BezierInfo-2/docs/js/graphics-element/lib/bezierjs/utils.js

import { Bezier } from "./bezier.js";

// math-inlining.
const { abs, cos, sin, acos, atan2, sqrt, pow } = Math;

// cube root function yielding real roots
function crt(v) {
  return v < 0 ? -pow(-v, 1 / 3) : pow(v, 1 / 3);
}

// trig constants
const pi = Math.PI,
  tau = 2 * pi,
  quart = pi / 2,
  // float precision significant decimal
  epsilon = 0.000001,
  // extremas used in bbox calculation and similar algorithms
  nMax = Number.MAX_SAFE_INTEGER || 9007199254740991,
  nMin = Number.MIN_SAFE_INTEGER || -9007199254740991,
  // a zero coordinate, which is surprisingly useful
  ZERO = { x: 0, y: 0, z: 0 };

// Bezier utility functions
const utils = {
  // Legendre-Gauss abscissae with n=24 (x_i values, defined at i=n as the roots of the nth order Legendre polynomial Pn(x))
  Tvalues: [
    -0.0640568928626056260850430826247450385909,
    0.0640568928626056260850430826247450385909,
    -0.1911188674736163091586398207570696318404,
    0.1911188674736163091586398207570696318404,
    -0.3150426796961633743867932913198102407864,
    0.3150426796961633743867932913198102407864,
    -0.4337935076260451384870842319133497124524,
    0.4337935076260451384870842319133497124524,
    -0.5454214713888395356583756172183723700107,
    0.5454214713888395356583756172183723700107,
    -0.6480936519369755692524957869107476266696,
    0.6480936519369755692524957869107476266696,
    -0.7401241915785543642438281030999784255232,
    0.7401241915785543642438281030999784255232,
    -0.8200019859739029219539498726697452080761,
    0.8200019859739029219539498726697452080761,
    -0.8864155270044010342131543419821967550873,
    0.8864155270044010342131543419821967550873,
    -0.9382745520027327585236490017087214496548,
    0.9382745520027327585236490017087214496548,
    -0.9747285559713094981983919930081690617411,
    0.9747285559713094981983919930081690617411,
    -0.9951872199970213601799974097007368118745,
    0.9951872199970213601799974097007368118745,
  ],

  // Legendre-Gauss weights with n=24 (w_i values, defined by a function linked to in the Bezier primer article)
  Cvalues: [
    0.1279381953467521569740561652246953718517,
    0.1279381953467521569740561652246953718517,
    0.1258374563468282961213753825111836887264,
    0.1258374563468282961213753825111836887264,
    0.121670472927803391204463153476262425607,
    0.121670472927803391204463153476262425607,
    0.1155056680537256013533444839067835598622,
    0.1155056680537256013533444839067835598622,
    0.1074442701159656347825773424466062227946,
    0.1074442701159656347825773424466062227946,
    0.0976186521041138882698806644642471544279,
    0.0976186521041138882698806644642471544279,
    0.086190161531953275917185202983742667185,
    0.086190161531953275917185202983742667185,
    0.0733464814110803057340336152531165181193,
    0.0733464814110803057340336152531165181193,
    0.0592985849154367807463677585001085845412,
    0.0592985849154367807463677585001085845412,
    0.0442774388174198061686027482113382288593,
    0.0442774388174198061686027482113382288593,
    0.0285313886289336631813078159518782864491,
    0.0285313886289336631813078159518782864491,
    0.0123412297999871995468056670700372915759,
    0.0123412297999871995468056670700372915759,
  ],

  arcfn: function (t, derivativeFn) {
    const d = derivativeFn(t);
    let l = d.x * d.x + d.y * d.y;
    if (typeof d.z !== "undefined") {
      l += d.z * d.z;
    }
    return sqrt(l);
  },

  compute: function (t, points, _3d) {
    // shortcuts
    if (t === 0) {
      points[0].t = 0;
      return points[0];
    }

    const order = points.length - 1;

    if (t === 1) {
      points[order].t = 1;
      return points[order];
    }

    const mt = 1 - t;
    let p = points;

