OtsLine3d

package org.opentrafficsim.core.geometry;

import java.awt.geom.Line2D;
import java.awt.geom.Path2D;
import java.awt.geom.PathIterator;
import java.awt.geom.Point2D;
import java.io.Serializable;
import java.util.ArrayList;
import java.util.Arrays;
import java.util.List;
import java.util.Locale;
import java.util.stream.Collectors;
import java.util.stream.DoubleStream;

import org.djunits.unit.DirectionUnit;
import org.djunits.value.vdouble.scalar.Direction;
import org.djunits.value.vdouble.scalar.Length;
import org.djutils.draw.point.Point3d;
import org.djutils.exceptions.Throw;
import org.djutils.logger.CategoryLogger;
import org.locationtech.jts.geom.Coordinate;
import org.locationtech.jts.geom.CoordinateSequence;
import org.locationtech.jts.geom.Geometry;
import org.locationtech.jts.geom.GeometryFactory;
import org.locationtech.jts.geom.LineString;
import org.locationtech.jts.linearref.LengthIndexedLine;

import nl.tudelft.simulation.dsol.animation.Locatable;

/**
 * Line with OtsPoint3d points, a cached length indexed line, a cached length, and a cached centroid (all calculated on first
 * use).
 * <p>
 * Copyright (c) 2013-2023 Delft University of Technology, PO Box 5, 2600 AA, Delft, the Netherlands. All rights reserved. <br>
 * BSD-style license. See <a href="https://opentrafficsim.org/docs/license.html">OpenTrafficSim License</a>.
 * </p>
 * @author <a href="https://github.com/averbraeck">Alexander Verbraeck</a>
 * @author <a href="https://tudelft.nl/staff/p.knoppers-1">Peter Knoppers</a>
 * @author <a href="https://www.citg.tudelft.nl">Guus Tamminga</a>
 * @author <a href="https://dittlab.tudelft.nl">Wouter Schakel</a>
 */
public class OtsLine3d implements Locatable, Serializable // XXX: DJ
{
    /** */
    private static final long serialVersionUID = 20150722L;

    /** The points of the line. */
    private OtsPoint3d[] points;

    /** The cumulative length of the line at point 'i'. */
    private double[] lengthIndexedLine = null;

    /** The cached length; will be calculated at time of construction. */
    private Length length; // XXX: DJ uses double

    /** The cached centroid; will be calculated when needed for the first time. */
    private OtsPoint3d centroid = null;

    /** The cached bounds; will be calculated when needed for the first time. */
    private Bounds bounds = null; // XXX: DJ

    /** The cached helper points for fractional projection; will be calculated when needed for the first time. */
    private OtsPoint3d[] fractionalHelperCenters = null;

    /** The cached helper directions for fractional projection; will be calculated when needed for the first time. */
    private Point2D.Double[] fractionalHelperDirections = null;

    /** Intersection of unit offset lines of first two segments. */
    private OtsPoint3d firstOffsetIntersection;

    /** Intersection of unit offset lines of last two segments. */
    private OtsPoint3d lastOffsetIntersection;

    /** Precision for fractional projection algorithm. */
    private static final double FRAC_PROJ_PRECISION = 2e-5 /* PK too fine 1e-6 */;

    /** Radius at each vertex. */
    private Length[] vertexRadii; // XXX: DJ

    /** Bounding of this OtsLine3d. */
    private Bounds envelope;

    /**
     * Construct a new OtsLine3d.
     * @param points OtsPoint3d...; the array of points to construct this OtsLine3d from.
     * @throws OtsGeometryException when the provided points do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final OtsPoint3d... points) throws OtsGeometryException
    {
        init(points);
    }

    /**
     * Construct a new OtsLine3d, and immediately make the length-indexed line.
     * @param pts OtsPoint3d...; the array of points to construct this OtsLine3d from.
     * @throws OtsGeometryException when the provided points do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    private void init(final OtsPoint3d... pts) throws OtsGeometryException
    {
        if (pts.length < 2)
        {
            throw new OtsGeometryException("Degenerate OtsLine3d; has " + pts.length + " point" + (pts.length != 1 ? "s" : ""));
        }
        this.lengthIndexedLine = new double[pts.length];
        this.lengthIndexedLine[0] = 0.0;
        for (int i = 1; i < pts.length; i++)
        {
            if (pts[i - 1].x == pts[i].x && pts[i - 1].y == pts[i].y && pts[i - 1].z == pts[i].z)
            {
                throw new OtsGeometryException(
                        "Degenerate OtsLine3d; point " + (i - 1) + " has the same x, y and z as point " + i);
            }
            this.lengthIndexedLine[i] = this.lengthIndexedLine[i - 1] + pts[i - 1].distanceSI(pts[i]);
        }
        this.points = pts;
        this.length = Length.instantiateSI(this.lengthIndexedLine[this.lengthIndexedLine.length - 1]); // XXX: DJ double
    }

    /** Which offsetLine method to use... */
    public enum OffsetMethod
    {
        /** Via JTS buffer. */
        JTS,

        /** Peter Knoppers. */
        PK;
    };

    /** Which offset line method to use... */
    public static final OffsetMethod OFFSETMETHOD = OffsetMethod.PK;

