refactor(studio): one great circle distance for the map bows and the road fetch

TravelElements ramped how far a leg bows by a local km helper and
roadRoute refused long legs by a second copy of it. Both now call legKm
from legKm.ts, the same formula with the same clamp.

client/src/components/Studio/TravelElements.tsx

@@ -8,6 +8,7 @@ import {
   type LucideIcon,
 } from 'lucide-react'
 import { fontStack } from './bookFonts'
+import { legKm } from './legKm'
 import { brightness } from './folioColour'
 import { COUNTRY_SHAPES, countryParts, countryWorldPath, projectMercator, unprojectMercator } from './countryShapes'
 import { projectOntoTiles, tileView, usableStaticUrl } from './mapTiles'
@@ -358,16 +359,6 @@ function MapView({ el, frameStyle, big = false }: {
    */
   const bowed = el.routeArc === 'bow' && trail.length === 0 && points.length > 1
 
-  /** Kilometres between two coordinates, for the ramp. */
-  const km = (a: { lat: number; lng: number }, b: { lat: number; lng: number }) => {
-    const r = (d: number) => (d * Math.PI) / 180
-    const dLat = r(b.lat - a.lat)
-    const dLng = r(b.lng - a.lng)
-    const h = Math.sin(dLat / 2) ** 2
-      + Math.cos(r(a.lat)) * Math.cos(r(b.lat)) * Math.sin(dLng / 2) ** 2
-    return 12742 * Math.asin(Math.min(1, Math.sqrt(h)))
-  }
-
   const meanLat = el.points.length
     ? el.points.reduce((sum, p) => sum + p.lat, 0) / el.points.length
     : 0
@@ -389,7 +380,7 @@ function MapView({ el, frameStyle, big = false }: {
       const d = Math.hypot(dx, dy)
       if (d < 1e-6) return { k: 0, nx: 0, ny: 0 }
 
-      const u = Math.min(1, Math.max(0, Math.log(km(a, b) / 150) / Math.log(10)))
+      const u = Math.min(1, Math.max(0, Math.log(legKm(a, b) / 150) / Math.log(10)))
       const ramp = u * u * (3 - 2 * u)
       // How much of the leg runs east-west, which is where a bow is truthful.
       const eastWest = 0.35 + 0.65 * Math.abs(dx / d)

client/src/components/Studio/legKm.test.ts

@@ -0,0 +1,24 @@
+import { describe, expect, it } from 'vitest';
+import { legKm } from './legKm';
+
+describe('legKm', () => {
+  it('is zero between a point and itself', () => {
+    expect(legKm({ lat: 48.1, lng: 11.6 }, { lat: 48.1, lng: 11.6 })).toBe(0);
+  });
+
+  it('measures a degree of latitude as about 111 km', () => {
+    expect(legKm({ lat: 0, lng: 0 }, { lat: 1, lng: 0 })).toBeCloseTo(111.19, 1);
+  });
+
+  it('is the same both ways', () => {
+    const a = { lat: 35.68, lng: 139.69 };
+    const b = { lat: 34.05, lng: -118.24 };
+    expect(legKm(a, b)).toBeCloseTo(legKm(b, a), 9);
+    expect(legKm(a, b)).toBeGreaterThan(8700);
+    expect(legKm(a, b)).toBeLessThan(8900);
+  });
+
+  it('stays defined for antipodal points', () => {
+    expect(legKm({ lat: 0, lng: 0 }, { lat: 0, lng: 180 })).toBeCloseTo((12742 * Math.PI) / 2, 3);
+  });
+});

client/src/components/Studio/legKm.ts

@@ -0,0 +1,14 @@
+/**
+ * Kilometres between two coordinates, along the great circle.
+ *
+ * Shared by the map element, which ramps how far a leg bows by it, and by the
+ * road fetch, which refuses to route a leg past its ceiling. The clamp keeps
+ * `asin` defined when rounding pushes two near antipodal points a hair past 1.
+ */
+export function legKm(a: { lat: number; lng: number }, b: { lat: number; lng: number }): number {
+  const r = (d: number) => (d * Math.PI) / 180;
+  const dLat = r(b.lat - a.lat);
+  const dLng = r(b.lng - a.lng);
+  const h = Math.sin(dLat / 2) ** 2 + Math.cos(r(a.lat)) * Math.cos(r(b.lat)) * Math.sin(dLng / 2) ** 2;
+  return 12742 * Math.asin(Math.min(1, Math.sqrt(h)));
+}

client/src/components/Studio/roadRoute.ts

@@ -1,4 +1,5 @@
 import { calculateRouteWithLegs } from '../Map/RouteCalculator'
+import { legKm } from './legKm'
 
 /**
  * The roads between a book's stops.
@@ -63,15 +64,6 @@ export interface RoadStop {
   lng: number
 }
 
-function km(a: RoadStop, b: RoadStop): number {
-  const r = (d: number) => (d * Math.PI) / 180
-  const dLat = r(b.lat - a.lat)
-  const dLng = r(b.lng - a.lng)
-  const h = Math.sin(dLat / 2) ** 2
-    + Math.cos(r(a.lat)) * Math.cos(r(b.lat)) * Math.sin(dLng / 2) ** 2
-  return 12742 * Math.asin(Math.min(1, Math.sqrt(h)))
-}
-
 /**
  * Thin a leg to what a printed line can show.
  *
@@ -107,7 +99,7 @@ export async function fetchRoads(
     const a = stops[i]
     const b = stops[i + 1]
 
-    if (km(a, b) > ROAD_CEILING_KM) {
+    if (legKm(a, b) > ROAD_CEILING_KM) {
       opts.onProgress?.(i + 1, legs)
       continue
     }