Phase 2: 3D live map driven by API WebSocket

- apps/live-map/src/hooks/useLiveState.ts: WebSocket subscription
  with exponential-backoff reconnect, polling fallback, status tracking
- apps/live-map/src/scene/Scene.tsx: refactored Three.js scene with
  per-frame orbit propagation, vessel marker color-coding by owner,
  orbit-line visibility tied to focus filters, smooth camera follow
  on selected vessel/body
- apps/live-map/src/scene/layout.ts: bodyPositionAt / vesselPositionAt
  helpers (heliocentric frame, walk up the parent chain), logScale
  helpers for the system view
- apps/live-map/src/scene/color.ts: per-body and per-owner color maps
- apps/live-map/src/panels/TimeControls.tsx: play/pause/reverse/reset
  buttons, ×1/×10/×100/×1k/×10k/×100k speeds, UT scrub slider,
  live-edge indicator (LIVE / Nh behind / Nh ahead)
- apps/live-map/src/panels/VesselList.tsx: vessel sidebar with click-
  to-track; color-coded by owner (KASA=blue, SPES=orange)
- apps/live-map/src/panels/FocusPanel.tsx: planet/moon/vessel orbit
  visibility toggles
- apps/live-map/src/panels/StatusPill.tsx: WS status (LIVE/POLLING/
  OFFLINE/STALE), body + vessel + message counts
- tests/scene.test.ts: 10 tests for layout helpers (periodicity,
  vessel-centered positioning, logScale round-trips)

End-to-end verified: mock publisher → API → live-map WebSocket →
scene re-renders with the new vessel positions and orbits.