    /**
     * Construct parallel line.<br>
     * TODO Let the Z-component of the result follow the Z-values of the reference line.
     * @param offset double; offset distance from the reference line; positive is LEFT, negative is RIGHT
     * @return OtsLine3d; the line that has the specified offset from the reference line
     */
    public final OtsLine3d offsetLine(final double offset)
    {
        try
        {
            switch (OFFSETMETHOD)
            {
                case PK:
                    return OtsOffsetLinePk.offsetLine(this, offset);

                case JTS:
                    return OtsBufferingJts.offsetLine(this, offset);

                default:
                    return null;
            }
        }
        catch (OtsGeometryException exception)
        {
            CategoryLogger.always().error(exception);
            return null;
        }
    }

    /**
     * Construct a line that is equal to this line except for segments that are shorter than the <cite>noiseLevel</cite>. The
     * result is guaranteed to start with the first point of this line and end with the last point of this line.
     * @param noiseLevel double; the minimum segment length that is <b>not</b> removed
     * @return OtsLine3d; the filtered line
     */
    public final OtsLine3d noiseFilteredLine(final double noiseLevel)
    {
        if (this.size() <= 2)
        {
            return this; // Except for some cached fields; an OtsLine3d is immutable; so safe to return
        }
        OtsPoint3d prevPoint = null;
        List<OtsPoint3d> list = null;
        for (int index = 0; index < this.size(); index++)
        {
            OtsPoint3d currentPoint = this.points[index];
            if (null != prevPoint && prevPoint.distanceSI(currentPoint) < noiseLevel)
            {
                if (null == list)
                {
                    // Found something to filter; copy this up to (and including) prevPoint
                    list = new ArrayList<>();
                    for (int i = 0; i < index; i++)
                    {
                        list.add(this.points[i]);
                    }
                }
                if (index == this.size() - 1)
                {
                    if (list.size() > 1)
                    {
                        // Replace the last point of the result by the last point of this OtsLine3d
                        list.set(list.size() - 1, currentPoint);
                    }
                    else
                    {
                        // Append the last point of this even though it is close to the first point than the noise value to
                        // comply with the requirement that first and last point of this are ALWAYS included in the result.
                        list.add(currentPoint);
                    }
                }
                continue; // Do not replace prevPoint by currentPoint
            }
            else if (null != list)
            {
                list.add(currentPoint);
            }
            prevPoint = currentPoint;
        }
        if (null == list)
        {
            return this;
        }
        if (list.size() == 2 && list.get(0).equals(list.get(1)))
        {
            // CategoryLogger.always().debug("Fixing up degenerate noiseFilteredLine by inserting an intermediate point");
            // Find something to insert along the way
            for (int index = 1; index < this.size() - 1; index++)
            {
                if (!this.points[index].equals(list.get(0)))
                {
                    list.add(1, this.points[index]);
                    break;
                }
            }
        }
        try
        {
            return new OtsLine3d(list);
        }
        catch (OtsGeometryException exception)
        {
            CategoryLogger.always().error(exception);
            throw new Error(exception);
        }
    }

    /**
     * Clean up a list of points that describe a polyLine by removing points that lie within epsilon distance of a more
     * straightened version of the line. <br>
     * TODO Test this code (currently untested).
     * @param epsilon double; maximal deviation
     * @param useHorizontalDistance boolean; if true; the horizontal distance is used; if false; the 3D distance is used
     * @return OtsLine3d; a new OtsLine3d containing all the remaining points
     */
    public final OtsLine3d noiseFilterRamerDouglasPeucker(final double epsilon, final boolean useHorizontalDistance) // XXX: DJ
    {
        try
        {
            // Apply the Ramer-Douglas-Peucker algorithm to the buffered points.
            // Adapted from https://en.wikipedia.org/wiki/Ramer%E2%80%93Douglas%E2%80%93Peucker_algorithm
            double maxDeviation = 0;
            int splitIndex = -1;
            int pointCount = size();
            OtsLine3d straight = new OtsLine3d(get(0), get(pointCount - 1));
            // Find the point with largest deviation from the straight line from start point to end point
            for (int i = 1; i < pointCount - 1; i++)
            {
                OtsPoint3d point = get(i);
                OtsPoint3d closest =
                        useHorizontalDistance ? point.closestPointOnLine2D(straight) : point.closestPointOnLine(straight);
                double deviation = useHorizontalDistance ? closest.horizontalDistanceSI(point) : closest.distanceSI(point);
                if (deviation > maxDeviation)
                {
                    splitIndex = i;
                    maxDeviation = deviation;
                }
            }
            if (maxDeviation <= epsilon)
            {
                // All intermediate points can be dropped. Return a new list containing only the first and last point.
                return straight;
            }
            // The largest deviation is larger than epsilon.
            // Split the polyLine at the point with the maximum deviation. Process each sub list recursively and concatenate the
            // results
            OtsLine3d first = new OtsLine3d(Arrays.copyOfRange(this.points, 0, splitIndex + 1))
                    .noiseFilterRamerDouglasPeucker(epsilon, useHorizontalDistance);
            OtsLine3d second = new OtsLine3d(Arrays.copyOfRange(this.points, splitIndex, this.points.length))
                    .noiseFilterRamerDouglasPeucker(epsilon, useHorizontalDistance);
            return concatenate(epsilon, first, second);
        }
        catch (OtsGeometryException exception)
        {
            CategoryLogger.always().error(exception); // Peter thinks this cannot happen ...
            return null;
        }
    }

