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import type {
PolygonConstraint,
VertexKind,
SimplePolygonSolveOptions,
PolygonData,
SolverError,
SegmentRef,
} from "./solver-types";
type Pt = { x: number; y: number };
function err(
kind: SolverError["kind"],
message: string,
details?: Record<string, unknown>
): SolverError {
return { kind, message, details };
}
// ---------- basic geometry ----------
function dist(a: Pt, b: Pt): number {
return Math.hypot(a.x - b.x, a.y - b.y);
}
function sub(a: Pt, b: Pt): Pt {
return { x: a.x - b.x, y: a.y - b.y };
}
function dot(a: Pt, b: Pt): number {
return a.x * b.x + a.y * b.y;
}
function cross(a: Pt, b: Pt): number {
return a.x * b.y - a.y * b.x;
}
function signedArea(poly: Pt[]): number {
let a = 0;
for (let i = 0; i < poly.length; i++) {
const j = (i + 1) % poly.length;
a += poly[i].x * poly[j].y - poly[j].x * poly[i].y;
}
return 0.5 * a;
}
// Signed turn angle in (-π, π], using prev->cur and cur->next
function signedTurn(prev: Pt, cur: Pt, next: Pt): number {
const e1 = sub(cur, prev);
const e2 = sub(next, cur);
return Math.atan2(cross(e1, e2), dot(e1, e2));
}
// Interior angle in (0, 2π): interior = π - turn (for CCW simple polygons)
function interiorAngle(prev: Pt, cur: Pt, next: Pt): number {
const turn = signedTurn(prev, cur, next);
const ang = Math.PI - turn;
const twoPi = 2 * Math.PI;
return ((ang % twoPi) + twoPi) % twoPi;
}
function pointSegmentDistance(p: Pt, a: Pt, b: Pt): number {
const ab = sub(b, a);
const ap = sub(p, a);
const ab2 = dot(ab, ab);
if (ab2 === 0) return dist(p, a);
let t = dot(ap, ab) / ab2;
t = Math.max(0, Math.min(1, t));
const proj = { x: a.x + t * ab.x, y: a.y + t * ab.y };
return dist(p, proj);
}
function segmentsProperlyIntersect(a: Pt, b: Pt, c: Pt, d: Pt): boolean {
const ab = sub(b, a);
const ac = sub(c, a);
const ad = sub(d, a);
const cd = sub(d, c);
const ca = sub(a, c);
const cb = sub(b, c);
const o1 = Math.sign(cross(ab, ac));
const o2 = Math.sign(cross(ab, ad));
const o3 = Math.sign(cross(cd, ca));
const o4 = Math.sign(cross(cd, cb));
return (
o1 !== 0 &&
o2 !== 0 &&
o3 !== 0 &&
o4 !== 0 &&
o1 !== o2 &&
o3 !== o4
);
}
function segmentSegmentDistance(a: Pt, b: Pt, c: Pt, d: Pt): number {
if (segmentsProperlyIntersect(a, b, c, d)) return 0;
return Math.min(
pointSegmentDistance(a, c, d),
pointSegmentDistance(b, c, d),
pointSegmentDistance(c, a, b),
pointSegmentDistance(d, a, b)
);
}
// ---------- segment helpers (edges + diagonals) ----------
function normalizeSegment(seg: SegmentRef): SegmentRef {
return seg.i < seg.j ? seg : { i: seg.j, j: seg.i };
}
function segmentKey(seg: SegmentRef): string {
const s = normalizeSegment(seg);
return `${s.i}-${s.j}`;
}
function segmentVec(pts: Pt[], seg: SegmentRef): Pt {
return { x: pts[seg.j].x - pts[seg.i].x, y: pts[seg.j].y - pts[seg.i].y };
}
function vecLen(v: Pt): number {
return Math.hypot(v.x, v.y);
}
// Undirected angle between vectors in [0, π]
function undirectedAngleBetween(u: Pt, v: Pt): number {
return Math.atan2(Math.abs(cross(u, v)), dot(u, v));
}
// ---------- variable packing/unpacking (gauge fixing) ----------
//
// Fix:
// p0 = (0,0)
// p1.y = 0
//
// Variables x length = 2N-3:
