Intersect.js

var EPSILON = 10e-12;
var TOLERANCE = 10e-6;
var MAX_RECURSE = 10;
var MAX_ITERATE = 20;

/**
 * This method is analogous to paperjs#PathItem.getIntersections
 */
function getIntersections2( path1, path2 ){
    // First check the bounds of the two paths. If they don't intersect,
    // we don't need to iterate through their curves.
    if (!path1.getBounds().touches(path2.getBounds()))
        return [];
    var locations = [],
        curves1 = path1.getCurves(),
        curves2 = path2.getCurves(),
        length2 = curves2.length,
        values2 = [];
    for (var i = 0; i < length2; i++)
        values2[i] = curves2[i].getValues();
    for (var i = 0, l = curves1.length; i < l; i++) {
        var curve1 = curves1[i],
            values1 = curve1.getValues();
        for (var j = 0; j < length2; j++){
            value2 = values2[j];
            var v1Linear = Curve.isLinear(values1);
            var v2Linear = Curve.isLinear(value2);
            if( v1Linear && v2Linear ){
                _getLineLineIntersection(values1, value2, curve1, curves2[j], locations);
            } else if ( v1Linear || v2Linear ){
                _getCurveLineIntersection(values1, value2, curve1, curves2[j], locations);
            } else {
                Curve.getIntersections2(values1, value2, curve1, curves2[j], locations);
            }
        }
    }
    return locations;
}