    /**
     * Create a line at linearly varying offset from this line. The offset may change linearly from its initial value at the
     * start of the reference line to its final offset value at the end of the reference line.
     * @param offsetAtStart double; offset at the start of the reference line (positive value is Left, negative value is Right)
     * @param offsetAtEnd double; offset at the end of the reference line (positive value is Left, negative value is Right)
     * @return Geometry; the Geometry of the line at linearly changing offset of the reference line
     * @throws OtsGeometryException when this method fails to create the offset line
     */
    public final OtsLine3d offsetLine(final double offsetAtStart, final double offsetAtEnd) throws OtsGeometryException // XXX:
                                                                                                                        // DJ
    {
        // CategoryLogger.trace(Cat.CORE, OTSGeometry.printCoordinates("#referenceLine: \nc1,0,0\n# offset at start is "
        // + offsetAtStart + " at end is " + offsetAtEnd + "\n#", referenceLine, "\n "));

        OtsLine3d offsetLineAtStart = offsetLine(offsetAtStart);
        if (offsetAtStart == offsetAtEnd)
        {
            return offsetLineAtStart; // offset does not change
        }
        // CategoryLogger.trace(Cat.CORE, OTSGeometry.printCoordinates("#offset line at start: \nc0,0,0\n#",
        // offsetLineAtStart, "\n "));
        OtsLine3d offsetLineAtEnd = offsetLine(offsetAtEnd);
        // CategoryLogger.trace(Cat.CORE, OTSGeometry.printCoordinates("#offset line at end: \nc0.7,0.7,0.7\n#",
        // offsetLineAtEnd, "\n "));
        Geometry startGeometry = offsetLineAtStart.getLineString();
        Geometry endGeometry = offsetLineAtEnd.getLineString();
        LengthIndexedLine first = new LengthIndexedLine(startGeometry);
        double firstLength = startGeometry.getLength();
        LengthIndexedLine second = new LengthIndexedLine(endGeometry);
        double secondLength = endGeometry.getLength();
        ArrayList<Coordinate> out = new ArrayList<>();
        Coordinate[] firstCoordinates = startGeometry.getCoordinates();
        Coordinate[] secondCoordinates = endGeometry.getCoordinates();
        int firstIndex = 0;
        int secondIndex = 0;
        Coordinate prevCoordinate = null;
        final double tooClose = 0.05; // 5 cm
        while (firstIndex < firstCoordinates.length && secondIndex < secondCoordinates.length)
        {
            double firstRatio = firstIndex < firstCoordinates.length ? first.indexOf(firstCoordinates[firstIndex]) / firstLength
                    : Double.MAX_VALUE;
            double secondRatio = secondIndex < secondCoordinates.length
                    ? second.indexOf(secondCoordinates[secondIndex]) / secondLength : Double.MAX_VALUE;
            double ratio;
            if (firstRatio < secondRatio)
            {
                ratio = firstRatio;
                firstIndex++;
            }
            else
            {
                ratio = secondRatio;
                secondIndex++;
            }
            Coordinate firstCoordinate = first.extractPoint(ratio * firstLength);
            Coordinate secondCoordinate = second.extractPoint(ratio * secondLength);
            Coordinate resultCoordinate = new Coordinate((1 - ratio) * firstCoordinate.x + ratio * secondCoordinate.x,
                    (1 - ratio) * firstCoordinate.y + ratio * secondCoordinate.y);
            if (null == prevCoordinate || resultCoordinate.distance(prevCoordinate) > tooClose)
            {
                out.add(resultCoordinate);
                prevCoordinate = resultCoordinate;
            }
        }
        Coordinate[] resultCoordinates = new Coordinate[out.size()];
        for (int index = 0; index < out.size(); index++)
        {
            resultCoordinates[index] = out.get(index);
        }
        return new OtsLine3d(resultCoordinates);
    }