    // constant?
    if (order === 0) {
      points[0].t = t;
      return points[0];
    }

    // linear?
    if (order === 1) {
      const ret = {
        x: mt * p[0].x + t * p[1].x,
        y: mt * p[0].y + t * p[1].y,
        t: t,
      };
      if (_3d) {
        ret.z = mt * p[0].z + t * p[1].z;
      }
      return ret;
    }

    // quadratic/cubic curve?
    if (order < 4) {
      let mt2 = mt * mt,
        t2 = t * t,
        a,
        b,
        c,
        d = 0;
      if (order === 2) {
        p = [p[0], p[1], p[2], ZERO];
        a = mt2;
        b = mt * t * 2;
        c = t2;
      } else if (order === 3) {
        a = mt2 * mt;
        b = mt2 * t * 3;
        c = mt * t2 * 3;
        d = t * t2;
      }
      const ret = {
        x: a * p[0].x + b * p[1].x + c * p[2].x + d * p[3].x,
        y: a * p[0].y + b * p[1].y + c * p[2].y + d * p[3].y,
        t: t,
      };
      if (_3d) {
        ret.z = a * p[0].z + b * p[1].z + c * p[2].z + d * p[3].z;
      }
      return ret;
    }

    // higher order curves: use de Casteljau's computation
    const dCpts = JSON.parse(JSON.stringify(points));
    while (dCpts.length > 1) {
      for (let i = 0; i < dCpts.length - 1; i++) {
        dCpts[i] = {
          x: dCpts[i].x + (dCpts[i + 1].x - dCpts[i].x) * t,
          y: dCpts[i].y + (dCpts[i + 1].y - dCpts[i].y) * t,
        };
        if (typeof dCpts[i].z !== "undefined") {
          dCpts[i] = dCpts[i].z + (dCpts[i + 1].z - dCpts[i].z) * t;
        }
      }
      dCpts.splice(dCpts.length - 1, 1);
    }
    dCpts[0].t = t;
    return dCpts[0];
  },

  computeWithRatios: function (t, points, ratios, _3d) {
    const mt = 1 - t,
      r = ratios,
      p = points;

    let f1 = r[0],
      f2 = r[1],
      f3 = r[2],
      f4 = r[3],
      d;

    // spec for linear
    f1 *= mt;
    f2 *= t;

    if (p.length === 2) {
      d = f1 + f2;
      return {
        x: (f1 * p[0].x + f2 * p[1].x) / d,
        y: (f1 * p[0].y + f2 * p[1].y) / d,
        z: !_3d ? false : (f1 * p[0].z + f2 * p[1].z) / d,
        t: t,
      };
    }

    // upgrade to quadratic
    f1 *= mt;
    f2 *= 2 * mt;
    f3 *= t * t;

    if (p.length === 3) {
      d = f1 + f2 + f3;
      return {
        x: (f1 * p[0].x + f2 * p[1].x + f3 * p[2].x) / d,
        y: (f1 * p[0].y + f2 * p[1].y + f3 * p[2].y) / d,
        z: !_3d ? false : (f1 * p[0].z + f2 * p[1].z + f3 * p[2].z) / d,
        t: t,
      };
    }

    // upgrade to cubic
    f1 *= mt;
    f2 *= 1.5 * mt;
    f3 *= 3 * mt;
    f4 *= t * t * t;

    if (p.length === 4) {
      d = f1 + f2 + f3 + f4;
      return {
        x: (f1 * p[0].x + f2 * p[1].x + f3 * p[2].x + f4 * p[3].x) / d,
        y: (f1 * p[0].y + f2 * p[1].y + f3 * p[2].y + f4 * p[3].y) / d,
        z: !_3d ? false : (f1 * p[0].z + f2 * p[1].z + f3 * p[2].z + f4 * p[3].z) / d,
        t: t,
      };
    }
  },

  derive: function (points, _3d) {
    const dpoints = [];
    for (let p = points, d = p.length, c = d - 1; d > 1; d--, c--) {
      const list = [];
      for (let j = 0, dpt; j < c; j++) {
        dpt = {
          x: c * (p[j + 1].x - p[j].x),
          y: c * (p[j + 1].y - p[j].y),
        };
        if (_3d) {
          dpt.z = c * (p[j + 1].z - p[j].z);
        }
        list.push(dpt);
      }
      dpoints.push(list);
      p = list;
    }
    return dpoints;
  },

