diff --git a/src/core/axis.ts b/src/core/axis.ts
new file mode 100644
index 0000000..b9b83f4
--- /dev/null
+++ b/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);
+}
diff --git a/tests/axis.test.ts b/tests/axis.test.ts
new file mode 100644
index 0000000..f84fa76
--- /dev/null
+++ b/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([]);
+  });
+});