// x[0] = p1.x
// then for i=2..N-1: (xi, yi)
function unpackVars(N: number, x: number[]): Pt[] {
const pts: Pt[] = new Array(N);
pts[0] = { x: 0, y: 0 };
pts[1] = { x: x[0], y: 0 };
let k = 1;
for (let i = 2; i < N; i++) {
pts[i] = { x: x[k], y: x[k + 1] };
k += 2;
}
return pts;
}
function packVars(pts: Pt[]): number[] {
const N = pts.length;
const x: number[] = new Array(2 * N - 3);
x[0] = pts[1].x;
let k = 1;
for (let i = 2; i < N; i++) {
x[k] = pts[i].x;
x[k + 1] = pts[i].y;
k += 2;
}
return x;
}
function gaugeFix(pts: Pt[]): Pt[] {
const N = pts.length;
// translate so p0 -> origin
const t = { x: pts[0].x, y: pts[0].y };
const p = pts.map((q) => ({ x: q.x - t.x, y: q.y - t.y }));
// rotate so p1 lies on +x axis
const p1 = p[1];
const ang = Math.atan2(p1.y, p1.x);
const c = Math.cos(-ang);
const s = Math.sin(-ang);
const r = p.map((q) => ({ x: c * q.x - s * q.y, y: s * q.x + c * q.y }));
// ensure p1.x positive
if (r[1].x < 0) {
for (let i = 0; i < N; i++) r[i] = { x: -r[i].x, y: -r[i].y };
}
r[1].y = 0;
r[0] = { x: 0, y: 0 };
return r;
}
function regularNgonInit(N: number): Pt[] {
const R = 1;
const pts: Pt[] = [];
for (let i = 0; i < N; i++) {
const t = (2 * Math.PI * i) / N;
pts.push({ x: R * Math.cos(t), y: R * Math.sin(t) });
}
return gaugeFix(pts);
}
function mulberry32(seed: number): () => number {
let a = seed >>> 0;
return function () {
a |= 0;
a = (a + 0x6d2b79f5) | 0;
let t = Math.imul(a ^ (a >>> 15), 1 | a);
t = (t + Math.imul(t ^ (t >>> 7), 61 | t)) ^ t;
return ((t ^ (t >>> 14)) >>> 0) / 4294967296;
};
}
function jitter(pts: Pt[], rng: () => number, scale = 0.05): Pt[] {
const out = pts.map((p) => ({
x: p.x + (rng() * 2 - 1) * scale,
y: p.y + (rng() * 2 - 1) * scale,
}));
return gaugeFix(out);
}
// ---------- residual building ----------
type ResidualBuild = {
residuals: number[]; // hard + weighted penalties
hardResiduals: number[]; // only numeric constraints
penaltyResiduals: number[]; // only penalties before weighting
};
function buildResiduals(
pts: Pt[],
vertexKinds: VertexKind[],
constraints: PolygonConstraint[],
simplicityEps: number
): ResidualBuild {
const N = pts.length;
const hard: number[] = [];
// hard numeric constraints
for (const c of constraints) {
if (c.type === "length") {
const seg = normalizeSegment(c.seg);
hard.push(vecLen(segmentVec(pts, seg)) - c.length);
} else if (c.type === "interiorAngle") {
const i = c.i;
const prev = pts[(i + N - 1) % N];
const cur = pts[i];
const next = pts[(i + 1) % N];
hard.push(interiorAngle(prev, cur, next) - c.angleRad);
} else if (c.type === "segmentLengthRatio") {
const sx = normalizeSegment(c.segX);
const sy = normalizeSegment(c.segY);
const lx = vecLen(segmentVec(pts, sx));
const ly = vecLen(segmentVec(pts, sy));
hard.push(lx - c.factor * ly);
} else {
// segmentRelativeAngle
const sa = normalizeSegment(c.segA);
const sb = normalizeSegment(c.segB);
const ua = segmentVec(pts, sa);
const ub = segmentVec(pts, sb);
const la = vecLen(ua);
const lb = vecLen(ub);
const minLen = 1e-12;
if (la < minLen || lb < minLen) {
hard.push(1e3);
} else {
hard.push(undirectedAngleBetween(ua, ub) - c.angleRad);
}
}
}
// penalties (inequalities / geometry validity)
