Voronoi Spiral

A spiral of points is created. Each i^th point is rotated by an angle i*2*PI*Ratio around the origin. The initial value of the Ratio is 0.38196601125 (which gives the "golden angle"). You can change the Ratio using the slider below. A better explanation about what's going on can find here: youtu.be/sj8Sg8qnjOg.

The final Voronoi diagram is based on the Delaunay triangulation of the points, using delaunator of Mapbox: github.com/mapbox/delaunator

#Delaunay #triangulation #voronoi

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// Voronoi Spiral. Created by Reinder Nijhoff 2019
// @reindernijhoff
//
// https://turtletoy.net/turtle/70b4fd8c25
//


Canvas.setpenopacity(1);

const canvas_size = 45;

const coords = [];
let delaunay;

const turtle = new Turtle();

function walk(i) {
    if (i == 0) {
        for (let j=0; j<6000; j++) {
            const Ratio = 0.38196601125; // min=0.01, max=1, step=0.001;
            const r = 4 * Math.sqrt(j);
            const a = j * 2 * Math.PI * Ratio;
            const x = Math.cos(a)*r;
            const y = Math.sin(a)*r;
            
//            const x = (Math.random()**2*2-1) * canvas_size;
//            const y = (Math.random()**2*2-1) * canvas_size;
            
            coords.push([x,y]);
        }
        delaunay = Delaunator.from(coords);
    }
    
    const t0i = i;
    const t0 = (t0i/3|0) * 3;
    const t1i = delaunay.halfedges[i];
    const t = delaunay.triangles;
    const c = delaunay.coords;
    
    if (t1i >= t0i) {
        const t1 = (t1i/3|0) * 3;
        const p0 = circumcenter(c[t[t0+0]*2+0], c[t[t0+0]*2+1], c[t[t0+1]*2+0], c[t[t0+1]*2+1], c[t[t0+2]*2+0], c[t[t0+2]*2+1]);
        const p1 = circumcenter(c[t[t1+0]*2+0], c[t[t1+0]*2+1], c[t[t1+1]*2+0], c[t[t1+1]*2+1], c[t[t1+2]*2+0], c[t[t1+2]*2+1]);

       turtle.jump(p0.x, p0.y);
        turtle.goto(p1.x, p1.y);
    } else if (t1i < 0) {
        const p0 = circumcenter(c[t[t0+0]*2+0], c[t[t0+0]*2+1], c[t[t0+1]*2+0], c[t[t0+1]*2+1], c[t[t0+2]*2+0], c[t[t0+2]*2+1]);
        const v0 = [c[t[t0+(t0i % 3)]*2+0], c[t[t0+(t0i % 3)]*2+1]];
        const v1 = [c[t[t0+((t0i + 1) % 3)]*2+0], c[t[t0+((t0i + 1) % 3)]*2+1]];

        const d = [(v0[1]-v1[1]), -(v0[0]-v1[0])];
        const l = Math.sqrt(d[0]**2 + d[1]**2);
        
        turtle.jump(p0.x, p0.y);
        turtle.goto(p0.x + 200*d[0]/l, p0.y + 200*d[1]/l);
    }
    
    if (i < coords.length) {
        turtle.jump(coords[i][0], coords[i][1]-.5);
        turtle.circle(.5);
    }
    
    return i < delaunay.halfedges.length - 1;
}

//
// https://github.com/mapbox/delaunator
//
// ISC License
//
// Copyright (c) 2017, Mapbox
//
// Permission to use, copy, modify, and/or distribute this software for any purpose
// with or without fee is hereby granted, provided that the above copyright notice
// and this permission notice appear in all copies.
//
// THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES WITH
// REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF MERCHANTABILITY AND
// FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY SPECIAL, DIRECT,
// INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES WHATSOEVER RESULTING FROM LOSS
// OF USE, DATA OR PROFITS, WHETHER IN AN ACTION OF CONTRACT, NEGLIGENCE OR OTHER
// TORTIOUS ACTION, ARISING OUT OF OR IN CONNECTION WITH THE USE OR PERFORMANCE OF
// THIS SOFTWARE.
//

const EPSILON = Math.pow(2, -52);
const EDGE_STACK = new Uint32Array(512);

class Delaunator {
    static from(points, getX = defaultGetX, getY = defaultGetY) {
        const n = points.length;
        const coords = new Float64Array(n * 2);

        for (let i = 0; i < n; i++) {
            const p = points[i];
            coords[2 * i] = getX(p);
            coords[2 * i + 1] = getY(p);
        }

        return new Delaunator(coords);
    }

    constructor(coords) {
        const n = coords.length >> 1;
        if (n > 0 && typeof coords[0] !== 'number') throw new Error('Expected coords to contain numbers.');

        this.coords = coords;

