tetrahedralLee.js

import {
  geoProjection as projection,
  geoStereographicRaw,
  geoCentroid
} from "d3-geo";
import { greatest } from "d3-array";
import { abs, asin, degrees, sqrt } from "./math.js";
import {
  complexAdd,
  complexMul,
  complexNorm,
  complexPow,
  complexSub,
} from "./complex.js";
import { solve2d } from "./newton.js";
import voronoi from "./polyhedral/voronoi.js";

export function leeRaw(lambda, phi) {
  const w = [-1 / 2, sqrt(3) / 2];
  let k = [0, 0],
    h = [0, 0],
    z = complexMul(geoStereographicRaw(lambda, phi), [sqrt(2), 0]);

  // rotate to have s ~= 1
  const sector = greatest([0, 1, 2], (i) => complexMul(z, complexPow(w, [i, 0]))[0]);
  const rot = complexPow(w, [sector, 0]);

  const n = complexNorm(z);

  if (n > 0.3) {
    // if |z| > 0.5, use the approx based on y = (1-z)
    // McIlroy formula 6 p6 and table for G page 16
    const y = complexSub([1, 0], complexMul(rot, z));

    // w1 = gamma(1/3) * gamma(1/2) / 3 / gamma(5/6);
    // https://bl.ocks.org/Fil/1aeff1cfda7188e9fbf037d8e466c95c
    const w1 = 1.4021821053254548;

    const G0 = [
      1.15470053837925, 0.192450089729875, 0.0481125224324687,
      0.010309826235529, 3.34114739114366e-4, -1.50351632601465e-3,
      -1.2304417796231e-3, -6.75190201960282e-4, -2.84084537293856e-4,
      -8.21205120500051e-5, -1.59257630018706e-6, 1.91691805888369e-5,
      1.73095888028726e-5, 1.03865580818367e-5, 4.70614523937179e-6,
      1.4413500104181e-6, 1.92757960170179e-8, -3.82869799649063e-7,
      -3.57526015225576e-7, -2.2175964844211e-7,
    ];

    let G = [0, 0];
    for (let i = G0.length; i--; ) {
      G = complexAdd([G0[i], 0], complexMul(G, y));
    }

    k = complexSub([w1, 0], complexMul(complexPow(y, 1 / 2), G));
    k = complexMul(k, rot);
    k = complexMul(k, rot);
  }

  if (n < 0.5) {
    // if |z| < 0.3
    // https://www.wolframalpha.com/input/?i=series+of+((1-z%5E3))+%5E+(-1%2F2)+at+z%3D0 (and ask for "more terms")
    // 1 + z^3/2 + (3 z^6)/8 + (5 z^9)/16 + (35 z^12)/128 + (63 z^15)/256 + (231 z^18)/1024 + O(z^21)
    // https://www.wolframalpha.com/input/?i=integral+of+1+%2B+z%5E3%2F2+%2B+(3+z%5E6)%2F8+%2B+(5+z%5E9)%2F16+%2B+(35+z%5E12)%2F128+%2B+(63+z%5E15)%2F256+%2B+(231+z%5E18)%2F1024
    // (231 z^19)/19456 + (63 z^16)/4096 + (35 z^13)/1664 + z^10/32 + (3 z^7)/56 + z^4/8 + z + constant
    const H0 = [1, 1 / 8, 3 / 56, 1 / 32, 35 / 1664, 63 / 4096, 231 / 19456];
    const z3 = complexPow(z, [3, 0]);
    for (let i = H0.length; i--; ) h = complexAdd([H0[i], 0], complexMul(h, z3));
    h = complexMul(h, z);
  }

  if (n < 0.3) return h;
  if (n > 0.5) return k;

  // in between 0.3 and 0.5, interpolate
  const t = (n - 0.3) / (0.5 - 0.3);
  return complexAdd(complexMul(k, [t, 0]), complexMul(h, [1 - t, 0]));
}

const leeSolver = solve2d(leeRaw);
leeRaw.invert = function (x, y) {
  if (x > 1.5) return false; // immediately avoid using the wrong face
  const p = leeSolver(x, y, x, y * 0.5);
  const q = leeRaw(p[0], p[1]);
  q[0] -= x;
  q[1] -= y;
  return (q[0] * q[0] + q[1] * q[1] < 1e-8)
    ? p
    : [-10, 0]; // far out of the face
};

const asin1_3 = asin(1 / 3);
const centers = [
  [0, 90],
  [-180, -asin1_3 * degrees],
  [-60, -asin1_3 * degrees],
  [60, -asin1_3 * degrees],
];
const tetrahedron = [
  [1, 2, 3],
  [0, 2, 1],
  [0, 3, 2],
  [0, 1, 3],
].map((face) => face.map((i) => centers[i]));

export default function (
  faceProjection = (face) => {
    const c = geoCentroid({ type: "MultiPoint", coordinates: face });
    const rotate = abs(c[1]) == 90 ? [0, -c[1], -30] : [-c[0], -c[1], 30];
    return projection(leeRaw).scale(1).translate([0, 0]).rotate(rotate);
  }
) {
  return voronoi([-1, 0, 0, 0], {
    features: tetrahedron.map((t) => ({
      type: "Feature",
      geometry: {type: "Polygon", coordinates: [[...t, t[0]]]}
    }))
  }, faceProjection)
    .rotate([30, 180]) // North Pole aspect
    .angle(30)
    .scale(118.662)
    .translate([480, 195.47]);
}