const pen: number[] = [];
// CCW area margin (this solver is still "SimpleNgon" but enforces CCW)
const area = signedArea(pts);
const areaMin = 1e-8;
pen.push(area < areaMin ? areaMin - area : 0);
// vertexKinds enforcement
const turnEps = 1e-6;
for (let i = 0; i < N; i++) {
const prev = pts[(i + N - 1) % N];
const cur = pts[i];
const next = pts[(i + 1) % N];
const t = signedTurn(prev, cur, next);
const kind = vertexKinds[i];
if (kind === "convex") {
pen.push(Math.max(0, turnEps - t));
} else if (kind === "reflex") {
pen.push(Math.max(0, t + turnEps));
} else {
// straight: |turn| <= eps
pen.push(Math.max(0, Math.abs(t) - turnEps));
}
}
// simplicity: penalize non-adjacent edge pairs that are too close/intersect
for (let i = 0; i < N; i++) {
const a = pts[i];
const b = pts[(i + 1) % N];
for (let j = i + 1; j < N; j++) {
const k = (j + 1) % N;
// skip same/adjacent edges (sharing a vertex)
if (j === i) continue;
if (j === (i + 1) % N) continue;
if (i === (j + 1) % N) continue;
const c = pts[j];
const d = pts[k];
const dmin = segmentSegmentDistance(a, b, c, d);
pen.push(Math.max(0, simplicityEps - dmin));
}
}
const penaltyWeight = 100;
const residuals = hard.concat(pen.map((v) => penaltyWeight * v));
return { residuals, hardResiduals: hard, penaltyResiduals: pen };
}
// ---------- numeric linear algebra ----------
function matMulAtA(J: number[][], lambda: number): number[][] {
const m = J.length;
const n = J[0].length;
const A: number[][] = Array.from({ length: n }, () => new Array(n).fill(0));
for (let i = 0; i < m; i++) {
const row = J[i];
for (let a = 0; a < n; a++) {
const ra = row[a];
for (let b = 0; b <= a; b++) {
A[a][b] += ra * row[b];
}
}
}
for (let a = 0; a < n; a++) {
for (let b = 0; b < a; b++) A[b][a] = A[a][b];
A[a][a] += lambda;
}
return A;
}
function matMulAtR(J: number[][], r: number[]): number[] {
const m = J.length;
const n = J[0].length;
const g = new Array(n).fill(0);
for (let i = 0; i < m; i++) {
const row = J[i];
const ri = r[i];
for (let a = 0; a < n; a++) g[a] += row[a] * ri;
}
return g;
}
// Solve A x = b via Gaussian elimination with partial pivoting
function solveLinearSystem(Ain: number[][], bin: number[]): number[] | null {
const n = bin.length;
const A = Ain.map((row) => row.slice());
const b = bin.slice();
for (let i = 0; i < n; i++) {
let piv = i;
let best = Math.abs(A[i][i]);
for (let r = i + 1; r < n; r++) {
const v = Math.abs(A[r][i]);
if (v > best) {
best = v;
piv = r;
}
}
if (best < 1e-14) return null;
if (piv !== i) {
[A[i], A[piv]] = [A[piv], A[i]];
[b[i], b[piv]] = [b[piv], b[i]];
}
const diag = A[i][i];
for (let r = i + 1; r < n; r++) {
const f = A[r][i] / diag;
if (f === 0) continue;
b[r] -= f * b[i];
for (let c = i; c < n; c++) A[r][c] -= f * A[i][c];
}
}
const x = new Array(n).fill(0);
for (let i = n - 1; i >= 0; i--) {
let s = b[i];
for (let c = i + 1; c < n; c++) s -= A[i][c] * x[c];
x[i] = s / A[i][i];
}
return x;
}
function dotVec(a: number[], b: number[]): number {
let s = 0;
for (let i = 0; i < a.length; i++) s += a[i] * b[i];
return s;
}
function norm2(v: number[]): number {
let s = 0;
for (const x of v) s += x * x;
return Math.sqrt(s);
}
function maxAbs(v: number[]): number {
let m = 0;