        // arrays that will store the triangulation graph
        const maxTriangles = 2 * n - 5;
        const triangles = this.triangles = new Uint32Array(maxTriangles * 3);
        const halfedges = this.halfedges = new Int32Array(maxTriangles * 3);

        // temporary arrays for tracking the edges of the advancing convex hull
        this._hashSize = Math.ceil(Math.sqrt(n));
        const hullPrev = this.hullPrev = new Uint32Array(n); // edge to prev edge
        const hullNext = this.hullNext = new Uint32Array(n); // edge to next edge
        const hullTri = this.hullTri = new Uint32Array(n); // edge to adjacent triangle
        const hullHash = new Int32Array(this._hashSize).fill(-1); // angular edge hash

        // populate an array of point indices; calculate input data bbox
        const ids = new Uint32Array(n);
        let minX = Infinity;
        let minY = Infinity;
        let maxX = -Infinity;
        let maxY = -Infinity;

        for (let i = 0; i < n; i++) {
            const x = coords[2 * i];
            const y = coords[2 * i + 1];
            if (x < minX) minX = x;
            if (y < minY) minY = y;
            if (x > maxX) maxX = x;
            if (y > maxY) maxY = y;
            ids[i] = i;
        }
        const cx = (minX + maxX) / 2;
        const cy = (minY + maxY) / 2;

        let minDist = Infinity;
        let i0, i1, i2;

        // pick a seed point close to the center
        for (let i = 0; i < n; i++) {
            const d = dist(cx, cy, coords[2 * i], coords[2 * i + 1]);
            if (d < minDist) {
                i0 = i;
                minDist = d;
            }
        }
        const i0x = coords[2 * i0];
        const i0y = coords[2 * i0 + 1];

        minDist = Infinity;

        // find the point closest to the seed
        for (let i = 0; i < n; i++) {
            if (i === i0) continue;
            const d = dist(i0x, i0y, coords[2 * i], coords[2 * i + 1]);
            if (d < minDist && d > 0) {
                i1 = i;
                minDist = d;
            }
        }
        let i1x = coords[2 * i1];
        let i1y = coords[2 * i1 + 1];

        let minRadius = Infinity;

        // find the third point which forms the smallest circumcircle with the first two
        for (let i = 0; i < n; i++) {
            if (i === i0 || i === i1) continue;
            const r = circumradius(i0x, i0y, i1x, i1y, coords[2 * i], coords[2 * i + 1]);
            if (r < minRadius) {
                i2 = i;
                minRadius = r;
            }
        }
        let i2x = coords[2 * i2];
        let i2y = coords[2 * i2 + 1];

        if (minRadius === Infinity) {
            throw new Error('No Delaunay triangulation exists for this input.');
        }

        // swap the order of the seed points for counter-clockwise orientation
        if (orient(i0x, i0y, i1x, i1y, i2x, i2y)) {
            const i = i1;
            const x = i1x;
            const y = i1y;
            i1 = i2;
            i1x = i2x;
            i1y = i2y;
            i2 = i;
            i2x = x;
            i2y = y;
        }

        const center = circumcenter(i0x, i0y, i1x, i1y, i2x, i2y);
        this._cx = center.x;
        this._cy = center.y;

        const dists = new Float64Array(n);
        for (let i = 0; i < n; i++) {
            dists[i] = dist(coords[2 * i], coords[2 * i + 1], center.x, center.y);
        }

        // sort the points by distance from the seed triangle circumcenter
        quicksort(ids, dists, 0, n - 1);

        // set up the seed triangle as the starting hull
        this.hullStart = i0;
        let hullSize = 3;

        hullNext[i0] = hullPrev[i2] = i1;
        hullNext[i1] = hullPrev[i0] = i2;
        hullNext[i2] = hullPrev[i1] = i0;

        hullTri[i0] = 0;
        hullTri[i1] = 1;
        hullTri[i2] = 2;

        hullHash[this._hashKey(i0x, i0y)] = i0;
        hullHash[this._hashKey(i1x, i1y)] = i1;
        hullHash[this._hashKey(i2x, i2y)] = i2;

        this.trianglesLen = 0;
        this._addTriangle(i0, i1, i2, -1, -1, -1);

        for (let k = 0, xp, yp; k < ids.length; k++) {
            const i = ids[k];
            const x = coords[2 * i];
            const y = coords[2 * i + 1];

            // skip near-duplicate points
            if (k > 0 && Math.abs(x - xp) <= EPSILON && Math.abs(y - yp) <= EPSILON) continue;
            xp = x;
            yp = y;

            // skip seed triangle points
            if (i === i0 || i === i1 || i === i2) continue;

