proj4/lib/common/vincenty.js

Proj4js is a JavaScript library to transform point coordinates from one coordinate system to another, including datum transformations.

/**
 * Calculates the inverse geodesic problem using Vincenty's formulae.
 * Computes the forward azimuth and ellipsoidal distance between two points
 * specified by latitude and longitude on the surface of an ellipsoid.
 *
 * @param {number} lat1 Latitude of the first point in radians.
 * @param {number} lon1 Longitude of the first point in radians.
 * @param {number} lat2 Latitude of the second point in radians.
 * @param {number} lon2 Longitude of the second point in radians.
 * @param {number} a Semi-major axis of the ellipsoid (meters).
 * @param {number} f Flattening of the ellipsoid.
 * @returns {{ azi1: number, s12: number }} An object containing:
 *   - azi1: Forward azimuth from the first point to the second point (radians).
 *   - s12: Ellipsoidal distance between the two points (meters).
 */
export function vincentyInverse(lat1, lon1, lat2, lon2, a, f) {
  const L = lon2 - lon1;
  const U1 = Math.atan((1 - f) * Math.tan(lat1));
  const U2 = Math.atan((1 - f) * Math.tan(lat2));
  const sinU1 = Math.sin(U1), cosU1 = Math.cos(U1);
  const sinU2 = Math.sin(U2), cosU2 = Math.cos(U2);

  let lambda = L, lambdaP, iterLimit = 100;
  let sinLambda, cosLambda, sinSigma, cosSigma, sigma, sinAlpha, cos2Alpha, cos2SigmaM, C;
  let uSq, A, B, deltaSigma, s;

  do {
    sinLambda = Math.sin(lambda);
    cosLambda = Math.cos(lambda);
    sinSigma = Math.sqrt(
      (cosU2 * sinLambda) * (cosU2 * sinLambda)
      + (cosU1 * sinU2 - sinU1 * cosU2 * cosLambda)
      * (cosU1 * sinU2 - sinU1 * cosU2 * cosLambda)
    );
    if (sinSigma === 0) {
      return { azi1: 0, s12: 0 }; // coincident points
    }
    cosSigma = sinU1 * sinU2 + cosU1 * cosU2 * cosLambda;
    sigma = Math.atan2(sinSigma, cosSigma);
    sinAlpha = cosU1 * cosU2 * sinLambda / sinSigma;
    cos2Alpha = 1 - sinAlpha * sinAlpha;
    cos2SigmaM = (cos2Alpha !== 0) ? (cosSigma - 2 * sinU1 * sinU2 / cos2Alpha) : 0;
    C = f / 16 * cos2Alpha * (4 + f * (4 - 3 * cos2Alpha));
    lambdaP = lambda;
    lambda = L + (1 - C) * f * sinAlpha
    * (sigma + C * sinSigma * (cos2SigmaM + C * cosSigma * (-1 + 2 * cos2SigmaM * cos2SigmaM)));
  } while (Math.abs(lambda - lambdaP) > 1e-12 && --iterLimit > 0);

  if (iterLimit === 0) {
    return { azi1: NaN, s12: NaN }; // formula failed to converge
  }

  uSq = cos2Alpha * (a * a - (a * (1 - f)) * (a * (1 - f))) / ((a * (1 - f)) * (a * (1 - f)));
  A = 1 + uSq / 16384 * (4096 + uSq * (-768 + uSq * (320 - 175 * uSq)));
  B = uSq / 1024 * (256 + uSq * (-128 + uSq * (74 - 47 * uSq)));
  deltaSigma = B * sinSigma * (cos2SigmaM + B / 4 * (cosSigma * (-1 + 2 * cos2SigmaM * cos2SigmaM)
    - B / 6 * cos2SigmaM * (-3 + 4 * sinSigma * sinSigma) * (-3 + 4 * cos2SigmaM * cos2SigmaM)));

  s = (a * (1 - f)) * A * (sigma - deltaSigma);

  // Forward azimuth
  const azi1 = Math.atan2(cosU2 * sinLambda, cosU1 * sinU2 - sinU1 * cosU2 * cosLambda);

  return { azi1, s12: s };
}

/**
 * Solves the direct geodetic problem using Vincenty's formulae.
 * Given a starting point, initial azimuth, and distance, computes the destination point on the ellipsoid.
 *
 * @param {number} lat1 Latitude of the starting point in radians.
 * @param {number} lon1 Longitude of the starting point in radians.
 * @param {number} azi1 Initial azimuth (forward azimuth) in radians.
 * @param {number} s12 Distance to travel from the starting point in meters.
 * @param {number} a Semi-major axis of the ellipsoid in meters.
 * @param {number} f Flattening of the ellipsoid.
 * @returns {{lat2: number, lon2: number}} The latitude and longitude (in radians) of the destination point.
 */
export function vincentyDirect(lat1, lon1, azi1, s12, a, f) {
  const U1 = Math.atan((1 - f) * Math.tan(lat1));
  const sinU1 = Math.sin(U1), cosU1 = Math.cos(U1);
  const sinAlpha1 = Math.sin(azi1), cosAlpha1 = Math.cos(azi1);

  const sigma1 = Math.atan2(sinU1, cosU1 * cosAlpha1);
  const sinAlpha = cosU1 * sinAlpha1;
  const cos2Alpha = 1 - sinAlpha * sinAlpha;
  const uSq = cos2Alpha * (a * a - (a * (1 - f)) * (a * (1 - f))) / ((a * (1 - f)) * (a * (1 - f)));
  const A = 1 + uSq / 16384 * (4096 + uSq * (-768 + uSq * (320 - 175 * uSq)));
  const B = uSq / 1024 * (256 + uSq * (-128 + uSq * (74 - 47 * uSq)));

  let sigma = s12 / ((a * (1 - f)) * A), sigmaP, iterLimit = 100;
  let cos2SigmaM, sinSigma, cosSigma, deltaSigma;

  do {
    cos2SigmaM = Math.cos(2 * sigma1 + sigma);
    sinSigma = Math.sin(sigma);
    cosSigma = Math.cos(sigma);
    deltaSigma = B * sinSigma * (cos2SigmaM + B / 4 * (cosSigma * (-1 + 2 * cos2SigmaM * cos2SigmaM)
      - B / 6 * cos2SigmaM * (-3 + 4 * sinSigma * sinSigma) * (-3 + 4 * cos2SigmaM * cos2SigmaM)));
    sigmaP = sigma;
    sigma = s12 / ((a * (1 - f)) * A) + deltaSigma;
  } while (Math.abs(sigma - sigmaP) > 1e-12 && --iterLimit > 0);

  if (iterLimit === 0) {
    return { lat2: NaN, lon2: NaN };
  }

  const tmp = sinU1 * sinSigma - cosU1 * cosSigma * cosAlpha1;
  const lat2 = Math.atan2(
    sinU1 * cosSigma + cosU1 * sinSigma * cosAlpha1,
    (1 - f) * Math.sqrt(sinAlpha * sinAlpha + tmp * tmp)
  );
  const lambda = Math.atan2(
    sinSigma * sinAlpha1,
    cosU1 * cosSigma - sinU1 * sinSigma * cosAlpha1
  );
  const C = f / 16 * cos2Alpha * (4 + f * (4 - 3 * cos2Alpha));
  const L = lambda - (1 - C) * f * sinAlpha
    * (sigma + C * sinSigma * (cos2SigmaM + C * cosSigma * (-1 + 2 * cos2SigmaM * cos2SigmaM)));
  const lon2 = lon1 + L;

  return { lat2, lon2 };
}