for (const x of v) m = Math.max(m, Math.abs(x));
return m;
}
function finiteDiffJacobian(
f: (x: number[]) => number[],
x: number[],
fx: number[]
): number[][] {
const n = x.length;
const m = fx.length;
const J: number[][] = Array.from({ length: m }, () => new Array(n).fill(0));
for (let j = 0; j < n; j++) {
const xj = x[j];
const h = 1e-6 * (1 + Math.abs(xj));
x[j] = xj + h;
const f2 = f(x);
x[j] = xj;
for (let i = 0; i < m; i++) {
J[i][j] = (f2[i] - fx[i]) / h;
}
}
return J;
}
// ---------- output ----------
function buildPolygonData(pts: Pt[]): PolygonData {
const N = pts.length;
const sideLengths: number[] = new Array(N);
const interiorAngles: number[] = new Array(N);
for (let i = 0; i < N; i++) {
sideLengths[i] = dist(pts[i], pts[(i + 1) % N]);
}
for (let i = 0; i < N; i++) {
const prev = pts[(i + N - 1) % N];
const cur = pts[i];
const next = pts[(i + 1) % N];
interiorAngles[i] = interiorAngle(prev, cur, next);
}
return { sides: N, sideLengths, interiorAngles };
}
// ---------- LM run ----------
function runLM(
N: number,
vertexKinds: VertexKind[],
constraints: PolygonConstraint[],
x0: number[],
cfg: {
maxIterations: number;
tol: number;
simplicityEps: number;
diagnostics: "none" | "basic" | "verbose";
}
):
| { ok: true; value: PolygonData }
| { ok: false; attempt?: { hardMax: number; penMax: number; hardNorm: number } } {
let x = x0.slice();
let lambda = 1e-3;
const f = (xx: number[]) => {
const pts = unpackVars(N, xx);
return buildResiduals(pts, vertexKinds, constraints, cfg.simplicityEps)
.residuals;
};
let r = f(x);
let cost = 0.5 * dotVec(r, r);
for (let iter = 0; iter < cfg.maxIterations; iter++) {
const pts = unpackVars(N, x);
const built = buildResiduals(pts, vertexKinds, constraints, cfg.simplicityEps);
const hardMax = maxAbs(built.hardResiduals);
const penMax = maxAbs(built.penaltyResiduals);
const hardNorm = norm2(built.hardResiduals);
if (hardMax <= cfg.tol && penMax <= 1e-9) {
return { ok: true, value: buildPolygonData(pts) };
}
const J = finiteDiffJacobian(f, x, r);
const A = matMulAtA(J, lambda);
const g = matMulAtR(J, r);
const dx = solveLinearSystem(A, g.map((v) => -v));
if (!dx) return { ok: false, attempt: { hardMax, penMax, hardNorm } };
const xNew = x.map((v, i) => v + dx[i]);
const rNew = f(xNew);
const costNew = 0.5 * dotVec(rNew, rNew);
if (costNew < cost) {
x = xNew;
r = rNew;
cost = costNew;
lambda = Math.max(1e-12, lambda * 0.33);
} else {
lambda = Math.min(1e12, lambda * 3.0);
}
if (cfg.diagnostics === "verbose" && iter % 25 === 0) {
// eslint-disable-next-line no-console
console.log(
`[iter ${iter}] cost=${cost.toExponential(3)} hardMax=${hardMax.toExponential(
3
)} penMax=${penMax.toExponential(3)} lambda=${lambda.toExponential(2)}`
);
}
}
const pts = unpackVars(N, x);
const built = buildResiduals(pts, vertexKinds, constraints, cfg.simplicityEps);
return {
ok: false,
attempt: {
hardMax: maxAbs(built.hardResiduals),
penMax: maxAbs(built.penaltyResiduals),
hardNorm: norm2(built.hardResiduals),
},
};
}
// ---------- public solver ----------
//
// NOTE: Despite the name, this enforces CCW via a penalty. If you later want
// to allow both orientations, remove the area penalty and add an option.