  between: function (v, m, M) {
    return (m <= v && v <= M) || utils.approximately(v, m) || utils.approximately(v, M);
  },

  approximately: function (a, b, precision) {
    return abs(a - b) <= (precision || epsilon);
  },

  length: function (derivativeFn) {
    const z = 0.5,
      len = utils.Tvalues.length;

    let sum = 0;

    for (let i = 0, t; i < len; i++) {
      t = z * utils.Tvalues[i] + z;
      sum += utils.Cvalues[i] * utils.arcfn(t, derivativeFn);
    }
    return z * sum;
  },

  map: function (v, ds, de, ts, te) {
    const d1 = de - ds,
      d2 = te - ts,
      v2 = v - ds,
      r = v2 / d1;
    return ts + d2 * r;
  },

  lerp: function (r, v1, v2) {
    const ret = {
      x: v1.x + r * (v2.x - v1.x),
      y: v1.y + r * (v2.y - v1.y),
    };
    if (!!v1.z && !!v2.z) {
      ret.z = v1.z + r * (v2.z - v1.z);
    }
    return ret;
  },

  pointToString: function (p) {
    let s = p.x + "/" + p.y;
    if (typeof p.z !== "undefined") {
      s += "/" + p.z;
    }
    return s;
  },

  pointsToString: function (points) {
    return "[" + points.map(utils.pointToString).join(", ") + "]";
  },

  copy: function (obj) {
    return JSON.parse(JSON.stringify(obj));
  },

  angle: function (o, v1, v2) {
    const dx1 = v1.x - o.x,
      dy1 = v1.y - o.y,
      dx2 = v2.x - o.x,
      dy2 = v2.y - o.y,
      cross = dx1 * dy2 - dy1 * dx2,
      dot = dx1 * dx2 + dy1 * dy2;
    return atan2(cross, dot);
  },

  // round as string, to avoid rounding errors
  round: function (v, d) {
    const s = "" + v;
    const pos = s.indexOf(".");
    return parseFloat(s.substring(0, pos + 1 + d));
  },

  dist: function (p1, p2) {
    const dx = p1.x - p2.x,
      dy = p1.y - p2.y;
    return sqrt(dx * dx + dy * dy);
  },

  closest: function (LUT, point) {
    let mdist = pow(2, 63),
      mpos,
      d;
    LUT.forEach(function (p, idx) {
      d = utils.dist(point, p);
      if (d < mdist) {
        mdist = d;
        mpos = idx;
      }
    });
    return { mdist: mdist, mpos: mpos };
  },

  abcratio: function (t, n) {
    // see ratio(t) note on http://pomax.github.io/bezierinfo/#abc
    if (n !== 2 && n !== 3) {
      return false;
    }
    if (typeof t === "undefined") {
      t = 0.5;
    } else if (t === 0 || t === 1) {
      return t;
    }
    const bottom = pow(t, n) + pow(1 - t, n),
      top = bottom - 1;
    return abs(top / bottom);
  },

  projectionratio: function (t, n) {
    // see u(t) note on http://pomax.github.io/bezierinfo/#abc
    if (n !== 2 && n !== 3) {
      return false;
    }
    if (typeof t === "undefined") {
      t = 0.5;
    } else if (t === 0 || t === 1) {
      return t;
    }
    const top = pow(1 - t, n),
      bottom = pow(t, n) + top;
    return top / bottom;
  },

  lli8: function (x1, y1, x2, y2, x3, y3, x4, y4) {
    const nx = (x1 * y2 - y1 * x2) * (x3 - x4) - (x1 - x2) * (x3 * y4 - y3 * x4),
      ny = (x1 * y2 - y1 * x2) * (y3 - y4) - (y1 - y2) * (x3 * y4 - y3 * x4),
      d = (x1 - x2) * (y3 - y4) - (y1 - y2) * (x3 - x4);
    if (d == 0) {
      return false;
    }
    return { x: nx / d, y: ny / d };
  },

