mathjs/lib/function/geometry/intersect.js

'use strict';

function factory (type, config, load, typed) {
  var matrix = load(require('../../type/matrix/function/matrix'));

  /**
   * Calculates the point of intersection of two lines in two or three dimensions
   * and of a line and a plane in three dimensions. The inputs are in the form of
   * arrays or 1 dimensional matrices. The line intersection functions return null
   * if the lines do not meet.
   *
   * Note: Fill the plane coefficients as `x + y + z = c` and not as `x + y + z + c = 0`.
   *
   * Syntax:
   *
   *    math.intersect(endPoint1Line1, endPoint2Line1, endPoint1Line2, endPoint2Line2)
   *    math.intersect(endPoint1, endPoint2, planeCoefficients)
   *
   * Examples:
   *
   *    math.intersect([0, 0], [10, 10], [10, 0], [0, 10]);              // Returns [5, 5]
   *    math.intersect([0, 0, 0], [10, 10, 0], [10, 0, 0], [0, 10, 0]);  // Returns [5, 5, 0]
   *    math.intersect([1, 0, 1],  [4, -2, 2], [1, 1, 1, 6]);            // Returns [7, -4, 3]
   *
   * @param  {Array | Matrix} w   Co-ordinates of first end-point of first line
   * @param  {Array | Matrix} x   Co-ordinates of second end-point of first line
   * @param  {Array | Matrix} y   Co-ordinates of first end-point of second line
   *                              OR Co-efficients of the plane's equation
   * @param  {Array | Matrix} z   Co-ordinates of second end-point of second line
   *                              OR null if the calculation is for line and plane
   * @return {Array}              Returns the point of intersection of lines/lines-planes
   */
  var intersect = typed('intersect', {
    'Array, Array, Array': function (x, y, plane) {
      if (!_3d(x)) { throw new TypeError('Array with 3 numbers expected for first argument'); }
      if (!_3d(y)) { throw new TypeError('Array with 3 numbers expected for second argument'); }
      if (!_4d(plane)) { throw new TypeError('Array with 4 numbers expected as third argument'); }

      return _intersectLinePlane(x[0], x[1], x[2], y[0], y[1], y[2], plane[0], plane[1], plane[2], plane[3]);
    },

    'Array, Array, Array, Array': function (w, x, y, z) {
      if (w.length === 2) {
        if (!_2d(w)) { throw new TypeError('Array with 2 numbers expected for first argument'); }
        if (!_2d(x)) { throw new TypeError('Array with 2 numbers expected for second argument'); }
        if (!_2d(y)) { throw new TypeError('Array with 2 numbers expected for third argument'); }
        if (!_2d(z)) { throw new TypeError('Array with 2 numbers expected for fourth argument'); }

        return _intersect2d(w[0], w[1], x[0], x[1], y[0], y[1], z[0], z[1]);
      }
      else if (w.length === 3) {
        if (!_3d(w)) { throw new TypeError('Array with 3 numbers expected for first argument'); }
        if (!_3d(x)) { throw new TypeError('Array with 3 numbers expected for second argument'); }
        if (!_3d(y)) { throw new TypeError('Array with 3 numbers expected for third argument'); }
        if (!_3d(z)) { throw new TypeError('Array with 3 numbers expected for fourth argument'); }

        return _intersect3d(w[0], w[1], w[2], x[0], x[1], x[2], y[0], y[1], y[2], z[0], z[1], z[2]);
      }
      else {
        throw new TypeError('Arrays with two or thee dimensional points expected');
      }
    },

    'Matrix, Matrix, Matrix': function (x, y, plane) {
      return matrix(intersect(x.valueOf(), y.valueOf(), plane.valueOf()));
    },

    'Matrix, Matrix, Matrix, Matrix': function (w, x, y, z) {
      // TODO: output matrix type should match input matrix type
      return matrix(intersect(w.valueOf(), x.valueOf(), y.valueOf(), z.valueOf()));
    }
  });

  return intersect;
}

function _2d(x) {
  return x.length === 2 && typeof x[0] === 'number' && typeof x[1] === 'number';
}

function _3d(x) {
  return x.length === 3 && typeof x[0] === 'number' && typeof x[1] === 'number' && typeof x[2] === 'number';
}

function _4d(x) {
  return x.length === 4 && typeof x[0] === 'number' && typeof x[1] === 'number' && typeof x[2] === 'number' && typeof x[3] === 'number';
}

function _intersect2d(x1, y1, x2, y2, x3, y3, x4, y4){
  var d1343 = (x1 - x3)*(x4 - x3) + (y1 - y3)*(y4 - y3);
  var d4321 = (x4 - x3)*(x2 - x1) + (y4 - y3)*(y2 - y1);
  var d1321 = (x1 - x3)*(x2 - x1) + (y1 - y3)*(y2 - y1);
  var d4343 = (x4 - x3)*(x4 - x3) + (y4 - y3)*(y4 - y3);
  var d2121 = (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1);
  var ta = ( d1343*d4321 - d1321*d4343 ) / ( d2121*d4343 - d4321*d4321 );
  var tb = ( d1343 + ta * d4321 ) / (d4343);

  var pax = x1 + ta * (x2 - x1);
  var pay = y1 + ta * (y2 - y1);
  var pbx = x3 + tb * (x4 - x3);
  var pby = y3 + tb * (y4 - y3);
  if (pax === pbx && pay === pby){
    return [pax, pay];
  }
  else{
    return null;
  }
}

function _intersect3d(x1, y1, z1, x2, y2, z2, x3, y3, z3, x4, y4, z4){
  var d1343 = (x1 - x3)*(x4 - x3) + (y1 - y3)*(y4 - y3) + (z1 - z3)*(z4 - z3);
  var d4321 = (x4 - x3)*(x2 - x1) + (y4 - y3)*(y2 - y1) + (z4 - z3)*(z2 - z1);
  var d1321 = (x1 - x3)*(x2 - x1) + (y1 - y3)*(y2 - y1) + (z1 - z3)*(z2 - z1);
  var d4343 = (x4 - x3)*(x4 - x3) + (y4 - y3)*(y4 - y3) + (z4 - z3)*(z4 - z3);
  var d2121 = (x2 - x1)*(x2 - x1) + (y2 - y1)*(y2 - y1) + (z2 - z1)*(z2 - z1);
  var ta = ( d1343*d4321 - d1321*d4343 ) / ( d2121*d4343 - d4321*d4321 );
  var tb = ( d1343 + ta * d4321 ) / (d4343);

  var pax = x1 + ta * (x2 - x1);
  var pay = y1 + ta * (y2 - y1);
  var paz = z1 + ta * (z2 - z1);
  var pbx = x3 + tb * (x4 - x3);
  var pby = y3 + tb * (y4 - y3);
  var pbz = z3 + tb * (z4 - z3);
  if (pax === pbx && pay === pby && paz === pbz){
    return [pax, pay, paz];
  }
  else{
    return null;
  }
}

function _intersectLinePlane(x1, y1, z1, x2, y2, z2, x, y, z, c){
  var t = (c - x1*x - y1*y - z1*z)/(x2*x + y2*y + z2*z - x1 - y1 - z1);
  var px = x1 + t * (x2 - x1);
  var py = y1 + t * (y2 - y1);
  var pz = z1 + t * (z2 - z1);
  return [px, py, pz];
  // TODO: Add cases when line is parallel to the plane:
  //       (a) no intersection,
  //       (b) line contained in plane
}

exports.name = 'intersect';
exports.factory = factory;