            // find a visible edge on the convex hull using edge hash
            let start = 0;
            for (let j = 0, key = this._hashKey(x, y); j < this._hashSize; j++) {
                start = hullHash[(key + j) % this._hashSize];
                if (start !== -1 && start !== hullNext[start]) break;
            }

            start = hullPrev[start];
            let e = start, q;
            while (q = hullNext[e], !orient(x, y, coords[2 * e], coords[2 * e + 1], coords[2 * q], coords[2 * q + 1])) {
                e = q;
                if (e === start) {
                    e = -1;
                    break;
                }
            }
            if (e === -1) continue; // likely a near-duplicate point; skip it

            // add the first triangle from the point
            let t = this._addTriangle(e, i, hullNext[e], -1, -1, hullTri[e]);

            // recursively flip triangles from the point until they satisfy the Delaunay condition
            hullTri[i] = this._legalize(t + 2);
            hullTri[e] = t; // keep track of boundary triangles on the hull
            hullSize++;

            // walk forward through the hull, adding more triangles and flipping recursively
            let n = hullNext[e];
            while (q = hullNext[n], orient(x, y, coords[2 * n], coords[2 * n + 1], coords[2 * q], coords[2 * q + 1])) {
                t = this._addTriangle(n, i, q, hullTri[i], -1, hullTri[n]);
                hullTri[i] = this._legalize(t + 2);
                hullNext[n] = n; // mark as removed
                hullSize--;
                n = q;
            }

            // walk backward from the other side, adding more triangles and flipping
            if (e === start) {
                while (q = hullPrev[e], orient(x, y, coords[2 * q], coords[2 * q + 1], coords[2 * e], coords[2 * e + 1])) {
                    t = this._addTriangle(q, i, e, -1, hullTri[e], hullTri[q]);
                    this._legalize(t + 2);
                    hullTri[q] = t;
                    hullNext[e] = e; // mark as removed
                    hullSize--;
                    e = q;
                }
            }

            // update the hull indices
            this.hullStart = hullPrev[i] = e;
            hullNext[e] = hullPrev[n] = i;
            hullNext[i] = n;

            // save the two new edges in the hash table
            hullHash[this._hashKey(x, y)] = i;
            hullHash[this._hashKey(coords[2 * e], coords[2 * e + 1])] = e;
        }

        this.hull = new Uint32Array(hullSize);
        for (let i = 0, e = this.hullStart; i < hullSize; i++) {
            this.hull[i] = e;
            e = hullNext[e];
        }
        this.hullPrev = this.hullNext = this.hullTri = null; // get rid of temporary arrays

        // trim typed triangle mesh arrays
        this.triangles = triangles.subarray(0, this.trianglesLen);
        this.halfedges = halfedges.subarray(0, this.trianglesLen);
    }

    _hashKey(x, y) {
        return Math.floor(pseudoAngle(x - this._cx, y - this._cy) * this._hashSize) % this._hashSize;
    }

    _legalize(a) {
        const {triangles, coords, halfedges} = this;

        let i = 0;
        let ar = 0;

        // recursion eliminated with a fixed-size stack
        while (true) {
            const b = halfedges[a];

            /* if the pair of triangles doesn't satisfy the Delaunay condition
             * (p1 is inside the circumcircle of [p0, pl, pr]), flip them,
             * then do the same check/flip recursively for the new pair of triangles
             *
             *           pl                    pl
             *          /||\                  /  \
             *       al/ || \bl            al/    \a
             *        /  ||  \              /      \
             *       /  a||b  \    flip    /___ar___\
             *     p0\   ||   /p1   =>   p0\---bl---/p1
             *        \  ||  /              \      /
             *       ar\ || /br             b\    /br
             *          \||/                  \  /
             *           pr                    pr
             */
            const a0 = a - a % 3;
            ar = a0 + (a + 2) % 3;

            if (b === -1) { // convex hull edge
                if (i === 0) break;
                a = EDGE_STACK[--i];
                continue;
            }

            const b0 = b - b % 3;
            const al = a0 + (a + 1) % 3;
            const bl = b0 + (b + 2) % 3;

            const p0 = triangles[ar];
            const pr = triangles[a];
            const pl = triangles[al];
            const p1 = triangles[bl];

            const illegal = inCircle(
                coords[2 * p0], coords[2 * p0 + 1],
                coords[2 * pr], coords[2 * pr + 1],
                coords[2 * pl], coords[2 * pl + 1],
                coords[2 * p1], coords[2 * p1 + 1]);

            if (illegal) {
                triangles[a] = p1;
                triangles[b] = p0;

                const hbl = halfedges[bl];