/**
 * This method is analogous to paperjs#Curve.getIntersections
 * @param  {[type]} v1
 * @param  {[type]} v2
 * @param  {[type]} curve1
 * @param  {[type]} curve2
 * @param  {[type]} locations
 * @param  {[type]} _v1t      - Only used for recusion
 * @param  {[type]} _v2t      - Only used for recusion
 */
paper.Curve.getIntersections2 = function( v1, v2, curve1, curve2, locations, _v1t, _v2t, _recurseDepth ) {
    _recurseDepth = _recurseDepth ? _recurseDepth + 1 : 1;
    // Avoid endless recursion.
    // Perhaps we should fall back to a more expensive method after this, but
    // so far endless recursion happens only when there is no real intersection and
    // the infinite fatline continue to intersect with the other curve outside its bounds!
    if( _recurseDepth > MAX_RECURSE ) return;
    // cache the original parameter range.
    _v1t = _v1t || { t1: 0, t2: 1 };
    _v2t = _v2t || { t1: 0, t2: 1 };
    var v1t = { t1: _v1t.t1, t2: _v1t.t2 };
    var v2t = { t1: _v2t.t1, t2: _v2t.t2 };
    // Get the clipped parts from the original curve, to avoid cumulative errors
    var _v1 = Curve.getPart( v1, v1t.t1, v1t.t2 );
    var _v2 = Curve.getPart( v2, v2t.t1, v2t.t2 );
// markCurve( _v1, '#f0f', true );
// markCurve( _v2, '#0ff', false );
    var nuT, parts, tmpt = { t1:null, t2:null }, iterate = 0;
    // Loop until both parameter range converge. We have to handle the degenerate case
    // seperately, where fat-line clipping can become numerically unstable when one of the
    // curves has converged to a point and the other hasn't.
    while( iterate < MAX_ITERATE &&
        ( Math.abs(v1t.t2 - v1t.t1) > TOLERANCE || Math.abs(v2t.t2 - v2t.t1) > TOLERANCE ) ){
        ++iterate;
        // First we clip v2 with v1's fat-line
        tmpt.t1 = v2t.t1; tmpt.t2 = v2t.t2;
        var intersects1 = _clipBezierFatLine( _v1, _v2, tmpt );
        // Stop if there are no possible intersections
        if( intersects1 === 0 ){
            return;
        } else if( intersects1 > 0 ){
            // Get the clipped parts from the original v2, to avoid cumulative errors
            // ...and reuse some objects.
            v2t.t1 = tmpt.t1; v2t.t2 = tmpt.t2;
            _v2 = Curve.getPart( v2, v2t.t1, v2t.t2 );
// markCurve( _v2, '#0ff', false );
            // Next we clip v1 with nuv2's fat-line
            tmpt.t1 = v1t.t1; tmpt.t2 = v1t.t2;
            var intersects2 = _clipBezierFatLine( _v2, _v1, tmpt );
            // Stop if there are no possible intersections
            if( intersects2 === 0 ){
                return;
            }else if( intersects1 > 0 ){
                // Get the clipped parts from the original v2, to avoid cumulative errors
                v1t.t1 = tmpt.t1; v1t.t2 = tmpt.t2;
                _v1 = Curve.getPart( v1, v1t.t1, v1t.t2 );
            }
// markCurve( _v1, '#f0f', true );
        }
        // Get the clipped parts from the original v1
        // Check if there could be multiple intersections
        if( intersects1 < 0 || intersects2 < 0 ){
            // Subdivide the curve which has converged the least from the original range [0,1],
            // which would be the curve with the largest parameter range after clipping
            if( v1t.t2 - v1t.t1 > v2t.t2 - v2t.t1 ){
                // subdivide _v1 and recurse
                nuT = ( _v1t.t1 + _v1t.t2 ) / 2.0;
                Curve.getIntersections2( v1, v2, curve1, curve2, locations, { t1: _v1t.t1, t2: nuT }, _v2t, _recurseDepth );
                Curve.getIntersections2( v1, v2, curve1, curve2, locations, { t1: nuT, t2: _v1t.t2 }, _v2t, _recurseDepth );
                return;
            } else {
                // subdivide _v2 and recurse
                nuT = ( _v2t.t1 + _v2t.t2 ) / 2.0;
                Curve.getIntersections2( v1, v2, curve1, curve2, locations, _v1t, { t1: _v2t.t1, t2: nuT }, _recurseDepth );
                Curve.getIntersections2( v1, v2, curve1, curve2, locations, _v1t, { t1: nuT, t2: _v2t.t2 }, _recurseDepth );
                return;
            }
        }
        // We need to bailout of clipping and try a numerically stable method if
        // any of the following are true.
        //  1. One of the parameter ranges is converged to a point.
        //  2. Both of the parameter ranges have converged reasonably well ( according to TOLERENCE ).
        //  3. One of the parameter range is converged enough so that it is *flat enough* to
        //      calculate line curve intersection implicitly.
        //
        // Check if one of the parameter range has converged completely to a point.
        // Now things could get only worse if we iterate more for the other
        // curve to converge if it hasn't yet happened so.
        if( Math.abs(v1t.t2 - v1t.t1) < EPSILON ){
            locations.push(new CurveLocation(curve1, v1t.t1, curve1.getPointAt(v1t.t1, true), curve2));
            return;
        }else if( Math.abs(v2t.t2 - v2t.t1) < EPSILON ){
            locations.push(new CurveLocation(curve1, null, curve2.getPointAt(v1t.t1, true), curve2));
            return;
        }
        // Check to see if both parameter ranges have converged or else,
        // see if either or both of the curves are flat enough to be treated as lines
        if( Math.abs(v1t.t2 - v1t.t1) <= TOLERANCE || Math.abs(v2t.t2 - v2t.t1) <= TOLERANCE ){
            locations.push(new CurveLocation(curve1, v1t.t1, curve1.getPointAt(v1t.t1, true), curve2));
            return;
        } else {
            var curve1Flat = Curve.isFlatEnough( _v1, /*#=*/ TOLERANCE );
            var curve2Flat = Curve.isFlatEnough( _v2, /*#=*/ TOLERANCE );
            if ( curve1Flat && curve2Flat ) {
                _getLineLineIntersection( _v1, _v2, curve1, curve2, locations );
                return;
            } else if( curve1Flat || curve2Flat ){
                // Use curve line intersection method while specifying which curve to be treated as line
                _getCurveLineIntersection( _v1, _v2, curve1, curve2, locations, curve1Flat );
                return;
            }
        }
    }
};

