5/4` reads radius `4.600000` and hole `2.600000`.
- The winding: on all 48 cases at all four sides the scanline mean meets the closed form, worst gap `1.73e-3` at the carpet's corners outside `5/8` at side 256, worst gap at side 2048 `4.84e-4` on the same case, and no reading outside the perimeter bound. At side 2048: `0.282984` against `0.282996` for one pencil inside `3/1`, `1.678772` against `1.678770` for four quarter-turn copies inside `7/3`, `9.103089` against `9.103448` for the carpet's fills inside `7/3`.
- The simple curve: at `b = 1` and one pencil the roulette is one simple closed curve below the loop threshold on both sides, so the cover falls like the pixel, `0.014844, 0.007489, 0.003762, 0.001866` inside `3/1`, and extrapolates to at most `0.000124` in absolute value over the 16 ratios `2/1` to `9/1` inside and outside.
- The simple curve's inside: the centre flood is the whole inside, meeting the signed form `rho (rho -+ d^2/r) / (rho + d)^2` to `8.70e-5` at worst over the same 16, `0.282998` against `0.282996` inside `3/1` and `0.801268` against `0.801333` outside `3/1`, where the inside form would read `0.531445` and miss by `0.270`.
- The cover: the carpet's fills inside `7/3` cover `0.800044` on a bar of `0.000343`, the ladder falling `0.814487, 0.807142, 0.803764, 0.801904` with the wall's own share falling `0.269255` to `0.038857`; its corners cover `0.765594`, four quarter-turn copies cover `0.629299`, two cover `0.443227` and one pencil covers `0.273092`. Outside `5/8` the carpet's fills cover `0.758414` and its corners `0.867284`.
- The ladder: successive differences fall by a factor of at most `0.626` on every case and both steps, which is the pixel law the Richardson limit rests on; the largest bar over the 48 is `0.000354`.
- The hole: the centre flood holds its inscribed disc on all 48 cases with no leak, the slack running from `0.000111` at four copies inside `7/3` to `0.461499` at one pencil outside `2/1`, and to `0.142005` at one pencil inside `4/1` over the inside cases alone.
- The candidates: 0 of 144 readings survive ten bars. The annulus overshoots the carpet's fills inside `7/3` by `0.0561` and one pencil inside `2/1` by `0.99721`; the sum of enclosed areas reads `9.103448` against a cover that cannot pass `1`; the single curve's form misses every cover, reading `0.282996` inside `3/1` where the cover is the wall alone and `0.139898` against `0.273092` for one pencil inside `7/3`.
- The collapse: two half-turn copies inside `5/2` are one curve under the coincidence law and read the same cover `0.213852` as the single pencil, while inside `7/3` they are two curves and read `0.443227`.
- `mrlynum::spirograph::cover`, `disc`, `signed_area`, `representatives`, `point` and `track` are the only outside calls; the crate tests are `the_simple_curve_holds_its_whole_inside_in_the_hole`, `the_disc_reads_the_innermost_curve_and_not_the_farthest_seat` and `the_mean_winding_is_the_signed_areas_over_the_disc`.