Show each body on its page turned as it is at the map's date, under its real Sun

The body page used to spin every body about its pole at 0.08 radians a second, under a light at
(4, 3, 5) whatever the date. A body with IAU elements is now drawn as it is at the map's clock,
using the same pole, prime meridian and map convention as the system view. It is shown pole up
under a Sun held at the light's old azimuth, so the camera still opens on the day side. The Sun's
height above the equator is the real one, and so is the face it lights. The clock's rate now
turns the page too: at 1 h/s Earth's sub-solar point moved 15.17 degrees in the 1.012 h of sky
one wall second carried. What the page gives up is the stars, which do not turn with the body.

bodyPageView in src/app/shared/rendering/body-orientation.ts takes the Sun's direction from where
the body is: a planet's own mean elements, a moon's planet's place plus its own offset. It sets
the sphere's rotation and the light's direction. Exoplanets, Eris, Haumea and Makemake keep the
old slow turn and light.

Saturn's rings now lie flat in its equator, the page's horizontal. They used to lean 17 degrees,
which put them out of the plane they orbit in.

Measured:
- Live app, clock pinned to 2025-06-01 12:00 UTC: the Sun stands over 0.433 W, 22.125 N on
  Earth's page, the same point as on its sphere in the system view.
- Unit test, raycast on the page's own sphere: Earth one light-time earlier is 0.09 degrees from
  Horizons' sub-solar longitude. Its latitude, put on the flattened Earth, is within 0.03.
- The Moon's sub-solar point is within 0.004 of Horizons'.
Three mutants each fail their named test: a moon lit as if it had no planet; the Sun not held at
the page's azimuth; the body left in the ICRF instead of the page's frame.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