/**
 * Clip curve V2 with fat-line of v1
 * @param  {Array}  v1 - Section of the first curve, for which we will make a fat-line
 * @param  {Array}  v2 - Section of the second curve; we will clip this curve with the fat-line of v1
 * @param  {Object} v2t - The parameter range of v2
 * @return {number} -> 0 -no Intersection, 1 -one intersection, -1 -more than one intersection
 */
function _clipBezierFatLine( v1, v2, v2t ){
    // first curve, P
    var p0x = v1[0], p0y = v1[1], p3x = v1[6], p3y = v1[7];
    var p1x = v1[2], p1y = v1[3], p2x = v1[4], p2y = v1[5];
    // second curve, Q
    var q0x = v2[0], q0y = v2[1], q3x = v2[6], q3y = v2[7];
    var q1x = v2[2], q1y = v2[3], q2x = v2[4], q2y = v2[5];
    // Calculate the fat-line L for P is the baseline l and two
    // offsets which completely encloses the curve P.
    var d1 = _getSignedDist( p0x, p0y, p3x, p3y, p1x, p1y ) || 0;
    var d2 = _getSignedDist( p0x, p0y, p3x, p3y, p2x, p2y ) || 0;
    var dmin, dmax;
    if( d1 * d2 > 0){
        // 3/4 * min{0, d1, d2}
        dmin = 0.75 * Math.min( 0, d1, d2 );
        dmax = 0.75 * Math.max( 0, d1, d2 );
    } else {
        // 4/9 * min{0, d1, d2}
        dmin = 0.4444444444444444 * Math.min( 0, d1, d2 );
        dmax = 0.4444444444444444 * Math.max( 0, d1, d2 );
    }
    // Calculate non-parametric bezier curve D(ti, di(t)) -
    // di(t) is the distance of Q from the baseline l of the fat-line,
    // ti is equally spaced in [0,1]
    var dq0 = _getSignedDist( p0x, p0y, p3x, p3y, q0x, q0y );
    var dq1 = _getSignedDist( p0x, p0y, p3x, p3y, q1x, q1y );
    var dq2 = _getSignedDist( p0x, p0y, p3x, p3y, q2x, q2y );
    var dq3 = _getSignedDist( p0x, p0y, p3x, p3y, q3x, q3y );
    // Find the minimum and maximum distances from l,
    // this is useful for checking whether the curves intersect with each other or not.
    var mindist = Math.min( dq0, dq1, dq2, dq3 );
    var maxdist = Math.max( dq0, dq1, dq2, dq3 );
    // If the fatlines don't overlap, we have no intersections!
    if( dmin > maxdist || dmax < mindist ){
        return 0;
    }
    var tmp;
    if( dq3 < dq0 ){
        tmp = dmin; dmin = dmax; dmax = tmp;
    }
    // Calculate the convex hull for non-parametric bezier curve D(ti, di(t))
    var Dt  = _convexhull( dq0, dq1, dq2, dq3 );
    // Now we clip the convex hulls for D(ti, di(t)) with dmin and dmax
    // for the coorresponding t values (tmin, tmax):
    // Portions of curve v2 before tmin and after tmax can safely be clipped away
    var tmaxdmin = -Infinity, ixd, ixdx, i, len, inv_m;
    var tmin = Infinity, tmax = -Infinity, Dtl, dtlx1, dtly1, dtlx2, dtly2;
    for (i = 0, len = Dt.length; i < len; i++) {
        Dtl = Dt[i];
        dtlx1 = Dtl[0]; dtly1 = Dtl[1]; dtlx2 = Dtl[2]; dtly2 = Dtl[3];
        if( dtly2 < dtly1 ){
            tmp = dtly2; dtly2 = dtly1; dtly1 = tmp;
            tmp = dtlx2; dtlx2 = dtlx1; dtlx1 = tmp;
        }
        // we know that (dtlx2 - dtlx1) is never 0
        inv_m =  (dtly2 - dtly1) / (dtlx2 - dtlx1);
        if( dmin >= dtly1 && dmin <= dtly2 ){
            ixdx = dtlx1 + (dmin - dtly1) / inv_m;
            if ( ixdx < tmin ) tmin = ixdx;
            if ( ixdx > tmaxdmin ) tmaxdmin = ixdx;
        }
        if( dmax >= dtly1 && dmax <= dtly2 ){
            ixdx = dtlx1 + (dmax - dtly1) / inv_m;
            if( ixdx > tmax ) tmax = ixdx;
            if( ixdx < tmin ) tmin = 0;
        }
    }
    // Return the parameter values for v2 for which we can be sure that the
    // intersection with v1 lies within.
    if(tmin === Infinity || tmax === -Infinity){
        return -1;
    }
    if( tmaxdmin > tmax ){ tmax = 1; }
// Debug: Plot the non-parametric graph and hull
// plotD_vs_t( 500, 110, Dt, [dq0, dq1, dq2, dq3], v1, dmin, dmax, tmin, tmax, 1.0 / ( tmax - tmin + 0.3 ) )
    // tmin and tmax are within the range (0, 1). We need to project it to the original
    // parameter range for v2.
    var v2tmin = v2t.t1;
    var tdiff = ( v2t.t2 - v2tmin );
    v2t.t1 = v2tmin + tmin * tdiff;
    v2t.t2 = v2tmin + tmax * tdiff;
    // If the new parameter range fails to converge by atleast 20% of the original range,
    // possibly we have multiple intersections. We need to subdivide one of the curves.
    if( (tdiff - ( v2t.t2 - v2t.t1 ))/tdiff < 0.2 ){
        return -1;
    }
    return 1;
}