    /**
     * Create a line at linearly varying offset from this line. The offset may change linearly from its initial value at the
     * start of the reference line via a number of intermediate offsets at intermediate positions to its final offset value at
     * the end of the reference line.
     * @param relativeFractions double[]; positional fractions for which the offsets have to be generated
     * @param offsets double[]; offsets at the relative positions (positive value is Left, negative value is Right)
     * @return Geometry; the Geometry of the line at linearly changing offset of the reference line
     * @throws OtsGeometryException when this method fails to create the offset line
     */
    public final OtsLine3d offsetLine(final double[] relativeFractions, final double[] offsets) throws OtsGeometryException // XXX:
                                                                                                                            // DJ
    {
        Throw.whenNull(relativeFractions, "relativeFraction may not be null");
        Throw.whenNull(offsets, "offsets may not be null");
        Throw.when(relativeFractions.length < 2, OtsGeometryException.class, "size of relativeFractions must be >= 2");
        Throw.when(relativeFractions.length != offsets.length, OtsGeometryException.class,
                "size of relativeFractions must be equal to size of offsets");
        Throw.when(relativeFractions[0] < 0, OtsGeometryException.class, "relativeFractions may not start before 0");
        Throw.when(relativeFractions[relativeFractions.length - 1] > 1, OtsGeometryException.class,
                "relativeFractions may not end beyond 1");
        List<Double> fractionsList = DoubleStream.of(relativeFractions).boxed().collect(Collectors.toList());
        List<Double> offsetsList = DoubleStream.of(offsets).boxed().collect(Collectors.toList());
        if (relativeFractions[0] != 0)
        {
            fractionsList.add(0, 0.0);
            offsetsList.add(0, 0.0);
        }
        if (relativeFractions[relativeFractions.length - 1] < 1.0)
        {
            fractionsList.add(1.0);
            offsetsList.add(0.0);
        }
        OtsLine3d[] offsetLine = new OtsLine3d[fractionsList.size()];
        for (int i = 0; i < fractionsList.size(); i++)
        {
            offsetLine[i] = offsetLine(offsetsList.get(i));
            System.out.println("# offset is " + offsetsList.get(i));
            System.out.println(offsetLine[i].toPlot());
        }
        List<Coordinate> out = new ArrayList<>();
        Coordinate prevCoordinate = null;
        final double tooClose = 0.05; // 5 cm
        // TODO make tooClose a parameter of this method.
        for (int i = 0; i < offsetsList.size() - 1; i++)
        {
            Throw.when(fractionsList.get(i + 1) <= fractionsList.get(i), OtsGeometryException.class,
                    "fractions must be in ascending order");
            Geometry startGeometry =
                    offsetLine[i].extractFractional(fractionsList.get(i), fractionsList.get(i + 1)).getLineString();
            Geometry endGeometry =
                    offsetLine[i + 1].extractFractional(fractionsList.get(i), fractionsList.get(i + 1)).getLineString();
            LengthIndexedLine first = new LengthIndexedLine(startGeometry);
            double firstLength = startGeometry.getLength();
            LengthIndexedLine second = new LengthIndexedLine(endGeometry);
            double secondLength = endGeometry.getLength();
            Coordinate[] firstCoordinates = startGeometry.getCoordinates();
            Coordinate[] secondCoordinates = endGeometry.getCoordinates();
            int firstIndex = 0;
            int secondIndex = 0;
            while (firstIndex < firstCoordinates.length && secondIndex < secondCoordinates.length)
            {
                double firstRatio = firstIndex < firstCoordinates.length
                        ? first.indexOf(firstCoordinates[firstIndex]) / firstLength : Double.MAX_VALUE;
                double secondRatio = secondIndex < secondCoordinates.length
                        ? second.indexOf(secondCoordinates[secondIndex]) / secondLength : Double.MAX_VALUE;
                double ratio;
                if (firstRatio < secondRatio)
                {
                    ratio = firstRatio;
                    firstIndex++;
                }
                else
                {
                    ratio = secondRatio;
                    secondIndex++;
                }
                Coordinate firstCoordinate = first.extractPoint(ratio * firstLength);
                Coordinate secondCoordinate = second.extractPoint(ratio * secondLength);
                Coordinate resultCoordinate = new Coordinate((1 - ratio) * firstCoordinate.x + ratio * secondCoordinate.x,
                        (1 - ratio) * firstCoordinate.y + ratio * secondCoordinate.y);
                if (null == prevCoordinate || resultCoordinate.distance(prevCoordinate) > tooClose)
                {
                    out.add(resultCoordinate);
                    prevCoordinate = resultCoordinate;
                }
            }
        }
        Coordinate[] resultCoordinates = new Coordinate[out.size()];
        for (int index = 0; index < out.size(); index++)
        {
            resultCoordinates[index] = out.get(index);
        }
        return new OtsLine3d(resultCoordinates);
    }

    /**
     * Concatenate several OtsLine3d instances.
     * @param lines OtsLine3d...; OtsLine3d... one or more OtsLine3d. The last point of the first
     *            &lt;strong&gt;must&lt;/strong&gt; match the first of the second, etc.
     * @return OtsLine3d
     * @throws OtsGeometryException if zero lines are given, or when there is a gap between consecutive lines
     */
    public static OtsLine3d concatenate(final OtsLine3d... lines) throws OtsGeometryException
    {
        return concatenate(0.0, lines);
    }

    /**
     * Concatenate two OtsLine3d instances. This method is separate for efficiency reasons.
     * @param toleranceSI double; the tolerance between the end point of a line and the first point of the next line
     * @param line1 OtsLine3d; first line
     * @param line2 OtsLine3d; second line
     * @return OtsLine3d
     * @throws OtsGeometryException if zero lines are given, or when there is a gap between consecutive lines
     */
    public static OtsLine3d concatenate(final double toleranceSI, final OtsLine3d line1, final OtsLine3d line2)
            throws OtsGeometryException
    {
        if (line1.getLast().distance(line2.getFirst()).si > toleranceSI)
        {
            throw new OtsGeometryException("Lines are not connected: " + line1.getLast() + " to " + line2.getFirst()
                    + " distance is " + line1.getLast().distance(line2.getFirst()).si + " > " + toleranceSI);
        }
        int size = line1.size() + line2.size() - 1;
        OtsPoint3d[] points = new OtsPoint3d[size];
        int nextIndex = 0;
        for (int j = 0; j < line1.size(); j++)
        {
            points[nextIndex++] = line1.get(j);
        }
        for (int j = 1; j < line2.size(); j++)
        {
            points[nextIndex++] = line2.get(j);
        }
        return new OtsLine3d(points);
    }