src/app/shared/rendering/body-orientation.spec.ts

@@ -0,0 +1,73 @@
+import * as THREE from 'three/webgpu';
+import { describe, expect, it } from 'vitest';
+
+import { BodyRecord } from '../models/body.model';
+import { bodyPageView } from './body-orientation';
+
+// Earth (the Earth-Moon barycentre's mean elements) and the Moon as bodies.json carries them.
+const EARTH: BodyRecord = {
+  id: 'earth', systemStarId: 0, name: 'Earth', kind: 'planet', radiusKm: 6371, orbitSource: 'test',
+  orbit: {semiMajorAxisAu: 1.00000018, eccentricity: 0.01673163, inclinationDeg: -0.00054346, longitudeOfAscendingNodeDeg: -5.11260389, argumentOfPeriapsisDeg: 108.04266274, meanAnomalyAtEpochDeg: -2.4631431299999917, epochJd: 2451545},
+  rates: {meanMotionDegPerDay: 0.9856091187759068, longitudeOfAscendingNodeDegPerDay: -0.000006604751813826146, argumentOfPeriapsisDegPerDay: 0.000015309819575633124, semiMajorAxisAuPerDay: -8.213552361396303e-13, eccentricityPerDay: -1.002327173169062e-9, inclinationDegPerDay: -3.6609938398357287e-7},
+  rotationalElements: {poleRaDeg: [0, -0.641, 0], poleDecDeg: [90, -0.557, 0], primeMeridianDeg: [190.147, 360.9856235, 0]}
+};
+const MOON: BodyRecord = {
+  id: 'moon', systemStarId: 0, name: 'Moon', kind: 'moon', radiusKm: 1737.4, orbitSource: 'test', parentBodyId: 'earth',
+  orbit: {semiMajorAxisAu: 0.0025695552897999907, eccentricity: 0.0554, inclinationDeg: 5.16, longitudeOfAscendingNodeDeg: 125.08, argumentOfPeriapsisDeg: 318.15, meanAnomalyAtEpochDeg: 135.27, epochJd: 2451545},
+  rates: {meanMotionDegPerDay: 13.176358, longitudeOfAscendingNodeDegPerDay: -0.052990660396105185, argumentOfPeriapsisDegPerDay: 0.164353223839846},
+  rotationalElements: {poleRaDeg: [269.9949, 0.0031, 0], poleDecDeg: [66.5392, 0.013, 0], primeMeridianDeg: [38.3213, 13.17635815, -1.4e-12], terms: [{angleDeg: [125.045, -1935.5364525], ra: -3.8787, dec: 1.5419, pm: 3.561}, {angleDeg: [250.089, -3871.072905], ra: -0.1204, dec: 0.0239, pm: 0.1208}, {angleDeg: [260.008, 475263.3328725], ra: 0.07, dec: -0.0278, pm: -0.0642}, {angleDeg: [176.625, 487269.629985], ra: -0.0172, dec: 0.0068, pm: 0.0158}, {angleDeg: [357.529, 35999.0509575], ra: 0, dec: 0, pm: 0.0252}]}
+};
+const BODIES = [EARTH, MOON];
+const JUNE_1_2025_NOON_UTC = 2460828.0;
+/** The page's light, at (4, 3, 5): 38.7 degrees round from the camera's side. */
+const SUN_AZIMUTH = Math.atan2(4, 5);
+
+/** The point of the page's sphere, as east longitude and latitude on its map, that faces the Sun. */
+function subSolarPoint(body: BodyRecord, jdUtc: number): { eastDeg: number; latDeg: number } {
+  const sphere = new THREE.Mesh(new THREE.SphereGeometry(1, 64, 32));
+  const sun = new THREE.Vector3();
+  expect(bodyPageView(body, BODIES, jdUtc, SUN_AZIMUTH, sphere.quaternion, sun)).toBe(true);
+  sphere.updateMatrixWorld();
+  const hit = new THREE.Raycaster(sun.clone().multiplyScalar(4), sun.clone().negate()).intersectObject(sphere)[0];
+  return { eastDeg: (hit.uv!.x - 0.5) * 360, latDeg: (hit.uv!.y - 0.5) * 180 };
+}
+
+describe('bodyPageView', () => {
+  it('lights the same face of Earth on its page: within 4 degrees of Greenwich at noon UTC, and where Horizons has it', () => {
+    expect(Math.abs(subSolarPoint(EARTH, JUNE_1_2025_NOON_UTC).eastDeg)).toBeLessThan(4);
+    // Horizons' sub-solar point from the Sun, 1.5795 E and 22.2604 N, is Earth as it was 8.43
+    // minutes before, when the light arriving then left the Sun; its latitude is geodetic, on the
+    // flattened Earth, where the sphere's is geocentric: 0.14 degrees apart at this latitude.
+    const horizons = subSolarPoint(EARTH, JUNE_1_2025_NOON_UTC - 8.43351424 / 1440);
+    const geodetic = (Math.atan(Math.tan((horizons.latDeg * Math.PI) / 180) / (1 - 1 / 298.257) ** 2) * 180) / Math.PI;
+    expect(Math.abs(horizons.eastDeg - 1.579501)).toBeLessThan(0.1);
+    expect(Math.abs(geodetic - 22.260426)).toBeLessThan(0.05);
+  });
+
+  it('takes a moon’s Sun from where it and its planet are: the Moon’s sub-solar point is Horizons’', () => {
+    const moon = subSolarPoint(MOON, JUNE_1_2025_NOON_UTC);
+    // 116.2859 E and 1.5030 N, seen from Earth's centre.
+    expect(Math.abs(moon.eastDeg - 116.285934)).toBeLessThan(0.1);
+    expect(Math.abs(moon.latDeg - 1.503004)).toBeLessThan(0.05);
+  });
+
+  it('keeps the pole up and the Sun where the page’s light stands, turning the body under it', () => {
+    const planet = new THREE.Quaternion();
+    const sun = new THREE.Vector3();
+    for (const hours of [0, 6, 12]) {
+      bodyPageView(EARTH, BODIES, JUNE_1_2025_NOON_UTC + hours / 24, SUN_AZIMUTH, planet, sun);
+      expect(new THREE.Vector3(0, 1, 0).applyQuaternion(planet).angleTo(new THREE.Vector3(0, 1, 0))).toBeLessThan(1e-9);
+      expect(Math.atan2(sun.x, sun.z)).toBeCloseTo(SUN_AZIMUTH, 9);
+    }
+  });
+
+  it('leaves a body with no elements to the page, as it was', () => {
+    const planet = new THREE.Quaternion(0.1, 0.2, 0.3, 0.9).normalize();
+    const before = planet.clone();
+    const sun = new THREE.Vector3(4, 3, 5);
+
+    expect(bodyPageView({ ...EARTH, rotationalElements: undefined }, BODIES, JUNE_1_2025_NOON_UTC, SUN_AZIMUTH, planet, sun)).toBe(false);
+    expect(planet.equals(before)).toBe(true);
+    expect(sun.toArray()).toEqual([4, 3, 5]);
+  });
+});