export function solveSimpleNgon(
vertexCount: number,
vertexKinds: VertexKind[],
constraints: PolygonConstraint[],
options?: SimplePolygonSolveOptions
): PolygonData | SolverError {
if (!Number.isInteger(vertexCount) || vertexCount < 3) {
return err("InvalidInput", "vertexCount must be an integer >= 3", {
vertexCount,
});
}
const N = vertexCount;
if (vertexKinds.length !== N) {
return err("InvalidInput", "vertexKinds must have length N", {
expected: N,
got: vertexKinds.length,
});
}
for (let i = 0; i < N; i++) {
const k = vertexKinds[i];
if (k !== "convex" && k !== "reflex" && k !== "straight") {
return err(
"InvalidInput",
`vertexKinds[${i}] must be "convex", "reflex", or "straight"`,
{ value: k }
);
}
}
if (constraints.length === 0) {
return err("InvalidInput", "At least one constraint is required");
}
// Validate constraints
const absoluteLengthSegs = new Set<string>();
const ratioKeys = new Set<string>();
const relAngleKeys = new Set<string>();
let hasAbsoluteLength = false;
for (let idx = 0; idx < constraints.length; idx++) {
const c = constraints[idx];
if (c.type === "length") {
hasAbsoluteLength = true;
const seg = normalizeSegment(c.seg);
if (
!Number.isInteger(seg.i) ||
seg.i < 0 ||
seg.i >= N ||
!Number.isInteger(seg.j) ||
seg.j < 0 ||
seg.j >= N
) {
return err("InvalidInput", `Invalid vertex index in length at ${idx}`, {
c,
});
}
if (seg.i === seg.j) {
return err("InvalidInput", `length.seg at ${idx} has i === j`, { c });
}
if (!(c.length > 0) || !Number.isFinite(c.length)) {
return err("InvalidInput", `length.length at ${idx} must be > 0`, {
c,
});
}
const key = segmentKey(seg);
if (absoluteLengthSegs.has(key)) {
return err("InvalidInput", `duplicate length constraint for segment ${key}`, { c });
}
absoluteLengthSegs.add(key);
}
else if (c.type === "interiorAngle") {
const { i, angleRad } = c;
if (!Number.isInteger(i) || i < 0 || i >= N) {
return err("InvalidInput", `Invalid vertex index in interiorAngle at ${idx}`, { c });
}
if (!Number.isFinite(angleRad) || !(angleRad > 0) || !(angleRad < 2 * Math.PI)) {
return err("InvalidInput", `interiorAngle.angleRad at ${idx} must satisfy 0 < angleRad < 2π (π allowed)`, { c });
}
const kind = vertexKinds[i];
if (kind === "convex" && !(angleRad < Math.PI)) {
return err("InvalidInput", `interiorAngle at ${idx} conflicts with vertexKinds[${i}] === "convex"`, { c });
}
if (kind === "reflex" && !(angleRad > Math.PI)) {
return err("InvalidInput", `interiorAngle at ${idx} conflicts with vertexKinds[${i}] === "reflex"`, { c });
}
if (kind === "straight" && angleRad !== Math.PI) {
return err("InvalidInput", `interiorAngle at ${idx} conflicts with vertexKinds[${i}] === "straight" (must be π)`, { c });
}
}
else if (c.type === "segmentLengthRatio") {
const sx = normalizeSegment(c.segX);
const sy = normalizeSegment(c.segY);
for (const [name, s] of [["segX", sx], ["segY", sy]] as const) {
if (
!Number.isInteger(s.i) ||
s.i < 0 ||
s.i >= N ||
!Number.isInteger(s.j) ||
s.j < 0 ||
s.j >= N
) {