    /**
     * Concatenate several OtsLine3d instances.
     * @param toleranceSI double; the tolerance between the end point of a line and the first point of the next line
     * @param lines OtsLine3d...; OtsLine3d... one or more OtsLine3d. The last point of the first
     *            &lt;strong&gt;must&lt;/strong&gt; match the first of the second, etc.
     * @return OtsLine3d
     * @throws OtsGeometryException if zero lines are given, or when there is a gap between consecutive lines
     */
    public static OtsLine3d concatenate(final double toleranceSI, final OtsLine3d... lines) throws OtsGeometryException
    {
        // CategoryLogger.trace(Cat.CORE, "Concatenating " + lines.length + " lines.");
        if (0 == lines.length)
        {
            throw new OtsGeometryException("Empty argument list");
        }
        else if (1 == lines.length)
        {
            return lines[0];
        }
        int size = lines[0].size();
        for (int i = 1; i < lines.length; i++)
        {
            if (lines[i - 1].getLast().distance(lines[i].getFirst()).si > toleranceSI)
            {
                throw new OtsGeometryException(
                        "Lines are not connected: " + lines[i - 1].getLast() + " to " + lines[i].getFirst() + " distance is "
                                + lines[i - 1].getLast().distance(lines[i].getFirst()).si + " > " + toleranceSI);
            }
            size += lines[i].size() - 1;
        }
        OtsPoint3d[] points = new OtsPoint3d[size];
        int nextIndex = 0;
        for (int i = 0; i < lines.length; i++)
        {
            OtsLine3d line = lines[i];
            for (int j = 0 == i ? 0 : 1; j < line.size(); j++)
            {
                points[nextIndex++] = line.get(j);
            }
        }
        return new OtsLine3d(points);
    }

    /**
     * Construct a new OtsLine3d with all points of this OtsLine3d in reverse order.
     * @return OtsLine3d; the new OtsLine3d
     */
    public final OtsLine3d reverse()
    {
        OtsPoint3d[] resultPoints = new OtsPoint3d[size()];
        int nextIndex = size();
        for (OtsPoint3d p : getPoints())
        {
            resultPoints[--nextIndex] = p;
        }
        try
        {
            return new OtsLine3d(resultPoints);
        }
        catch (OtsGeometryException exception)
        {
            // Cannot happen
            throw new RuntimeException(exception);
        }
    }

    /**
     * Construct a new OtsLine3d covering the indicated fraction of this OtsLine3d.
     * @param start double; starting point, valid range [0..<cite>end</cite>)
     * @param end double; ending point, valid range (<cite>start</cite>..1]
     * @return OtsLine3d; the new OtsLine3d
     * @throws OtsGeometryException when start &gt;= end, or start &lt; 0, or end &gt; 1
     */
    public final OtsLine3d extractFractional(final double start, final double end) throws OtsGeometryException
    {
        if (start < 0 || start >= end || end > 1)
        {
            throw new OtsGeometryException(
                    "Bad interval (start=" + start + ", end=" + end + ", this is " + this.toString() + ")");
        }
        return extract(start * this.length.si, end * this.length.si);
    }

    /**
     * Create a new OtsLine3d that covers a sub-section of this OtsLine3d.
     * @param start Length; the length along this OtsLine3d where the sub-section starts, valid range [0..<cite>end</cite>)
     * @param end Length; length along this OtsLine3d where the sub-section ends, valid range
     *            (<cite>start</cite>..<cite>length</cite> (length is the length of this OtsLine3d)
     * @return OtsLine3d; the selected sub-section
     * @throws OtsGeometryException when start &gt;= end, or start &lt; 0, or end &gt; length
     */
    public final OtsLine3d extract(final Length start, final Length end) throws OtsGeometryException // XXX: DJ
    {
        return extract(start.si, end.si);
    }

    /**
     * Create a new OtsLine3d that covers a sub-section of this OtsLine3d.
     * @param start double; length along this OtsLine3d where the sub-section starts, valid range [0..<cite>end</cite>)
     * @param end double; length along this OtsLine3d where the sub-section ends, valid range
     *            (<cite>start</cite>..<cite>length</cite> (length is the length of this OtsLine3d)
     * @return OtsLine3d; the selected sub-section
     * @throws OtsGeometryException when start &gt;= end, or start &lt; 0, or end &gt; length
     */
    public final OtsLine3d extract(final double start, final double end) throws OtsGeometryException
    {
        if (Double.isNaN(start) || Double.isNaN(end) || start < 0 || start >= end || end > getLengthSI())
        {
            throw new OtsGeometryException(
                    "Bad interval (" + start + ".." + end + "; length of this OtsLine3d is " + this.getLengthSI() + ")");
        }
        double cumulativeLength = 0;
        double nextCumulativeLength = 0;
        double segmentLength = 0;
        int index = 0;
        List<OtsPoint3d> pointList = new ArrayList<>();
        // CategoryLogger.trace(Cat.CORE, "interval " + start + ".." + end);
        while (start > cumulativeLength)
        {
            OtsPoint3d fromPoint = this.points[index];
            index++;
            OtsPoint3d toPoint = this.points[index];
            segmentLength = fromPoint.distanceSI(toPoint);
            cumulativeLength = nextCumulativeLength;
            nextCumulativeLength = cumulativeLength + segmentLength;
            if (nextCumulativeLength >= start)
            {
                break;
            }
        }
        if (start == nextCumulativeLength)
        {
            pointList.add(this.points[index]);
        }
        else
        {
            pointList.add(OtsPoint3d.interpolate((start - cumulativeLength) / segmentLength, this.points[index - 1],
                    this.points[index]));
            if (end > nextCumulativeLength)
            {
                pointList.add(this.points[index]);
            }
        }
        while (end > nextCumulativeLength)
        {
            OtsPoint3d fromPoint = this.points[index];
            index++;
            if (index >= this.points.length)
            {
                break; // rounding error
            }
            OtsPoint3d toPoint = this.points[index];
            segmentLength = fromPoint.distanceSI(toPoint);
            cumulativeLength = nextCumulativeLength;
            nextCumulativeLength = cumulativeLength + segmentLength;
            if (nextCumulativeLength >= end)
            {
                break;
            }
            pointList.add(toPoint);
        }
        if (end == nextCumulativeLength)
        {
            pointList.add(this.points[index]);
        }
        else
        {
            OtsPoint3d point = OtsPoint3d.interpolate((end - cumulativeLength) / segmentLength, this.points[index - 1],
                    this.points[index]);
            // can be the same due to rounding
            if (!point.equals(pointList.get(pointList.size() - 1)))
            {
                pointList.add(point);
            }
        }
        try
        {
            return new OtsLine3d(pointList);
        }
        catch (OtsGeometryException exception)
        {
            CategoryLogger.always().error(exception, "interval " + start + ".." + end + " too short");
            throw new OtsGeometryException("interval " + start + ".." + end + "too short");
        }
    }