  lli4: function (p1, p2, p3, p4) {
    const x1 = p1.x,
      y1 = p1.y,
      x2 = p2.x,
      y2 = p2.y,
      x3 = p3.x,
      y3 = p3.y,
      x4 = p4.x,
      y4 = p4.y;
    return utils.lli8(x1, y1, x2, y2, x3, y3, x4, y4);
  },

  lli: function (v1, v2) {
    return utils.lli4(v1, v1.c, v2, v2.c);
  },

  makeline: function (p1, p2) {
    const x1 = p1.x,
      y1 = p1.y,
      x2 = p2.x,
      y2 = p2.y,
      dx = (x2 - x1) / 3,
      dy = (y2 - y1) / 3;
    return new Bezier(x1, y1, x1 + dx, y1 + dy, x1 + 2 * dx, y1 + 2 * dy, x2, y2);
  },

  findbbox: function (sections) {
    let mx = nMax,
      my = nMax,
      MX = nMin,
      MY = nMin;
    sections.forEach(function (s) {
      const bbox = s.bbox();
      if (mx > bbox.x.min) mx = bbox.x.min;
      if (my > bbox.y.min) my = bbox.y.min;
      if (MX < bbox.x.max) MX = bbox.x.max;
      if (MY < bbox.y.max) MY = bbox.y.max;
    });
    return {
      x: { min: mx, mid: (mx + MX) / 2, max: MX, size: MX - mx },
      y: { min: my, mid: (my + MY) / 2, max: MY, size: MY - my },
    };
  },

  shapeintersections: function (s1, bbox1, s2, bbox2, curveIntersectionThreshold) {
    if (!utils.bboxoverlap(bbox1, bbox2)) return [];
    const intersections = [];
    const a1 = [s1.startcap, s1.forward, s1.back, s1.endcap];
    const a2 = [s2.startcap, s2.forward, s2.back, s2.endcap];
    a1.forEach(function (l1) {
      if (l1.virtual) return;
      a2.forEach(function (l2) {
        if (l2.virtual) return;
        const iss = l1.intersects(l2, curveIntersectionThreshold);
        if (iss.length > 0) {
          iss.c1 = l1;
          iss.c2 = l2;
          iss.s1 = s1;
          iss.s2 = s2;
          intersections.push(iss);
        }
      });
    });
    return intersections;
  },

  makeshape: function (forward, back, curveIntersectionThreshold) {
    const bpl = back.points.length;
    const fpl = forward.points.length;
    const start = utils.makeline(back.points[bpl - 1], forward.points[0]);
    const end = utils.makeline(forward.points[fpl - 1], back.points[0]);
    const shape = {
      startcap: start,
      forward: forward,
      back: back,
      endcap: end,
      bbox: utils.findbbox([start, forward, back, end]),
    };
    shape.intersections = function (s2) {
      return utils.shapeintersections(shape, shape.bbox, s2, s2.bbox, curveIntersectionThreshold);
    };
    return shape;
  },

  getminmax: function (curve, d, list) {
    if (!list) return { min: 0, max: 0 };
    let min = nMax,
      max = nMin,
      t,
      c;
    if (list.indexOf(0) === -1) {
      list = [0].concat(list);
    }
    if (list.indexOf(1) === -1) {
      list.push(1);
    }
    for (let i = 0, len = list.length; i < len; i++) {
      t = list[i];
      c = curve.get(t);
      if (c[d] < min) {
        min = c[d];
      }
      if (c[d] > max) {
        max = c[d];
      }
    }
    return { min: min, mid: (min + max) / 2, max: max, size: max - min };
  },

  align: function (points, line) {
    const tx = line.p1.x,
      ty = line.p1.y,
      a = -atan2(line.p2.y - ty, line.p2.x - tx),
      d = function (v) {
        return {
          x: (v.x - tx) * cos(a) - (v.y - ty) * sin(a),
          y: (v.x - tx) * sin(a) + (v.y - ty) * cos(a),
        };
      };
    return points.map(d);
  },

