@cesium/engine/Source/Core/QuarticRealPolynomial.js

CesiumJS is a JavaScript library for creating 3D globes and 2D maps in a web browser without a plugin.

import CubicRealPolynomial from "./CubicRealPolynomial.js";
import DeveloperError from "./DeveloperError.js";
import CesiumMath from "./Math.js";
import QuadraticRealPolynomial from "./QuadraticRealPolynomial.js";

/**
 * Defines functions for 4th order polynomial functions of one variable with only real coefficients.
 *
 * @namespace QuarticRealPolynomial
 */
const QuarticRealPolynomial = {};

/**
 * Provides the discriminant of the quartic equation from the supplied coefficients.
 *
 * @param {number} a The coefficient of the 4th order monomial.
 * @param {number} b The coefficient of the 3rd order monomial.
 * @param {number} c The coefficient of the 2nd order monomial.
 * @param {number} d The coefficient of the 1st order monomial.
 * @param {number} e The coefficient of the 0th order monomial.
 * @returns {number} The value of the discriminant.
 */
QuarticRealPolynomial.computeDiscriminant = function (a, b, c, d, e) {
  //>>includeStart('debug', pragmas.debug);
  if (typeof a !== "number") {
    throw new DeveloperError("a is a required number.");
  }
  if (typeof b !== "number") {
    throw new DeveloperError("b is a required number.");
  }
  if (typeof c !== "number") {
    throw new DeveloperError("c is a required number.");
  }
  if (typeof d !== "number") {
    throw new DeveloperError("d is a required number.");
  }
  if (typeof e !== "number") {
    throw new DeveloperError("e is a required number.");
  }
  //>>includeEnd('debug');

  const a2 = a * a;
  const a3 = a2 * a;
  const b2 = b * b;
  const b3 = b2 * b;
  const c2 = c * c;
  const c3 = c2 * c;
  const d2 = d * d;
  const d3 = d2 * d;
  const e2 = e * e;
  const e3 = e2 * e;

  const discriminant =
    b2 * c2 * d2 -
    4.0 * b3 * d3 -
    4.0 * a * c3 * d2 +
    18 * a * b * c * d3 -
    27.0 * a2 * d2 * d2 +
    256.0 * a3 * e3 +
    e *
      (18.0 * b3 * c * d -
        4.0 * b2 * c3 +
        16.0 * a * c2 * c2 -
        80.0 * a * b * c2 * d -
        6.0 * a * b2 * d2 +
        144.0 * a2 * c * d2) +
    e2 *
      (144.0 * a * b2 * c -
        27.0 * b2 * b2 -
        128.0 * a2 * c2 -
        192.0 * a2 * b * d);
  return discriminant;
};

function original(a3, a2, a1, a0) {
  const a3Squared = a3 * a3;

  const p = a2 - (3.0 * a3Squared) / 8.0;
  const q = a1 - (a2 * a3) / 2.0 + (a3Squared * a3) / 8.0;
  const r =
    a0 -
    (a1 * a3) / 4.0 +
    (a2 * a3Squared) / 16.0 -
    (3.0 * a3Squared * a3Squared) / 256.0;

  // Find the roots of the cubic equations:  h^6 + 2 p h^4 + (p^2 - 4 r) h^2 - q^2 = 0.
  const cubicRoots = CubicRealPolynomial.computeRealRoots(
    1.0,
    2.0 * p,
    p * p - 4.0 * r,
    -q * q,
  );

  if (cubicRoots.length > 0) {
    const temp = -a3 / 4.0;