    /**
     * Build an array of OtsPoint3d from an array of Coordinate.
     * @param coordinates Coordinate[]; the coordinates
     * @return OtsPoint3d[]
     */
    private static OtsPoint3d[] coordinatesToOtsPoint3d(final Coordinate[] coordinates) // XXX: DJ
    {
        OtsPoint3d[] result = new OtsPoint3d[coordinates.length];
        for (int i = 0; i < coordinates.length; i++)
        {
            result[i] = new OtsPoint3d(coordinates[i]);
        }
        return result;
    }

    /**
     * Create an OtsLine3d, while cleaning repeating successive points.
     * @param points OtsPoint3d...; the coordinates of the line as OtsPoint3d
     * @return the line
     * @throws OtsGeometryException when number of points &lt; 2
     */
    public static OtsLine3d createAndCleanOtsLine3d(final OtsPoint3d... points) throws OtsGeometryException
    {
        if (points.length < 2)
        {
            throw new OtsGeometryException(
                    "Degenerate OtsLine3d; has " + points.length + " point" + (points.length != 1 ? "s" : ""));
        }
        return createAndCleanOtsLine3d(new ArrayList<>(Arrays.asList(points)));
    }

    /**
     * Create an OtsLine3d, while cleaning repeating successive points.
     * @param pointList List&lt;OtsPoint3d&gt;; list of the coordinates of the line as OtsPoint3d; any duplicate points in this
     *            list are removed (this method may modify the provided list)
     * @return OtsLine3d; the line
     * @throws OtsGeometryException when number of non-equal points &lt; 2
     */
    public static OtsLine3d createAndCleanOtsLine3d(final List<OtsPoint3d> pointList) throws OtsGeometryException
    {
        // clean successive equal points
        int i = 1;
        while (i < pointList.size())
        {
            if (pointList.get(i - 1).equals(pointList.get(i)))
            {
                pointList.remove(i);
            }
            else
            {
                i++;
            }
        }
        return new OtsLine3d(pointList);
    }

    /**
     * Construct a new OtsLine3d from an array of Coordinate.
     * @param coordinates Coordinate[]; the array of coordinates to construct this OtsLine3d from
     * @throws OtsGeometryException when the provided points do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final Coordinate[] coordinates) throws OtsGeometryException // XXX: DJ
    {
        this(coordinatesToOtsPoint3d(coordinates));
    }

    /**
     * Construct a new OtsLine3d from a LineString.
     * @param lineString LineString; the lineString to construct this OtsLine3d from.
     * @throws OtsGeometryException when the provided LineString does not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final LineString lineString) throws OtsGeometryException // XXX: DJ
    {
        this(lineString.getCoordinates());
    }

    /**
     * Construct a new OtsLine3d from a Geometry.
     * @param geometry Geometry; the geometry to construct this OtsLine3d from
     * @throws OtsGeometryException when the provided Geometry do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final Geometry geometry) throws OtsGeometryException // XXX: DJ
    {
        this(geometry.getCoordinates());
    }

    /**
     * Construct a new OtsLine3d from a List&lt;OtsPoint3d&gt;.
     * @param pointList List&lt;OtsPoint3d&gt;; the list of points to construct this OtsLine3d from.
     * @throws OtsGeometryException when the provided points do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final List<OtsPoint3d> pointList) throws OtsGeometryException
    {
        this(pointList.toArray(new OtsPoint3d[pointList.size()]));
    }

    /**
     * Construct a new OtsShape (closed shape) from a Path2D.
     * @param path Path2D; the Path2D to construct this OtsLine3d from.
     * @throws OtsGeometryException when the provided points do not constitute a valid line (too few points or identical
     *             adjacent points)
     */
    public OtsLine3d(final Path2D path) throws OtsGeometryException
    {
        List<OtsPoint3d> pl = new ArrayList<>();
        for (PathIterator pi = path.getPathIterator(null); !pi.isDone(); pi.next())
        {
            double[] p = new double[6];
            int segType = pi.currentSegment(p);
            if (segType == PathIterator.SEG_MOVETO || segType == PathIterator.SEG_LINETO)
            {
                pl.add(new OtsPoint3d(p[0], p[1]));
            }
            else if (segType == PathIterator.SEG_CLOSE)
            {
                if (!pl.get(0).equals(pl.get(pl.size() - 1)))
                {
                    pl.add(new OtsPoint3d(pl.get(0).x, pl.get(0).y));
                }
                break;
            }
        }
        init(pl.toArray(new OtsPoint3d[pl.size() - 1]));
    }