  roots: function (points, line) {
    line = line || { p1: { x: 0, y: 0 }, p2: { x: 1, y: 0 } };

    const order = points.length - 1;
    const aligned = utils.align(points, line);
    const reduce = function (t) {
      return 0 <= t && t <= 1;
    };

    if (order === 2) {
      const a = aligned[0].y,
        b = aligned[1].y,
        c = aligned[2].y,
        d = a - 2 * b + c;
      if (d !== 0) {
        const m1 = -sqrt(b * b - a * c),
          m2 = -a + b,
          v1 = -(m1 + m2) / d,
          v2 = -(-m1 + m2) / d;
        return [v1, v2].filter(reduce);
      } else if (b !== c && d === 0) {
        return [(2 * b - c) / (2 * b - 2 * c)].filter(reduce);
      }
      return [];
    }

    // see http://www.trans4mind.com/personal_development/mathematics/polynomials/cubicAlgebra.htm
    const pa = aligned[0].y,
      pb = aligned[1].y,
      pc = aligned[2].y,
      pd = aligned[3].y;

    let d = -pa + 3 * pb - 3 * pc + pd,
      a = 3 * pa - 6 * pb + 3 * pc,
      b = -3 * pa + 3 * pb,
      c = pa;

    if (utils.approximately(d, 0)) {
      // this is not a cubic curve.
      if (utils.approximately(a, 0)) {
        // in fact, this is not a quadratic curve either.
        if (utils.approximately(b, 0)) {
          // in fact in fact, there are no solutions.
          return [];
        }
        // linear solution:
        return [-c / b].filter(reduce);
      }
      // quadratic solution:
      const q = sqrt(b * b - 4 * a * c),
        a2 = 2 * a;
      return [(q - b) / a2, (-b - q) / a2].filter(reduce);
    }

    // at this point, we know we need a cubic solution:

    a /= d;
    b /= d;
    c /= d;

    const p = (3 * b - a * a) / 3,
      p3 = p / 3,
      q = (2 * a * a * a - 9 * a * b + 27 * c) / 27,
      q2 = q / 2,
      discriminant = q2 * q2 + p3 * p3 * p3;

    let u1, v1, x1, x2, x3;
    if (discriminant < 0) {
      const mp3 = -p / 3,
        mp33 = mp3 * mp3 * mp3,
        r = sqrt(mp33),
        t = -q / (2 * r),
        cosphi = t < -1 ? -1 : t > 1 ? 1 : t,
        phi = acos(cosphi),
        crtr = crt(r),
        t1 = 2 * crtr;
      x1 = t1 * cos(phi / 3) - a / 3;
      x2 = t1 * cos((phi + tau) / 3) - a / 3;
      x3 = t1 * cos((phi + 2 * tau) / 3) - a / 3;
      return [x1, x2, x3].filter(reduce);
    } else if (discriminant === 0) {
      u1 = q2 < 0 ? crt(-q2) : -crt(q2);
      x1 = 2 * u1 - a / 3;
      x2 = -u1 - a / 3;
      return [x1, x2].filter(reduce);
    } else {
      const sd = sqrt(discriminant);
      u1 = crt(-q2 + sd);
      v1 = crt(q2 + sd);
      return [u1 - v1 - a / 3].filter(reduce);
    }
  },

  droots: function (p) {
    // quadratic roots are easy
    if (p.length === 3) {
      const a = p[0],
        b = p[1],
        c = p[2],
        d = a - 2 * b + c;
      if (d !== 0) {
        const m1 = -sqrt(b * b - a * c),
          m2 = -a + b,
          v1 = -(m1 + m2) / d,
          v2 = -(-m1 + m2) / d;
        return [v1, v2];
      } else if (b !== c && d === 0) {
        return [(2 * b - c) / (2 * (b - c))];
      }
      return [];
    }

    // linear roots are even easier
    if (p.length === 2) {
      const a = p[0],
        b = p[1];
      if (a !== b) {
        return [a / (a - b)];
      }
      return [];
    }

    return [];
  },

  curvature: function (t, d1, d2, _3d, kOnly) {
    let num,
      dnm,
      adk,
      dk,
      k = 0,
      r = 0;