/**
 * Calculate the convex hull for the non-paramertic bezier curve D(ti, di(t)).
 * The ti is equally spaced across [0..1] — [0, 1/3, 2/3, 1] for
 * di(t), [dq0, dq1, dq2, dq3] respectively. In other words our CVs for the curve are
 * already sorted in the X axis in the increasing order. Calculating convex-hull is
 * much easier than a set of arbitrary points.
 */
function _convexhull( dq0, dq1, dq2, dq3 ){
    var distq1 = _getSignedDist( 0.0, dq0, 1.0, dq3, 0.3333333333333333, dq1 );
    var distq2 = _getSignedDist( 0.0, dq0, 1.0, dq3, 0.6666666666666666, dq2 );
    // Check if [1/3, dq1] and [2/3, dq2] are on the same side of line [0,dq0, 1,dq3]
    if( distq1 * distq2 < 0 ) {
        // dq1 and dq2 lie on different sides on [0, q0, 1, q3]
        // Convexhull is a quadrilateral and line [0, q0, 1, q3] is NOT part of the convexhull
        // so we are pretty much done here.
        Dt = [
            [ 0.0, dq0, 0.3333333333333333, dq1 ],
            [ 0.3333333333333333, dq1, 1.0, dq3 ],
            [ 0.6666666666666666, dq2, 0.0, dq0 ],
            [ 1.0, dq3, 0.6666666666666666, dq2 ]
        ];
    } else {
        // dq1 and dq2 lie on the same sides on [0, q0, 1, q3]
        // Convexhull can be a triangle or a quadrilateral and
        // line [0, q0, 1, q3] is part of the convexhull.
        // Check if the hull is a triangle or a quadrilateral
        var dqmin, dqmax, dqapex1, dqapex2;
        distq1 = Math.abs(distq1);
        distq2 = Math.abs(distq2);
        var vqa1a2x, vqa1a2y, vqa1Maxx, vqa1Maxy, vqa1Minx, vqa1Miny;
        if( distq1 > distq2 ){
            dqmin = [ 0.6666666666666666, dq2 ];
            dqmax = [ 0.3333333333333333, dq1 ];
            // apex is dq3 and the other apex point is dq0
            // vector dqapex->dqapex2 or the base vector which is already part of c-hull
            vqa1a2x = 1.0, vqa1a2y = dq3 - dq0;
            // vector dqapex->dqmax
            vqa1Maxx = 0.6666666666666666, vqa1Maxy = dq3 - dq1;
            // vector dqapex->dqmin
            vqa1Minx = 0.3333333333333333, vqa1Miny = dq3 - dq2;
        } else {
            dqmin = [ 0.3333333333333333, dq1 ];
            dqmax = [ 0.6666666666666666, dq2 ];
            // apex is dq0 in this case, and the other apex point is dq3
            // vector dqapex->dqapex2 or the base vector which is already part of c-hull
            vqa1a2x = -1.0, vqa1a2y = dq0 - dq3;
            // vector dqapex->dqmax
            vqa1Maxx = -0.6666666666666666, vqa1Maxy = dq0 - dq2;
            // vector dqapex->dqmin
            vqa1Minx = -0.3333333333333333, vqa1Miny = dq0 - dq1;
        }
        // compare cross products of these vectors to determine, if
        // point is in triangles [ dq3, dqMax, dq0 ] or [ dq0, dqMax, dq3 ]
        var vcrossa1a2_a1Min = vqa1a2x * vqa1Miny - vqa1a2y * vqa1Minx;
        var vcrossa1Max_a1Min = vqa1Maxx * vqa1Miny - vqa1Maxy * vqa1Minx;
        if( vcrossa1Max_a1Min * vcrossa1a2_a1Min < 0 ){
            // Point [2/3, dq2] is inside the triangle and the convex hull is a triangle
            Dt = [
                [ 0.0, dq0, dqmax[0], dqmax[1] ],
                [ dqmax[0], dqmax[1], 1.0, dq3 ],
                [ 1.0, dq3, 0.0, dq0 ]
            ];
        } else {
            // Convexhull is a quadrilateral and we need all lines in the correct order where
            // line [0, q0, 1, q3] is part of the convex hull
            Dt = [
                [ 0.0, dq0, 0.3333333333333333, dq1 ],
                [ 0.3333333333333333, dq1, 0.6666666666666666, dq2 ],
                [ 0.6666666666666666, dq2, 1.0, dq3 ],
                [ 1.0, dq3, 0.0, dq0 ]
            ];
        }
    }
    return Dt;
}