apps/live-map/tests/scene.test.ts

@@ -0,0 +1,166 @@
+import { describe, it, expect } from 'vitest';
+import {
+  bodyPositionAt,
+  vesselPositionAt,
+  findBodyMu,
+  logScale,
+  inverseLogScale,
+} from '../src/scene/layout.js';
+import type { CelestialBody, Vessel } from '@kerbal-rt/shared-types';
+
+const kerbol: CelestialBody = {
+  id: 'kerbol',
+  name: 'Kerbol',
+  kind: 'star',
+  parentId: null,
+  radius: 261_600_000,
+  sphereOfInfluence: 1e30,
+  gravitationalParameter: 1.172e18,
+  rotationPeriod: 432_000,
+  axialTilt: 0,
+  orbit: {
+    semiMajorAxis: 0,
+    eccentricity: 0,
+    inclination: 0,
+    longitudeOfAscendingNode: 0,
+    argumentOfPeriapsis: 0,
+    meanAnomalyAtEpoch: 0,
+    epoch: 0,
+  },
+};
+
+const kerbin: CelestialBody = {
+  id: 'kerbin',
+  name: 'Kerbin',
+  kind: 'planet',
+  parentId: 'kerbol',
+  radius: 600_000,
+  sphereOfInfluence: 84_159_286,
+  gravitationalParameter: 3.5316e12,
+  rotationPeriod: 21_600,
+  axialTilt: 0,
+  orbit: {
+    semiMajorAxis: 13_599_840_256,
+    eccentricity: 0, // circular for predictable test math
+    inclination: 0,
+    longitudeOfAscendingNode: 0,
+    argumentOfPeriapsis: 0,
+    meanAnomalyAtEpoch: 0,
+    epoch: 0,
+  },
+};
+
+const mun: CelestialBody = {
+  id: 'mun',
+  name: 'Mun',
+  kind: 'moon',
+  parentId: 'kerbin',
+  radius: 200_000,
+  sphereOfInfluence: 2_429_559,
+  gravitationalParameter: 6.514e10,
+  rotationPeriod: 138_984,
+  axialTilt: 0,
+  orbit: {
+    semiMajorAxis: 12_000_000,
+    eccentricity: 0,
+    inclination: 0,
+    longitudeOfAscendingNode: 0,
+    argumentOfPeriapsis: 0,
+    meanAnomalyAtEpoch: 0,
+    epoch: 0,
+  },
+};
+
+const bodies: CelestialBody[] = [kerbol, kerbin, mun];
+
+describe('bodyPositionAt', () => {
+  it('returns origin for the star', () => {
+    const p = bodyPositionAt(bodies, 'kerbol', 100);
+    expect(p.x).toBe(0);
+    expect(p.y).toBe(0);
+    expect(p.z).toBe(0);
+  });
+
+  it('returns a non-zero position for a planet at t > 0', () => {
+    const p = bodyPositionAt(bodies, 'kerbin', 1_000_000);
+    const dist = Math.hypot(p.x, p.y, p.z);
+    // SMA is 13.6 billion meters, so position is on that order
+    expect(dist).toBeGreaterThan(1e10);
+    expect(dist).toBeLessThan(2e10);
+  });
+
+  it('returns origin for a body not in the catalog', () => {
+    const p = bodyPositionAt(bodies, 'unknown', 1);
+    expect(p).toEqual({ x: 0, y: 0, z: 0 });
+  });
+
+  it('is periodic (returns near the same position after one full orbit)', () => {
+    // Kerbin orbits Kerbol, so the period is computed with kerbol's μ
+    const p1 = bodyPositionAt(bodies, 'kerbin', 0);
+    const period =
+      (2 * Math.PI) / Math.sqrt(kerbol.gravitationalParameter / Math.pow(13_599_840_256, 3));
+    const p2 = bodyPositionAt(bodies, 'kerbin', period);
+    expect(p1.x).toBeCloseTo(p2.x, -2);
+    expect(p1.y).toBeCloseTo(p2.y, -2);
+  });
+});
+
+describe('vesselPositionAt', () => {
+  const vessel: Vessel = {
+    id: 'v1',
+    name: 'Test',
+    type: 'Probe',
+    owner: 'KASA',
+    situation: 'ORBITING',
+    status: 'ACTIVE',
+    orbit: {
+      semiMajorAxis: 7_000_000,
+      eccentricity: 0,
+      inclination: 0,
+      longitudeOfAscendingNode: 0,
+      argumentOfPeriapsis: 0,
+      meanAnomalyAtEpoch: 0,
+      epoch: 0,
+    },
+    referenceBodyId: 'kerbin',
+    createdAt: '2026-01-01T00:00:00Z',
+    retiredAt: null,
+  };
+
+  it('returns the kerbin-centered position when reference body is kerbin', () => {
+    // At t=0, vessel is at (7e6, 0, 0) in kerbin frame
+    const p = vesselPositionAt(bodies, vessel, 0);
+    // kerbin is at (sma, 0, 0) = (13.6e9, 0, 0)
+    // so vessel should be at roughly (13.6e9 + 7e6, 0, 0)
+    expect(p.x).toBeCloseTo(13_599_840_256 + 7_000_000, -2);
+    expect(p.y).toBeCloseTo(0, 2);
+  });
+});
+
+describe('findBodyMu', () => {
+  it("returns the body's own gravitational parameter", () => {
+    expect(findBodyMu(bodies, 'kerbin')).toBe(kerbin.gravitationalParameter);
+  });
+  it('returns 0 for null id', () => {
+    expect(findBodyMu(bodies, null)).toBe(0);
+  });
+  it('returns 0 for unknown id', () => {
+    expect(findBodyMu(bodies, 'nope')).toBe(0);
+  });
+});
+
+describe('logScale', () => {
+  it('is the inverse of inverseLogScale', () => {
+    for (const t of [0, 4, 8, 12]) {
+      const d = logScale(t);
+      expect(inverseLogScale(d)).toBeCloseTo(t, 6);
+    }
+  });
+  it('produces reasonable distances for camera placement', () => {
+    // t=4 → ~5.5e9 m (good for showing Kerbin at 13.6e9)
+    expect(logScale(4)).toBeGreaterThan(1e9);
+    expect(logScale(4)).toBeLessThan(1e10);
+    // t=10 → ~22e12 m (good for showing all planets)
+    expect(logScale(10)).toBeGreaterThan(1e12);
+  });
+});