    /**
     * Construct a Coordinate array and fill it with the points of this OtsLine3d.
     * @return an array of Coordinates corresponding to this OTSLine
     */
    public final Coordinate[] getCoordinates() // XXX: DJ
    {
        Coordinate[] result = new Coordinate[size()];
        for (int i = 0; i < size(); i++)
        {
            result[i] = this.points[i].getCoordinate();
        }
        return result;
    }

    /**
     * Construct a LineString from this OtsLine3d.
     * @return a LineString corresponding to this OTSLine
     */
    public final LineString getLineString() // XXX: DJ
    {
        GeometryFactory factory = new GeometryFactory();
        Coordinate[] coordinates = getCoordinates();
        CoordinateSequence cs = factory.getCoordinateSequenceFactory().create(coordinates);
        return new LineString(cs, factory);
    }

    /**
     * Return the number of points in this OtsLine3d.
     * @return the number of points on the line
     */
    public final int size()
    {
        return this.points.length;
    }

    /**
     * Return the first point of this OtsLine3d.
     * @return the first point on the line
     */
    public final OtsPoint3d getFirst()
    {
        return this.points[0];
    }

    /**
     * Return the last point of this OtsLine3d.
     * @return the last point on the line
     */
    public final OtsPoint3d getLast()
    {
        return this.points[size() - 1];
    }

    /**
     * Return one point of this OtsLine3d.
     * @param i int; the index of the point to retrieve
     * @return OTSPoint3d; the i-th point of the line
     * @throws OtsGeometryException when i &lt; 0 or i &gt; the number of points
     */
    public final OtsPoint3d get(final int i) throws OtsGeometryException
    {
        if (i < 0 || i > size() - 1)
        {
            throw new OtsGeometryException("OtsLine3d.get(i=" + i + "); i<0 or i>=size(), which is " + size());
        }
        return this.points[i];
    }

    /**
     * Return the length of this OtsLine3d as a double value in SI units. (Assumes that the coordinates of the points
     * constituting this line are expressed in meters.)
     * @return the length of the line in SI units
     */
    public final double getLengthSI()
    {
        return this.length.si;
    }

    /**
     * Return the length of this OtsLine3d in meters. (Assuming that the coordinates of the points constituting this line are
     * expressed in meters.)
     * @return the length of the line
     */
    public final Length getLength() // XXX: DJ
    {
        return this.length;
    }

    /**
     * Return an array of OtsPoint3d that represents this OtsLine3d. <strong>Do not modify the result.</strong>
     * @return the points of this line
     */
    public final OtsPoint3d[] getPoints()
    {
        return this.points;
    }

    /**
     * Make the length indexed line if it does not exist yet, and cache it.
     */
    private synchronized void makeLengthIndexedLine()
    {
        if (this.lengthIndexedLine == null)
        {
            this.lengthIndexedLine = new double[this.points.length];
            this.lengthIndexedLine[0] = 0.0;
            for (int i = 1; i < this.points.length; i++)
            {
                this.lengthIndexedLine[i] = this.lengthIndexedLine[i - 1] + this.points[i - 1].distanceSI(this.points[i]);
            }
        }
    }

    /**
     * Get the location at a position on the line, with its direction. Position can be below 0 or more than the line length. In
     * that case, the position will be extrapolated in the direction of the line at its start or end.
     * @param position Length; the position on the line for which to calculate the point on, before, of after the line
     * @return a directed point
     */
    public final DirectedPoint getLocationExtended(final Length position) // XXX: DJ
    {
        return getLocationExtendedSI(position.getSI());
    }

    /**
     * Get the location at a position on the line, with its direction. Position can be below 0 or more than the line length. In
     * that case, the position will be extrapolated in the direction of the line at its start or end.
     * @param positionSI double; the position on the line for which to calculate the point on, before, of after the line, in SI
     *            units
     * @return a directed point
     */
    public final synchronized DirectedPoint getLocationExtendedSI(final double positionSI) // XXX: DJ name + super(p)
    {
        makeLengthIndexedLine();
        if (positionSI >= 0.0 && positionSI <= getLengthSI())
        {
            try
            {
                return getLocationSI(positionSI);
            }
            catch (OtsGeometryException exception)
            {
                // cannot happen
            }
        }

        // position before start point -- extrapolate
        if (positionSI < 0.0)
        {
            double len = positionSI;
            double fraction = len / (this.lengthIndexedLine[1] - this.lengthIndexedLine[0]);
            OtsPoint3d p1 = this.points[0];
            OtsPoint3d p2 = this.points[1];
            return new DirectedPoint(p1.x + fraction * (p2.x - p1.x), p1.y + fraction * (p2.y - p1.y),
                    p1.z + fraction * (p2.z - p1.z), 0.0, 0.0, Math.atan2(p2.y - p1.y, p2.x - p1.x));
        }