    // Use the largest positive root.
    const hSquared = cubicRoots[cubicRoots.length - 1];

    if (Math.abs(hSquared) < CesiumMath.EPSILON14) {
      // y^4 + p y^2 + r = 0.
      const roots = QuadraticRealPolynomial.computeRealRoots(1.0, p, r);

      if (roots.length === 2) {
        const root0 = roots[0];
        const root1 = roots[1];

        let y;
        if (root0 >= 0.0 && root1 >= 0.0) {
          const y0 = Math.sqrt(root0);
          const y1 = Math.sqrt(root1);

          return [temp - y1, temp - y0, temp + y0, temp + y1];
        } else if (root0 >= 0.0 && root1 < 0.0) {
          y = Math.sqrt(root0);
          return [temp - y, temp + y];
        } else if (root0 < 0.0 && root1 >= 0.0) {
          y = Math.sqrt(root1);
          return [temp - y, temp + y];
        }
      }
      return [];
    } else if (hSquared > 0.0) {
      const h = Math.sqrt(hSquared);

      const m = (p + hSquared - q / h) / 2.0;
      const n = (p + hSquared + q / h) / 2.0;

      // Now solve the two quadratic factors:  (y^2 + h y + m)(y^2 - h y + n);
      const roots1 = QuadraticRealPolynomial.computeRealRoots(1.0, h, m);
      const roots2 = QuadraticRealPolynomial.computeRealRoots(1.0, -h, n);

      if (roots1.length !== 0) {
        roots1[0] += temp;
        roots1[1] += temp;

        if (roots2.length !== 0) {
          roots2[0] += temp;
          roots2[1] += temp;

          if (roots1[1] <= roots2[0]) {
            return [roots1[0], roots1[1], roots2[0], roots2[1]];
          } else if (roots2[1] <= roots1[0]) {
            return [roots2[0], roots2[1], roots1[0], roots1[1]];
          } else if (roots1[0] >= roots2[0] && roots1[1] <= roots2[1]) {
            return [roots2[0], roots1[0], roots1[1], roots2[1]];
          } else if (roots2[0] >= roots1[0] && roots2[1] <= roots1[1]) {
            return [roots1[0], roots2[0], roots2[1], roots1[1]];
          } else if (roots1[0] > roots2[0] && roots1[0] < roots2[1]) {
            return [roots2[0], roots1[0], roots2[1], roots1[1]];
          }
          return [roots1[0], roots2[0], roots1[1], roots2[1]];
        }
        return roots1;
      }

      if (roots2.length !== 0) {
        roots2[0] += temp;
        roots2[1] += temp;

        return roots2;
      }
      return [];
    }
  }
  return [];
}

function neumark(a3, a2, a1, a0) {
  const a1Squared = a1 * a1;
  const a2Squared = a2 * a2;
  const a3Squared = a3 * a3;

  const p = -2.0 * a2;
  const q = a1 * a3 + a2Squared - 4.0 * a0;
  const r = a3Squared * a0 - a1 * a2 * a3 + a1Squared;

  const cubicRoots = CubicRealPolynomial.computeRealRoots(1.0, p, q, r);

  if (cubicRoots.length > 0) {
    // Use the most positive root
    const y = cubicRoots[0];

    const temp = a2 - y;
    const tempSquared = temp * temp;

    const g1 = a3 / 2.0;
    const h1 = temp / 2.0;

    const m = tempSquared - 4.0 * a0;
    const mError = tempSquared + 4.0 * Math.abs(a0);

    const n = a3Squared - 4.0 * y;
    const nError = a3Squared + 4.0 * Math.abs(y);

    let g2;
    let h2;

    if (y < 0.0 || m * nError < n * mError) {
      const squareRootOfN = Math.sqrt(n);
      g2 = squareRootOfN / 2.0;
      h2 = squareRootOfN === 0.0 ? 0.0 : (a3 * h1 - a1) / squareRootOfN;
    } else {
      const squareRootOfM = Math.sqrt(m);
      g2 = squareRootOfM === 0.0 ? 0.0 : (a3 * h1 - a1) / squareRootOfM;
      h2 = squareRootOfM / 2.0;
    }

    let G;
    let g;
    if (g1 === 0.0 && g2 === 0.0) {
      G = 0.0;
      g = 0.0;
    } else if (CesiumMath.sign(g1) === CesiumMath.sign(g2)) {
      G = g1 + g2;
      g = y / G;
    } else {
      g = g1 - g2;
      G = y / g;
    }

    let H;
    let h;
    if (h1 === 0.0 && h2 === 0.0) {
      H = 0.0;
      h = 0.0;
    } else if (CesiumMath.sign(h1) === CesiumMath.sign(h2)) {
      H = h1 + h2;
      h = a0 / H;
    } else {
      h = h1 - h2;
      H = a0 / h;
    }

