Tooling
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Rainer Leit
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// Sphere mesh construction: Fibonacci sphere → Delaunay → close pole → SphereMesh.
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// Adapted from Red Blob Games sphere-mesh.js.
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let _Delaunator = null;
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export function setDelaunator(D) { _Delaunator = D; }
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// Fibonacci sphere with jitter — evenly-distributed points using the
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// Fibonacci spiral. Jitter randomises positions for more organic Voronoi cells.
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export function generateFibonacciSphere(N, jitter, rng) {
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const r_xyz = new Float32Array(3 * N);
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const s = 3.6 / Math.sqrt(N);
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const dlong = Math.PI * (3 - Math.sqrt(5));
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const dz = 2.0 / N;
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for (let k = 0, lng = 0, z = 1 - dz / 2; k < N; k++, z -= dz) {
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const r = Math.sqrt(1 - z * z);
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let latDeg = Math.asin(z) * 180 / Math.PI;
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let lonDeg = lng * 180 / Math.PI;
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if (jitter > 0) {
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const jLat = (rng() - rng());
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const jLon = (rng() - rng());
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const nextZ = Math.max(-1, z - dz * 2 * Math.PI * r / s);
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latDeg += jitter * jLat * (latDeg - Math.asin(nextZ) * 180 / Math.PI);
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lonDeg += jitter * jLon * (s / r * 180 / Math.PI);
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}
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const latR = latDeg * Math.PI / 180;
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const lonR = lonDeg * Math.PI / 180;
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r_xyz[3*k] = Math.cos(latR) * Math.cos(lonR);
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r_xyz[3*k+1] = Math.cos(latR) * Math.sin(lonR);
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r_xyz[3*k+2] = Math.sin(latR);
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lng += dlong;
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}
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return r_xyz;
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}
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// Stereographic projection (for Delaunay on a sphere).
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// Projects every point from the "north pole" (0,0,1) onto a plane.
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export function stereographicProjection(r_xyz, N) {
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const flat = new Float64Array(2 * N);
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for (let i = 0; i < N; i++) {
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const z = r_xyz[3*i+2];
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// Clamp denominator to prevent Infinity when a jittered point lands
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// on or near the projection pole (z ≈ 1). The exact projected position
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// doesn't matter for near-pole points — addPoleToMesh corrects connectivity.
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const denom = Math.max(1e-12, 1 - z);
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flat[2*i] = r_xyz[3*i] / denom;
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flat[2*i+1] = r_xyz[3*i+1] / denom;
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}
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return flat;
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}
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// Add pole back into mesh — close the mesh by connecting hull edges to the pole.
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export function addPoleToMesh(poleId, triangles, halfedges) {
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const numSides = triangles.length;
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const next = s => (s % 3 === 2) ? s - 2 : s + 1;
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let numUnpaired = 0, firstUnpaired = -1;
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const pointToSide = [];
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for (let s = 0; s < numSides; s++) {
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if (halfedges[s] === -1) {
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numUnpaired++;
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pointToSide[triangles[s]] = s;
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firstUnpaired = s;
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}
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}
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const nt = new Int32Array(numSides + 3 * numUnpaired);
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const nh = new Int32Array(numSides + 3 * numUnpaired);
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nt.set(triangles);
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nh.set(halfedges);
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for (let i = 0, s = firstUnpaired;
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i < numUnpaired;
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i++, s = pointToSide[nt[next(s)]]) {
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const ns = numSides + 3 * i;
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nh[s] = ns;
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nh[ns] = s;
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nt[ns] = nt[next(s)];
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nt[ns + 1] = nt[s];
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nt[ns + 2] = poleId;
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const k = numSides + (3 * i + 4) % (3 * numUnpaired);
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nh[ns + 2] = k;
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nh[k] = ns + 2;
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}
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return { triangles: nt, halfedges: nh };
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}
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// Lightweight dual-mesh helper wrapping Delaunator output.
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// Regions = Voronoi cells, Triangles = Delaunay triangles, Sides = half-edges.
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export class SphereMesh {
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constructor(triangles, halfedges, numRegions) {
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this.triangles = triangles;
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this.halfedges = halfedges;
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this.numRegions = numRegions;
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this.numSides = triangles.length;
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this.numTriangles = (triangles.length / 3) | 0;
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this._r_s = new Int32Array(numRegions).fill(-1);
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for (let s = 0; s < this.numSides; s++) {
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const r = triangles[s];
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if (this._r_s[r] === -1) this._r_s[r] = s;
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}
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// Pre-compute flat adjacency lists for r_circulate_r and r_circulate_t.
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// Replaces per-call half-edge traversal with cache-friendly array reads.
