feat(core): polyline coping axis with per-stretch frames (bent coping stage 1)

- buildAxis: IP1 → IP4 → IP3 as stretches with their own frames; arc length s
  from IP1; mitre normal u1 + u2 at the kink.
- toAxis / fromAxis: world ↔ (k, s, n) round trip; stretch chosen by the mitre plane.
- copingSpan (centred on L_IP/2) and nearKink (columns within the Level half-width).
- Straight P1 regression against spec §12 bearing values; bent round trip and continuity.
- Not wired into buildPier yet; the model still refuses bent coping (미실시).

Co-Authored-By: Claude Haiku 5.5 <noreply@anthropic.com>

tests/axis.test.ts

@@ -0,0 +1,99 @@
+import { describe, expect, it } from 'vitest';
+import { DIM, solveIp4 } from '../src/core/geometry';
+import { buildAxis, copingSpan, fromAxis, kinkArc, nearKink, toAxis } from '../src/core/axis';
+import type { PierInput } from '../src/core/types';
+import { loadP1 } from './helpers';
+
+const pt = (p: PierInput['ips'][number]) => ({ e: p.x, n: p.y });
+const ip = (p: PierInput, name: string) => pt(p.ips.find((q) => q.name === name)!);
+
+describe('straight axis (regression against spec §12)', () => {
+  const p1 = loadP1();
+  const axis = buildAxis(ip(p1, 'IP1'), ip(p1, 'IP3'), null);
+  const bp1 = copingSpan(axis, p1.copingLengthMm / 1000).sStart;
+
+  it('bearing s and n relative to BP1 match spec §12', () => {
+    const front = p1.bearings.filter((b) => b.row === '전열').sort((a, b) => a.no - b.no);
+    const back = p1.bearings.filter((b) => b.row === '후열').sort((a, b) => a.no - b.no);
+    const fs = front.map((b) => toAxis(axis, { e: b.x, n: b.y }));
+    const bs = back.map((b) => toAxis(axis, { e: b.x, n: b.y }));
+    expect(fs[0].s - bp1).toBeCloseTo(0.943, 3);
+    expect(fs[fs.length - 1].s - bp1).toBeCloseTo(10.965, 3);
+    expect(bs[0].s - bp1).toBeCloseTo(1.029, 3);
+    expect(bs[bs.length - 1].s - bp1).toBeCloseTo(11.05, 3);
+    expect(fs[0].n).toBeCloseTo(-0.698, 3);
+    expect(bs[0].n).toBeCloseTo(0.6, 3);
+  });
+
+  it('straight axis: every point is on stretch 0 and round-trips', () => {
+    for (const b of p1.bearings) {
+      const p = { e: b.x, n: b.y };
+      const q = toAxis(axis, p);
+      expect(q.k).toBe(0);
+      const back = fromAxis(axis, q);
+      expect(back.e).toBeCloseTo(p.e, 9);
+      expect(back.n).toBeCloseTo(p.n, 9);
+    }
+  });
+});
+
+// Synthetic bend: IP1 (0, 0), IP2 (1, 0), IP3 (6, 2), 60° right turn. The P1 sheet is straight (IP1, IP2, IP3 collinear),
+// so P1 cannot exercise a bend. The kink falls at about s = 7.15 m, inside a 6 m coping centred on the axis.
+describe('bent axis IP1 → IP4 → IP3', () => {
+  const p1 = loadP1();
+  const ip1 = { e: 0, n: 0 };
+  const ip2 = { e: 1, n: 0 };
+  const ip3 = { e: 6, n: 2 };
+  const kink = solveIp4(ip1, ip2, ip3, 60)!;
+  const axis = buildAxis(ip1, ip3, kink);
+  const a = kinkArc(axis)!;
+
+  it('kink lies on the IP1→IP2 line, at s ≈ 7.15, and the arc length adds the two stretches', () => {
+    const cross = (kink.e - ip1.e) * (ip2.n - ip1.n) - (kink.n - ip1.n) * (ip2.e - ip1.e);
+    expect(cross).toBeCloseTo(0, 9);
+    expect(a).toBeCloseTo(7.15, 1);
+    const len2 = Math.hypot(ip3.e - kink.e, ip3.n - kink.n);
+    expect(axis.length).toBeCloseTo(a + len2, 9);
+  });
+
+  it('the kink is the continuity point: s = a on both stretches at n = 0', () => {
+    const from0 = fromAxis(axis, { k: 0, s: a, n: 0 });
+    const from1 = fromAxis(axis, { k: 1, s: a, n: 0 });
+    expect(from0.e).toBeCloseTo(kink.e, 9);
+    expect(from0.n).toBeCloseTo(kink.n, 9);
+    expect(from1.e).toBeCloseTo(kink.e, 9);
+    expect(from1.n).toBeCloseTo(kink.n, 9);
+  });
+
+  it('every bearing keeps its input world coordinates through toAxis → fromAxis', () => {
+    for (const b of p1.bearings) {
+      const p = { e: b.x, n: b.y };
+      const back = fromAxis(axis, toAxis(axis, p));
+      expect(back.e).toBeCloseTo(p.e, 9);
+      expect(back.n).toBeCloseTo(p.n, 9);
+    }
+  });
+
+  it('a point just before the mitre plane is on stretch 0, just after it on stretch 1', () => {
+    const u1 = axis.segments[0].u;
+    const u2 = axis.segments[1].u;
+    const before = { e: kink.e - 0.01 * u1.e, n: kink.n - 0.01 * u1.n };
+    const after = { e: kink.e + 0.01 * u2.e, n: kink.n + 0.01 * u2.n };
+    expect(toAxis(axis, before).k).toBe(0);
+    expect(toAxis(axis, after).k).toBe(1);
+  });
+
+  it('coping centred on the axis: span is L_IP/2 ± L/2 and the kink lies inside it', () => {
+    const L = 6;
+    const span = copingSpan(axis, L);
+    expect(span.sStart).toBeCloseTo(axis.length / 2 - L / 2, 9);
+    expect(span.sStart < a && a < span.sEnd).toBe(true);
+  });
+
+  it('columns within the Level half-width of the kink are reported', () => {
+    const half = DIM.levelWidth / 2;
+    const columns = [a - 0.5, a + 5];
+    expect(nearKink(axis, columns, half)).toEqual([a - 0.5]);
+    expect(nearKink(axis, [a + 5], half)).toEqual([]);
+  });
+});