                // edge swapped on the other side of the hull (rare); fix the halfedge reference
                if (hbl === -1) {
                    let e = this.hullStart;
                    do {
                        if (this.hullTri[e] === bl) {
                            this.hullTri[e] = a;
                            break;
                        }
                        e = this.hullNext[e];
                    } while (e !== this.hullStart);
                }
                this._link(a, hbl);
                this._link(b, halfedges[ar]);
                this._link(ar, bl);

                const br = b0 + (b + 1) % 3;

                // don't worry about hitting the cap: it can only happen on extremely degenerate input
                if (i < EDGE_STACK.length) {
                    EDGE_STACK[i++] = br;
                }
            } else {
                if (i === 0) break;
                a = EDGE_STACK[--i];
            }
        }

        return ar;
    }

    _link(a, b) {
        this.halfedges[a] = b;
        if (b !== -1) this.halfedges[b] = a;
    }

    // add a new triangle given vertex indices and adjacent half-edge ids
    _addTriangle(i0, i1, i2, a, b, c) {
        const t = this.trianglesLen;

        this.triangles[t] = i0;
        this.triangles[t + 1] = i1;
        this.triangles[t + 2] = i2;

        this._link(t, a);
        this._link(t + 1, b);
        this._link(t + 2, c);

        this.trianglesLen += 3;

        return t;
    }
}

// monotonically increases with real angle, but doesn't need expensive trigonometry
function pseudoAngle(dx, dy) {
    const p = dx / (Math.abs(dx) + Math.abs(dy));
    return (dy > 0 ? 3 - p : 1 + p) / 4; // [0..1]
}

function dist(ax, ay, bx, by) {
    const dx = ax - bx;
    const dy = ay - by;
    return dx * dx + dy * dy;
}

function orient(px, py, qx, qy, rx, ry) {
    return (qy - py) * (rx - qx) - (qx - px) * (ry - qy) < 0;
}

function inCircle(ax, ay, bx, by, cx, cy, px, py) {
    const dx = ax - px;
    const dy = ay - py;
    const ex = bx - px;
    const ey = by - py;
    const fx = cx - px;
    const fy = cy - py;

    const ap = dx * dx + dy * dy;
    const bp = ex * ex + ey * ey;
    const cp = fx * fx + fy * fy;

    return dx * (ey * cp - bp * fy) -
           dy * (ex * cp - bp * fx) +
           ap * (ex * fy - ey * fx) < 0;
}

function circumradius(ax, ay, bx, by, cx, cy) {
    const dx = bx - ax;
    const dy = by - ay;
    const ex = cx - ax;
    const ey = cy - ay;

    const bl = dx * dx + dy * dy;
    const cl = ex * ex + ey * ey;
    const d = 0.5 / (dx * ey - dy * ex);

    const x = (ey * bl - dy * cl) * d;
    const y = (dx * cl - ex * bl) * d;

    return x * x + y * y;
}

function circumcenter(ax, ay, bx, by, cx, cy) {
    const dx = bx - ax;
    const dy = by - ay;
    const ex = cx - ax;
    const ey = cy - ay;

    const bl = dx * dx + dy * dy;
    const cl = ex * ex + ey * ey;
    const d = 0.5 / (dx * ey - dy * ex);

    const x = ax + (ey * bl - dy * cl) * d;
    const y = ay + (dx * cl - ex * bl) * d;

    return {x, y};
}

function quicksort(ids, dists, left, right) {
    if (right - left <= 20) {
        for (let i = left + 1; i <= right; i++) {
            const temp = ids[i];
            const tempDist = dists[temp];
            let j = i - 1;
            while (j >= left && dists[ids[j]] > tempDist) ids[j + 1] = ids[j--];
            ids[j + 1] = temp;
        }
    } else {
        const median = (left + right) >> 1;
        let i = left + 1;
        let j = right;
        swap(ids, median, i);
        if (dists[ids[left]] > dists[ids[right]]) swap(ids, left, right);
        if (dists[ids[i]] > dists[ids[right]]) swap(ids, i, right);
        if (dists[ids[left]] > dists[ids[i]]) swap(ids, left, i);

        const temp = ids[i];
        const tempDist = dists[temp];
        while (true) {
            do i++; while (dists[ids[i]] < tempDist);
            do j--; while (dists[ids[j]] > tempDist);
            if (j < i) break;
            swap(ids, i, j);
        }
        ids[left + 1] = ids[j];
        ids[j] = temp;

        if (right - i + 1 >= j - left) {
            quicksort(ids, dists, i, right);
            quicksort(ids, dists, left, j - 1);
        } else {
            quicksort(ids, dists, left, j - 1);
            quicksort(ids, dists, i, right);
        }
    }
}

function swap(arr, i, j) {
    const tmp = arr[i];
    arr[i] = arr[j];
    arr[j] = tmp;
}

function defaultGetX(p) {
    return p[0];
}
function defaultGetY(p) {
    return p[1];
}