    //
    // We're using the following formula for curvature:
    //
    //              x'y" - y'x"
    //   k(t) = ------------------
    //           (x'² + y'²)^(3/2)
    //
    // from https://en.wikipedia.org/wiki/Radius_of_curvature#Definition
    //
    // With it corresponding 3D counterpart:
    //
    //          sqrt( (y'z" - y"z')² + (z'x" - z"x')² + (x'y" - x"y')²)
    //   k(t) = -------------------------------------------------------
    //                     (x'² + y'² + z'²)^(3/2)
    //

    const d = utils.compute(t, d1);
    const dd = utils.compute(t, d2);
    const qdsum = d.x * d.x + d.y * d.y;

    if (_3d) {
      num = sqrt(pow(d.y * dd.z - dd.y * d.z, 2) + pow(d.z * dd.x - dd.z * d.x, 2) + pow(d.x * dd.y - dd.x * d.y, 2));
      dnm = pow(qdsum + d.z * d.z, 3 / 2);
    } else {
      num = d.x * dd.y - d.y * dd.x;
      dnm = pow(qdsum, 3 / 2);
    }

    if (num === 0 || dnm === 0) {
      return { k: 0, r: 0 };
    }

    k = num / dnm;
    r = dnm / num;

    // We're also computing the derivative of kappa, because
    // there is value in knowing the rate of change for the
    // curvature along the curve. And we're just going to
    // ballpark it based on an epsilon.
    if (!kOnly) {
      // compute k'(t) based on the interval before, and after it,
      // to at least try to not introduce forward/backward pass bias.
      const pk = utils.curvature(t - 0.001, d1, d2, _3d, true).k;
      const nk = utils.curvature(t + 0.001, d1, d2, _3d, true).k;
      dk = (nk - k + (k - pk)) / 2;
      adk = (abs(nk - k) + abs(k - pk)) / 2;
    }

    return { k: k, r: r, dk: dk, adk: adk };
  },

  inflections: function (points) {
    if (points.length < 4) return [];

    // FIXME: TODO: add in inflection abstraction for quartic+ curves?

    const p = utils.align(points, { p1: points[0], p2: points.slice(-1)[0] }),
      a = p[2].x * p[1].y,
      b = p[3].x * p[1].y,
      c = p[1].x * p[2].y,
      d = p[3].x * p[2].y,
      v1 = 18 * (-3 * a + 2 * b + 3 * c - d),
      v2 = 18 * (3 * a - b - 3 * c),
      v3 = 18 * (c - a);

    if (utils.approximately(v1, 0)) {
      if (!utils.approximately(v2, 0)) {
        let t = -v3 / v2;
        if (0 <= t && t <= 1) return [t];
      }
      return [];
    }

    const trm = v2 * v2 - 4 * v1 * v3,
      sq = Math.sqrt(trm),
      d2 = 2 * v1;

    if (utils.approximately(d2, 0)) return [];

    return [(sq - v2) / d2, -(v2 + sq) / d2].filter(function (r) {
      return 0 <= r && r <= 1;
    });
  },

  bboxoverlap: function (b1, b2) {
    const dims = ["x", "y"],
      len = dims.length;

    for (let i = 0, dim, l, t, d; i < len; i++) {
      dim = dims[i];
      l = b1[dim].mid;
      t = b2[dim].mid;
      d = (b1[dim].size + b2[dim].size) / 2;
      if (abs(l - t) >= d) return false;
    }
    return true;
  },

