133 lines
6 KiB
Markdown
133 lines
6 KiB
Markdown
# Restricted-area box: rectangle in kilometres, not in degrees
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**Goal:** the no-fly area filed with the regulator must be a usable rectangle at
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every latitude, the poles included.
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## Problem
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The area was a rectangle in latitude/longitude, padded by a margin in kilometres.
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It cannot be made to work near a pole, and the failure is geometric, not a bug:
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> Every meridian passes through the pole, so any lat/lon rectangle that contains
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> a pole spans all 360 degrees of longitude.
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Measured on a real launch from 89.99 N, 0 E (GFS 2026-08-03 06Z, 5 km margin):
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| Form | Area |
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| --- | --- |
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| lat/lon rectangle (south 88.93, north 90, west -180, east 180) | **44 200 km²** |
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| corridor actually flown, 13.7 × 123.4 km | **1 695 km²** |
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A 26× over-claim. The balloon also sweeps 79 degrees of longitude while
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descending one degree of latitude, because near a pole a short displacement
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crosses many meridians — so even without the pole clamp the degree form is wide.
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There is no latitude threshold below which the degree form is safe. Above it the
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box contains the pole and balloons; below it the box fails to contain the
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trajectory. The form itself is what fails.
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Rendering made this visible: at the pole the box drew as a circle following a
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parallel. That was a correct picture of a wrong shape.
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## Decision
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Define the area as a **rectangle in kilometres**, axis-aligned to east/north at
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its own centre, filed as four lat/lon corners joined by great circles.
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Rejected alternatives:
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- **Rectangle in degrees plus a circle near the pole.** A circle (centre +
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radius) is the standard ICAO NOTAM form, but this needs two filing forms and a
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latitude threshold to switch between them — the crutch this replaces.
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- **Circle everywhere.** One form, always filable, but a circle around a long
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corridor claims far more area than a rectangle.
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- **Rotate the rectangle to the track's heading.** Tighter for a diagonal track,
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up to 2×, but it puts an azimuth in the filing and the corners stop reading as
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north/south/east/west. Not taken; revisit only if area pressure appears.
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## Design
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```ts
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interface BoundingBox {
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corners: [LatLngTuple, LatLngTuple, LatLngTuple, LatLngTuple]; // NW, NE, SE, SW
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centre: LatLngTuple;
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widthKm: number;
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heightKm: number;
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}
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```
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`computeBoundingBox(path, marginKm)` keeps its signature.
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1. **Local frame.** Earth-centred radial/east/north unit vectors at a point. All
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three are unit length and orthogonal at every latitude including the poles, so
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nothing divides by `cos(latitude)`. Same construction as the predictor's
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integrator, `internal/numerics/spherical.go`, for the same reason.
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2. **Project** each path point to kilometres east/north by azimuthal
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equidistant: exact in distance from the origin at any range.
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3. **Two passes.** The first frame, on the track's mean direction, only locates
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the box centre; the second frames on that centre, which makes the corners
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symmetric about it and keeps projection error smallest where the corners are.
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4. **Pad** the half-extents by the margin and unproject the four corners.
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### Margin is a floor
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A great-circle edge bows **away** from the frame origin relative to its chord in
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this projection: `y_mid = hy · (1 + ρ² sin²β / 3)`, positive. Check by
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inspection — the equator in an azimuthal-equidistant projection centred on the
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pole is a circle at `R·π/2`, while the chord between two of its points 90° apart
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would sag to `0.707·R·π/2`. So the filed quad **contains** the projected
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rectangle and the requested clearance is never eaten. No correction is applied.
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(The first draft of this design asserted the opposite sign and specified a
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sagitta correction. It would have inflated the area for no reason.)
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### The box may cross a pole, and must
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A launch 1.1 km from the pole with a 5 km margin needs coverage 3.9 km past the
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pole. The north edge therefore passes over it and comes down the far side, which
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puts the two north corners ~180° apart in longitude — for the measured case,
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-43.87 and -158.44. Correct, and unreadable from the corners alone, so the panel
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also reports `width × height @ centre`.
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### Drawing
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`boundingBoxRing` samples 16 points per edge along the great circle (slerp
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between corner vectors). The drawn shape is then the filed shape and does not
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depend on the renderer's interpolation mode, and no single segment is long enough
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to land degenerate on the antimeridian — which is what previously stopped
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Cesium's render loop with "All attribute lists must have the same number of
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attributes" in its `splitLongitude` pass.
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## Removed
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`KM_PER_DEG_LAT` and the `cos(latitude)` margin division, the latitude clamps,
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the `circumpolar` branch, `LineOptions.arc`, and the `ArcType.RHUMB` branch in
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`cesium-scene.ts` — no parallels remain in the box, so rhumb lines have no
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remaining caller. Net less code, and no branch on latitude anywhere.
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## Verification
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`tests/unit/boundingBox.spec.ts` runs in Node against the pure module:
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- every trajectory point clears all four filed edges by the full margin, measured
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as distance to the edge's great circle — at 52.2 N over 500 km (where a
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wrong-signed bow would have been 5.1 km, the whole margin) and on the real
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polar track;
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- the polar box is under 3 000 km² where the degree form gave 44 200;
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- a meridional track of the same length in kilometres gives the same box at
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52.2 N and at 89.99 N — the test that fails if any latitude branch returns;
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- the drawn ring closes and no segment spans 90° of longitude.
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`tests/e2e/bbox.spec.ts` covers what only a browser answers: `scene.renderError`
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stays empty while a polar box is drawn, and the panel reports a corridor-sized
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area rather than a cap.
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Measured after the change: polar 13.3 × 123.4 km @ 89.4835, 78.3014; mid-latitude
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61.4 × 68.9 km @ 52.4654, 0.4651, with the northernmost track point clearing the
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north edge by 4.98 km against 5.0 asked.
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## Not addressed
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- **WGS84.** Everything here uses a spherical earth, R = 6371 km, matching the
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predictor. The ellipsoid remains deferred there too.
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- **Altitude.** The box is the lat/lon footprint; it does not encode a ceiling.
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