    // Now solve the two quadratic factors:  (y^2 + G y + H)(y^2 + g y + h);
    const roots1 = QuadraticRealPolynomial.computeRealRoots(1.0, G, H);
    const roots2 = QuadraticRealPolynomial.computeRealRoots(1.0, g, h);

    if (roots1.length !== 0) {
      if (roots2.length !== 0) {
        if (roots1[1] <= roots2[0]) {
          return [roots1[0], roots1[1], roots2[0], roots2[1]];
        } else if (roots2[1] <= roots1[0]) {
          return [roots2[0], roots2[1], roots1[0], roots1[1]];
        } else if (roots1[0] >= roots2[0] && roots1[1] <= roots2[1]) {
          return [roots2[0], roots1[0], roots1[1], roots2[1]];
        } else if (roots2[0] >= roots1[0] && roots2[1] <= roots1[1]) {
          return [roots1[0], roots2[0], roots2[1], roots1[1]];
        } else if (roots1[0] > roots2[0] && roots1[0] < roots2[1]) {
          return [roots2[0], roots1[0], roots2[1], roots1[1]];
        }
        return [roots1[0], roots2[0], roots1[1], roots2[1]];
      }
      return roots1;
    }
    if (roots2.length !== 0) {
      return roots2;
    }
  }
  return [];
}

/**
 * Provides the real valued roots of the quartic polynomial with the provided coefficients.
 *
 * @param {number} a The coefficient of the 4th order monomial.
 * @param {number} b The coefficient of the 3rd order monomial.
 * @param {number} c The coefficient of the 2nd order monomial.
 * @param {number} d The coefficient of the 1st order monomial.
 * @param {number} e The coefficient of the 0th order monomial.
 * @returns {number[]} The real valued roots.
 */
QuarticRealPolynomial.computeRealRoots = function (a, b, c, d, e) {
  //>>includeStart('debug', pragmas.debug);
  if (typeof a !== "number") {
    throw new DeveloperError("a is a required number.");
  }
  if (typeof b !== "number") {
    throw new DeveloperError("b is a required number.");
  }
  if (typeof c !== "number") {
    throw new DeveloperError("c is a required number.");
  }
  if (typeof d !== "number") {
    throw new DeveloperError("d is a required number.");
  }
  if (typeof e !== "number") {
    throw new DeveloperError("e is a required number.");
  }
  //>>includeEnd('debug');

  if (Math.abs(a) < CesiumMath.EPSILON15) {
    return CubicRealPolynomial.computeRealRoots(b, c, d, e);
  }
  const a3 = b / a;
  const a2 = c / a;
  const a1 = d / a;
  const a0 = e / a;

  let k = a3 < 0.0 ? 1 : 0;
  k += a2 < 0.0 ? k + 1 : k;
  k += a1 < 0.0 ? k + 1 : k;
  k += a0 < 0.0 ? k + 1 : k;

  switch (k) {
    case 0:
      return original(a3, a2, a1, a0);
    case 1:
      return neumark(a3, a2, a1, a0);
    case 2:
      return neumark(a3, a2, a1, a0);
    case 3:
      return original(a3, a2, a1, a0);
    case 4:
      return original(a3, a2, a1, a0);
    case 5:
      return neumark(a3, a2, a1, a0);
    case 6:
      return original(a3, a2, a1, a0);
    case 7:
      return original(a3, a2, a1, a0);
    case 8:
      return neumark(a3, a2, a1, a0);
    case 9:
      return original(a3, a2, a1, a0);
    case 10:
      return original(a3, a2, a1, a0);
    case 11:
      return neumark(a3, a2, a1, a0);
    case 12:
      return original(a3, a2, a1, a0);
    case 13:
      return original(a3, a2, a1, a0);
    case 14:
      return original(a3, a2, a1, a0);
    case 15:
      return original(a3, a2, a1, a0);
    default:
      return undefined;
  }
};
export default QuarticRealPolynomial;