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const adjCount = new Int32Array(numRegions);
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for (let r = 0; r < numRegions; r++) {
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const s0 = this._r_s[r];
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if (s0 === -1) continue;
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let s = s0;
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do {
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adjCount[r]++;
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s = this._next(this.halfedges[s]);
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} while (s !== s0);
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}
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this._adjOffset = new Int32Array(numRegions + 1);
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for (let r = 0; r < numRegions; r++) {
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this._adjOffset[r + 1] = this._adjOffset[r] + adjCount[r];
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}
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const totalAdj = this._adjOffset[numRegions];
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this._adjList = new Int32Array(totalAdj); // neighbor regions
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this._adjTriList = new Int32Array(totalAdj); // neighbor triangles
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for (let r = 0; r < numRegions; r++) {
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const s0 = this._r_s[r];
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if (s0 === -1) continue;
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let s = s0;
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let idx = this._adjOffset[r];
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do {
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this._adjList[idx] = this.s_end_r(s);
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this._adjTriList[idx] = this.s_inner_t(s);
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idx++;
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s = this._next(this.halfedges[s]);
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} while (s !== s0);
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}
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// Public aliases for direct adjacency iteration (avoids r_circulate_r copy overhead)
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this.adjOffset = this._adjOffset;
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this.adjList = this._adjList;
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}
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_next(s) { return (s % 3 === 2) ? s - 2 : s + 1; }
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s_begin_r(s){ return this.triangles[s]; }
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s_end_r(s) { return this.triangles[this._next(s)]; }
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s_inner_t(s){ return (s / 3) | 0; }
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s_outer_t(s){ return (this.halfedges[s] / 3) | 0; }
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r_circulate_r(out, r) {
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const start = this._adjOffset[r];
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const end = this._adjOffset[r + 1];
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const len = end - start;
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out.length = len;
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for (let i = 0; i < len; i++) out[i] = this._adjList[start + i];
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return out;
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}
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r_circulate_t(out, r) {
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const start = this._adjOffset[r];
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const end = this._adjOffset[r + 1];
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const len = end - start;
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out.length = len;
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for (let i = 0; i < len; i++) out[i] = this._adjTriList[start + i];
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return out;
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}
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}
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// Build sphere — Fibonacci points → Delaunay → close pole.
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export function buildSphere(N, jitter, rng) {
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const r_xyz = generateFibonacciSphere(N, jitter, rng);
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const flat = stereographicProjection(r_xyz, N);
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const delaunay = new _Delaunator(flat);
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const poleXYZ = new Float32Array(3 * (N + 1));
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poleXYZ.set(r_xyz);
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poleXYZ[3*N] = 0; poleXYZ[3*N+1] = 0; poleXYZ[3*N+2] = 1;
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const closed = addPoleToMesh(N, delaunay.triangles, delaunay.halfedges);
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const mesh = new SphereMesh(closed.triangles, closed.halfedges, N + 1);
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return { mesh, r_xyz: poleXYZ };
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}
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// Pre-compute Euclidean distance between each region and its neighbors.
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// Indexed by the same adjacency slot as adjList: neighborDist[i] is the
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// distance from region r to adjList[i] where adjOffset[r] <= i < adjOffset[r+1].
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export function computeNeighborDist(mesh, r_xyz) {
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const { adjOffset, adjList } = mesh;
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const neighborDist = new Float32Array(adjList.length);
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for (let r = 0; r < mesh.numRegions; r++) {
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const x = r_xyz[3*r], y = r_xyz[3*r+1], z = r_xyz[3*r+2];
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for (let i = adjOffset[r]; i < adjOffset[r+1]; i++) {
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const nb = adjList[i];
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const dx = x - r_xyz[3*nb], dy = y - r_xyz[3*nb+1], dz = z - r_xyz[3*nb+2];
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neighborDist[i] = Math.sqrt(dx*dx + dy*dy + dz*dz);
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}
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}
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return neighborDist;
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}
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// Triangle centres (= Voronoi vertices on the sphere).
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export function generateTriangleCenters(mesh, r_xyz) {
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const { numTriangles } = mesh;
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const t_xyz = new Float32Array(3 * numTriangles);
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for (let t = 0; t < numTriangles; t++) {
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const s0 = 3 * t;
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const a = mesh.s_begin_r(s0),
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b = mesh.s_begin_r(s0 + 1),
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c = mesh.s_begin_r(s0 + 2);
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t_xyz[3*t] = (r_xyz[3*a] + r_xyz[3*b] + r_xyz[3*c]) / 3;
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t_xyz[3*t+1] = (r_xyz[3*a+1]+r_xyz[3*b+1]+r_xyz[3*c+1]) / 3;
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t_xyz[3*t+2] = (r_xyz[3*a+2]+r_xyz[3*b+2]+r_xyz[3*c+2]) / 3;
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}
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return t_xyz;
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}
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