function drawFatline( v1 ) {
    function signum(num) {
        return ( num > 0 )? 1 : ( num < 0 )? -1 : 0;
    }
    var l = new Line( [v1[0], v1[1]], [v1[6], v1[7]], false );
    var p1 = new Point( v1[2], v1[3] ), p2 = new Point( v1[4], v1[5] );
    var d1 = l.getSide( p1 ) * l.getDistance( p1 ) || 0;
    var d2 = l.getSide( p2 ) * l.getDistance( p2 ) || 0;
    var dmin, dmax;
    if( d1 * d2 > 0){
        // 3/4 * min{0, d1, d2}
        dmin = 0.75 * Math.min( 0, d1, d2 );
        dmax = 0.75 * Math.max( 0, d1, d2 );
    } else {
        // 4/9 * min{0, d1, d2}
        dmin = 4 * Math.min( 0, d1, d2 ) / 9.0;
        dmax = 4 * Math.max( 0, d1, d2 ) / 9.0;
    }
    var ll = new Path.Line( v1[0], v1[1], v1[6], v1[7] );
    window.__p3.push( ll );
    window.__p3[window.__p3.length-1].style.strokeColor = new Color( 0,0,0.9, 0.8);
    var lp1 = ll.segments[0].point;
    var lp2 = ll.segments[1].point;
    var pm = l.vector, pm1 = pm.rotate( signum( dmin ) * -90 ), pm2 = pm.rotate( signum( dmax ) * -90 );
    var p11 = lp1.add( pm1.normalize( Math.abs(dmin) ) );
    var p12 = lp2.add( pm1.normalize( Math.abs(dmin) ) );
    var p21 = lp1.add( pm2.normalize( Math.abs(dmax) ) );
    var p22 = lp2.add( pm2.normalize( Math.abs(dmax) ) );
    window.__p3.push( new Path.Line( p11, p12 ) );
    window.__p3[window.__p3.length-1].style.strokeColor = new Color( 0,0,0.9);
    window.__p3.push( new Path.Line( p21, p22 ) );
    window.__p3[window.__p3.length-1].style.strokeColor = new Color( 0,0,0.9);
}

function plotD_vs_t( x, y, arr, arr2, v, dmin, dmax, tmin, tmax, yscale, tvalue ){
    yscale = yscale || 1;
    new Path.Line( x, y-100, x, y+100 ).style.strokeColor = '#aaa';
    new Path.Line( x, y, x + 200, y ).style.strokeColor = '#aaa';

    var clr = (tvalue)? '#a00' : '#00a';
    if( window.__p3 ) window.__p3.map(function(a){a.remove();});

    window.__p3 = [];

    drawFatline( v );

    window.__p3.push( new Path.Line( x, y + dmin * yscale, x + 200, y + dmin * yscale ) );
    window.__p3[window.__p3.length-1].style.strokeColor = '#000'
    window.__p3.push( new Path.Line( x, y + dmax * yscale, x + 200, y + dmax * yscale ) );
    window.__p3[window.__p3.length-1].style.strokeColor = '#000'
    window.__p3.push( new Path.Line( x + tmin * 190, y-100, x + tmin * 190, y+100 ) );
    window.__p3[window.__p3.length-1].style.strokeColor = clr
    window.__p3.push( new Path.Line( x + tmax * 190, y-100, x + tmax * 190, y+100 ) );
    window.__p3[window.__p3.length-1].style.strokeColor = clr