return err("InvalidInput", `Invalid vertex index in segmentLengthRatio.${name} at ${idx}`, { c });
}
if (s.i === s.j) return err("InvalidInput", `segmentLengthRatio.${name} at ${idx} has i === j`, { c });
}
if (!Number.isFinite(c.factor) || !(c.factor > 0)) {
return err("InvalidInput", `segmentLengthRatio.factor at ${idx} must be > 0`, { c });
}
const key = `${segmentKey(sx)}=${c.factor}*${segmentKey(sy)}`;
if (ratioKeys.has(key)) return err("InvalidInput", `Duplicate segmentLengthRatio constraint`, { c });
ratioKeys.add(key);
}
else {
// segmentRelativeAngle
const sa = normalizeSegment(c.segA);
const sb = normalizeSegment(c.segB);
for (const [name, s] of [["segA", sa], ["segB", sb]] as const) {
if (
!Number.isInteger(s.i) ||
s.i < 0 ||
s.i >= N ||
!Number.isInteger(s.j) ||
s.j < 0 ||
s.j >= N
) {
return err("InvalidInput", `Invalid vertex index in segmentRelativeAngle.${name} at ${idx}`, { c });
}
if (s.i === s.j) return err("InvalidInput", `segmentRelativeAngle.${name} at ${idx} has i === j`, { c });
}
if (!Number.isFinite(c.angleRad) || c.angleRad < 0 || c.angleRad > Math.PI) {
return err("InvalidInput", `segmentRelativeAngle.angleRad at ${idx} must satisfy 0 <= angleRad <= π`, { c });
}
const key = `angle(${segmentKey(sa)},${segmentKey(sb)})=${c.angleRad}`;
if (relAngleKeys.has(key)) return err("InvalidInput", `Duplicate segmentRelativeAngle constraint`, { c });
relAngleKeys.add(key);
}
}
// Fix global scale: need at least one absolute length.
if (!hasAbsoluteLength) {
return err("InvalidInput", "At least one absolute 'length' constraint is required to fix global scale");
}
const maxIterations = options?.maxIterations ?? 3000;
const tol = options?.tolerance ?? 1e-10;
const simplicityEps = options?.simplicityEpsilon ?? 1e-6;
const restarts = options?.restarts ?? 0;
const diag = options?.diagnostics ?? "none";
// init
let initPts: Pt[];
const init = options?.initialization ?? ({ kind: "regularNgon" } as const);
if (init.kind === "custom") {
if (init.initialVertices.length !== N) {
return err("InvalidInput", "custom initialization must provide N vertices", {
expected: N,
got: init.initialVertices.length,
});
}
initPts = gaugeFix(init.initialVertices.map((p) => ({ x: p.x, y: p.y })));
} else {
initPts = regularNgonInit(N);
}
const seed = (options?.randomSeed ?? 123456789) | 0;
const rng = mulberry32(seed);
let best: { hardMax: number; penMax: number; hardNorm: number } | null = null;
const trials = 1 + Math.max(0, restarts);
for (let trial = 0; trial < trials; trial++) {
const startPts = trial === 0 ? initPts : jitter(initPts, rng, 0.15);
const startX = packVars(startPts);
const res = runLM(N, vertexKinds, constraints, startX, {
maxIterations,
tol,
simplicityEps,
diagnostics: diag,
});
if (res.ok) return res.value;
if (res.attempt) {
const a = res.attempt;
if (
!best ||
a.hardMax < best.hardMax ||
(a.hardMax === best.hardMax && a.penMax < best.penMax)
) {
best = a;
}
}
}
return err(
"Infeasible",
"Could not satisfy constraints and geometric requirements within tolerance",
best ?? undefined
);
}
|