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>

src/core/axis.ts

@@ -0,0 +1,98 @@
+import type { Vec2 } from './geometry';
+
+/**
+ * Coping axis as a polyline IP1 → (IP4) → IP3 (spec §4.1). Each stretch is straight and has its own frame.
+ * Coordinates: k = stretch index, s = arc length from IP1 (the same s for both stretches), n = offset to the right of u.
+ * The lateral offset uses the same convention as the straight model: nrm = (u.n, −u.e).
+ */
+export interface AxisSegment {
+  start: Vec2;
+  u: Vec2;
+  nrm: Vec2;
+  /** Arc length at the start of this stretch. */
+  s0: number;
+}
+
+export interface Axis {
+  segments: AxisSegment[];
+  /** Total arc length L_IP. */
+  length: number;
+  /** Bend point IP4, or null for a straight axis. */
+  kink: Vec2 | null;
+  /** Mitre normal u1 + u2 (unit) at the kink; null for a straight axis. A point is on stretch 0 when (p − kink)·mitre ≤ 0. */
+  mitre: Vec2 | null;
+}
+
+export interface AxisPoint {
+  k: number;
+  s: number;
+  n: number;
+}
+
+const dist = (a: Vec2, b: Vec2) => Math.hypot(b.e - a.e, b.n - a.n);
+const sub = (a: Vec2, b: Vec2): Vec2 => ({ e: a.e - b.e, n: a.n - b.n });
+const dot = (a: Vec2, b: Vec2) => a.e * b.e + a.n * b.n;
+const unit = (v: Vec2): Vec2 => {
+  const len = Math.hypot(v.e, v.n);
+  if (len === 0) throw new Error('축 길이가 0입니다.');
+  return { e: v.e / len, n: v.n / len };
+};
+
+function frame(start: Vec2, end: Vec2): { u: Vec2; nrm: Vec2 } {
+  const u = unit(sub(end, start));
+  return { u, nrm: { e: u.n, n: -u.e } };
+}
+
+export function buildAxis(ip1: Vec2, ip3: Vec2, kink: Vec2 | null): Axis {
+  if (!kink) {
+    const f = frame(ip1, ip3);
+    return { segments: [{ start: ip1, ...f, s0: 0 }], length: dist(ip1, ip3), kink: null, mitre: null };
+  }
+  const a = dist(ip1, kink);
+  const f1 = frame(ip1, kink);
+  const f2 = frame(kink, ip3);
+  return {
+    segments: [
+      { start: ip1, ...f1, s0: 0 },
+      { start: kink, ...f2, s0: a },
+    ],
+    length: a + dist(kink, ip3),
+    kink,
+    mitre: unit({ e: f1.u.e + f2.u.e, n: f1.u.n + f2.u.n }),
+  };
+}
+
+/** Stretch and local coordinates of a world point. Points on the stretch-0 side of the mitre plane belong to stretch 0. */
+export function toAxis(axis: Axis, p: Vec2): AxisPoint {
+  const k = axis.mitre && axis.kink && dot(sub(p, axis.kink), axis.mitre) > 0 ? 1 : 0;
+  const seg = axis.segments[k];
+  const d = sub(p, seg.start);
+  return { k, s: seg.s0 + dot(d, seg.u), n: dot(d, seg.nrm) };
+}
+
+/** World point of local coordinates (inverse of toAxis for the same k). */
+export function fromAxis(axis: Axis, q: AxisPoint): Vec2 {
+  const seg = axis.segments[q.k];
+  const ds = q.s - seg.s0;
+  return {
+    e: seg.start.e + ds * seg.u.e + q.n * seg.nrm.e,
+    n: seg.start.n + ds * seg.u.n + q.n * seg.nrm.n,
+  };
+}
+
+/** Arc-length span of the coping of length L, centred on the axis (spec §4.2: centre at L_IP / 2). */
+export function copingSpan(axis: Axis, L: number): { sStart: number; sEnd: number } {
+  return { sStart: axis.length / 2 - L / 2, sEnd: axis.length / 2 + L / 2 };
+}
+
+/** Arc length of the kink, or null for a straight axis. */
+export function kinkArc(axis: Axis): number | null {
+  return axis.kink ? axis.segments[1].s0 : null;
+}
+
+/** Positions (arc length) closer than minDistance to the kink. Used to refuse columns at the bend. */
+export function nearKink(axis: Axis, sPositions: number[], minDistance: number): number[] {
+  const a = kinkArc(axis);
+  if (a === null) return [];
+  return sPositions.filter((s) => Math.abs(s - a) < minDistance);
+}