    for (var i = 0; i < arr.length; i++) {
        window.__p3.push( new Path.Line( new Point( x + arr[i][0] * 190, y + arr[i][1] * yscale ),
         new Point( x + arr[i][2] * 190, y + arr[i][3] * yscale ) ) );
        window.__p3[window.__p3.length-1].style.strokeColor = '#999';
    }
    var pnt = [];
    var arr2x = [ 0.0, 0.333333333, 0.6666666666, 1.0 ];
    for (var i = 0; i < arr2.length; i++) {
        pnt.push( new Point( x + arr2x[i] * 190, y + arr2[i] * yscale ) );
        window.__p3.push( new Path.Circle( pnt[pnt.length-1], 2 ) );
        window.__p3[window.__p3.length-1].style.fillColor = '#000'
    }
    // var pth = new Path( pnt[0], pnt[1], pnt[2], pnt[3] );
    // pth.closed = true;
    window.__p3.push( new Path( new Segment(pnt[0], null, pnt[1].subtract(pnt[0])), new Segment( pnt[3], pnt[2].subtract(pnt[3]), null ) ) );
    window.__p3[window.__p3.length-1].style.strokeColor = clr
    view.draw();
}

// This is basically an "unrolled" version of #Line.getDistance() with sign
// May be a static method could be better!
var _getSignedDist = function( a1x, a1y, a2x, a2y, bx, by ){
    var vx = a2x - a1x, vy = a2y - a1y;
    var m = vy / vx, b = a1y - ( m * a1x );
    return ( by - ( m * bx ) - b ) / Math.sqrt( m*m + 1 );
};

/**
 * Intersections between curve and line becomes rather simple here mostly
 * because of paperjs Numerical class. We can rotate the curve and line so that
 * the line is on X axis, and solve the implicit equations for X axis and the curve
 */
var _getCurveLineIntersection = function( v1, v2, curve1, curve2, locations, _other ){
    var i, root, point, vc = v1, vl = v2;
    var other = ( _other === undefined )? Curve.isLinear( v1 ) : _other;
    if( other ){
        vl = v1;
        vc = v2;
    }
    var l1x = vl[0], l1y = vl[1], l2x = vl[6], l2y = vl[7];
    // rotate both the curve and line around l1 so that line is on x axis
    var lvx = l2x - l1x, lvy = l2y - l1y;
    // Angle with x axis (1, 0)
    var angle = Math.atan2( -lvy, lvx ), sina = Math.sin( angle ), cosa = Math.cos( angle );
    // rotated line and curve values
    // (rl1x, rl1y) = (0, 0)
    var rl2x = lvx * cosa - lvy * sina, rl2y = lvy * cosa + lvx * sina;
    var rvc = [];
    for( i=0; i<8; i+=2 ){
        var vcx = vc[i] - l1x, vcy = vc[i+1] - l1y;
        rvc.push( vcx * cosa - vcy * sina );
        rvc.push( vcy * cosa + vcx * sina );
    }
    var roots = [];
    Curve.solveCubic(rvc, 1, 0, roots);
    i = roots.length;
    while( i-- ){
        root = roots[i];
        if( root >= 0 && root <= 1 ){
            point = Curve.evaluate(rvc, root, true, 0);
            // We do have a point on the infinite line. Check if it falls on the line *segment*.
            if( point.x  >= 0 && point.x <= rl2x ){
                // The actual intersection point
                point = Curve.evaluate(vc, root, true, 0);
                if( other ) root = null;
                var first = locations[0],
                last = locations[locations.length - 1];
                if ((!first || !point.equals(first._point))
                        && (!last || !point.equals(last._point)))
                    locations.push( new CurveLocation( curve1, root, point, curve2 ) );
            }
        }
    }
};

var _getLineLineIntersection = function( v1, v2, curve1, curve2, locations ){
    var point = Line.intersect(
                        v1[0], v1[1], v1[6], v1[7],
                        v2[0], v2[1], v2[6], v2[7], false);
    if (point) {
        // Avoid duplicates when hitting segments (closed paths too)
        var first = locations[0],
            last = locations[locations.length - 1];
        if ((!first || !point.equals(first._point))
                && (!last || !point.equals(last._point)))
            // Passing null for parameter leads to lazy determination
            // of parameter values in CurveLocation#getParameter()
            // only once they are requested.
            locations.push(new CurveLocation(curve1, null, point, curve2));
    }
};