  expandbox: function (bbox, _bbox) {
    if (_bbox.x.min < bbox.x.min) {
      bbox.x.min = _bbox.x.min;
    }
    if (_bbox.y.min < bbox.y.min) {
      bbox.y.min = _bbox.y.min;
    }
    if (_bbox.z && _bbox.z.min < bbox.z.min) {
      bbox.z.min = _bbox.z.min;
    }
    if (_bbox.x.max > bbox.x.max) {
      bbox.x.max = _bbox.x.max;
    }
    if (_bbox.y.max > bbox.y.max) {
      bbox.y.max = _bbox.y.max;
    }
    if (_bbox.z && _bbox.z.max > bbox.z.max) {
      bbox.z.max = _bbox.z.max;
    }
    bbox.x.mid = (bbox.x.min + bbox.x.max) / 2;
    bbox.y.mid = (bbox.y.min + bbox.y.max) / 2;
    if (bbox.z) {
      bbox.z.mid = (bbox.z.min + bbox.z.max) / 2;
    }
    bbox.x.size = bbox.x.max - bbox.x.min;
    bbox.y.size = bbox.y.max - bbox.y.min;
    if (bbox.z) {
      bbox.z.size = bbox.z.max - bbox.z.min;
    }
  },

  pairiteration: function (c1, c2, curveIntersectionThreshold) {
    const c1b = c1.bbox(),
      c2b = c2.bbox(),
      r = 100000,
      threshold = curveIntersectionThreshold || 0.5;

    if (c1b.x.size + c1b.y.size < threshold && c2b.x.size + c2b.y.size < threshold) {
      return [(((r * (c1._t1 + c1._t2)) / 2) | 0) / r + "/" + (((r * (c2._t1 + c2._t2)) / 2) | 0) / r];
    }

    let cc1 = c1.split(0.5),
      cc2 = c2.split(0.5),
      pairs = [
        { left: cc1.left, right: cc2.left },
        { left: cc1.left, right: cc2.right },
        { left: cc1.right, right: cc2.right },
        { left: cc1.right, right: cc2.left },
      ];

    pairs = pairs.filter(function (pair) {
      return utils.bboxoverlap(pair.left.bbox(), pair.right.bbox());
    });

    let results = [];

    if (pairs.length === 0) return results;

    pairs.forEach(function (pair) {
      results = results.concat(utils.pairiteration(pair.left, pair.right, threshold));
    });

    results = results.filter(function (v, i) {
      return results.indexOf(v) === i;
    });

    return results;
  },

  getccenter: function (p1, p2, p3) {
    const dx1 = p2.x - p1.x,
      dy1 = p2.y - p1.y,
      dx2 = p3.x - p2.x,
      dy2 = p3.y - p2.y,
      dx1p = dx1 * cos(quart) - dy1 * sin(quart),
      dy1p = dx1 * sin(quart) + dy1 * cos(quart),
      dx2p = dx2 * cos(quart) - dy2 * sin(quart),
      dy2p = dx2 * sin(quart) + dy2 * cos(quart),
      // chord midpoints
      mx1 = (p1.x + p2.x) / 2,
      my1 = (p1.y + p2.y) / 2,
      mx2 = (p2.x + p3.x) / 2,
      my2 = (p2.y + p3.y) / 2,
      // midpoint offsets
      mx1n = mx1 + dx1p,
      my1n = my1 + dy1p,
      mx2n = mx2 + dx2p,
      my2n = my2 + dy2p,
      // intersection of these lines:
      arc = utils.lli8(mx1, my1, mx1n, my1n, mx2, my2, mx2n, my2n),
      r = utils.dist(arc, p1);

    // arc start/end values, over mid point:
    let s = atan2(p1.y - arc.y, p1.x - arc.x),
      m = atan2(p2.y - arc.y, p2.x - arc.x),
      e = atan2(p3.y - arc.y, p3.x - arc.x),
      _;

    // determine arc direction (cw/ccw correction)
    if (s < e) {
      // if s<m<e, arc(s, e)
      // if m<s<e, arc(e, s + tau)
      // if s<e<m, arc(e, s + tau)
      if (s > m || m > e) {
        s += tau;
      }
      if (s > e) {
        _ = e;
        e = s;
        s = _;
      }
    } else {
      // if e<m<s, arc(e, s)
      // if m<e<s, arc(s, e + tau)
      // if e<s<m, arc(s, e + tau)
      if (e < m && m < s) {
        _ = e;
        e = s;
        s = _;
      } else {
        e += tau;
      }
    }
    // assign and done.
    arc.s = s;
    arc.e = e;
    arc.r = r;
    return arc;
  },

  numberSort: function (a, b) {
    return a - b;
  },
};

export { utils };