        // position beyond end point -- extrapolate
        int n1 = this.lengthIndexedLine.length - 1;
        int n2 = this.lengthIndexedLine.length - 2;
        double len = positionSI - getLengthSI();
        double fraction = len / (this.lengthIndexedLine[n1] - this.lengthIndexedLine[n2]);
        while (Double.isInfinite(fraction))
        {
            if (--n2 < 0)
            {
                CategoryLogger.always().error("lengthIndexedLine of {} is invalid", this);
                OtsPoint3d p = this.points[n1];
                return new DirectedPoint(p.x, p.y, p.z, 0.0, 0.0, 0.0); // Bogus direction
            }
            fraction = len / (this.lengthIndexedLine[n1] - this.lengthIndexedLine[n2]);
        }
        OtsPoint3d p1 = this.points[n2];
        OtsPoint3d p2 = this.points[n1];
        return new DirectedPoint(p2.x + fraction * (p2.x - p1.x), p2.y + fraction * (p2.y - p1.y),
                p2.z + fraction * (p2.z - p1.z), 0.0, 0.0, Math.atan2(p2.y - p1.y, p2.x - p1.x));
    }

    /**
     * Get the location at a fraction of the line, with its direction. Fraction should be between 0.0 and 1.0.
     * @param fraction double; the fraction for which to calculate the point on the line
     * @return a directed point
     * @throws OtsGeometryException when fraction less than 0.0 or more than 1.0.
      */
     public final DirectedPoint getLocationFraction(final double fraction) throws OtsGeometryException
     {
         if (fraction < 0.0 || fraction > 1.0)
         {
             throw new OtsGeometryException("getLocationFraction for line: fraction < 0.0 or > 1.0. fraction = " + fraction);
         }
         return getLocationSI(fraction * getLengthSI());
     }

     /**
      * Get the location at a fraction of the line, with its direction. Fraction should be between 0.0 and 1.0.
      * @param fraction double; the fraction for which to calculate the point on the line
      * @param tolerance double; the delta from 0.0 and 1.0 that will be forgiven
      * @return a directed point
      * @throws OtsGeometryException when fraction less than 0.0 or more than 1.0.
      */
     public final DirectedPoint getLocationFraction(final double fraction, final double tolerance) throws OtsGeometryException
     {
         if (fraction < -tolerance || fraction > 1.0 + tolerance)
         {
             throw new OtsGeometryException(
                     "getLocationFraction for line: fraction < 0.0 - tolerance or > 1.0 + tolerance; fraction = " + fraction);
         }
         double f = fraction < 0 ? 0.0 : fraction > 1.0 ? 1.0 : fraction;
         return getLocationSI(f * getLengthSI());
     }

     /**
      * Get the location at a fraction of the line (or outside the line), with its direction.
      * @param fraction double; the fraction for which to calculate the point on the line
      * @return a directed point
      */
     public final DirectedPoint getLocationFractionExtended(final double fraction)
     {
         return getLocationExtendedSI(fraction * getLengthSI());
     }

     /**
      * Get the location at a position on the line, with its direction. Position should be between 0.0 and line length.
      * @param position Length; the position on the line for which to calculate the point on the line
      * @return a directed point
      * @throws OtsGeometryException when position less than 0.0 or more than line length.
      */
     public final DirectedPoint getLocation(final Length position) throws OtsGeometryException // XXX: DJ
     {
         return getLocationSI(position.getSI());
     }

     /**
      * Binary search for a position on the line.
      * @param pos double; the position to look for.
      * @return the index below the position; the position is between points[index] and points[index+1]
      * @throws OtsGeometryException when index could not be found
      */
     private int find(final double pos) throws OtsGeometryException
     {
         if (pos == 0)
         {
             return 0;
         }

         int lo = 0;
         int hi = this.lengthIndexedLine.length - 1;
         while (lo <= hi)
         {
             if (hi == lo)
             {
                 return lo;
             }
             int mid = lo + (hi - lo) / 2;
             if (pos < this.lengthIndexedLine[mid])
             {
                 hi = mid - 1;
             }
             else if (pos > this.lengthIndexedLine[mid + 1])
             {
                 lo = mid + 1;
             }
             else
             {
                 return mid;
             }
         }
         throw new OtsGeometryException(
                 "Could not find position " + pos + " on line with length indexes: " + Arrays.toString(this.lengthIndexedLine));
     }

     /**
      * Get the location at a position on the line, with its direction. Position should be between 0.0 and line length.
      * @param positionSI double; the position on the line for which to calculate the point on the line
      * @return a directed point
      * @throws OtsGeometryException when position less than 0.0 or more than line length.
      */
     public final synchronized DirectedPoint getLocationSI(final double positionSI) throws OtsGeometryException // XXX: DJ
                                                                                                                // without SI
     {
         makeLengthIndexedLine();
         if (positionSI < 0.0 || positionSI > getLengthSI())
         {
             throw new OtsGeometryException("getLocationSI for line: position < 0.0 or > line length. Position = " + positionSI
                     + " m. Length = " + getLengthSI() + " m.");
         }

         // handle special cases: position == 0.0, or position == length
         if (positionSI == 0.0)
         {
             OtsPoint3d p1 = this.points[0];
             OtsPoint3d p2 = this.points[1];
             return new DirectedPoint(p1.x, p1.y, p1.z, 0.0, 0.0, Math.atan2(p2.y - p1.y, p2.x - p1.x));
         }
         if (positionSI == getLengthSI())
         {
             OtsPoint3d p1 = this.points[this.points.length - 2];
             OtsPoint3d p2 = this.points[this.points.length - 1];
             return new DirectedPoint(p2.x, p2.y, p2.z, 0.0, 0.0, Math.atan2(p2.y - p1.y, p2.x - p1.x));
         }

         // find the index of the line segment, use binary search
         int index = find(positionSI);
         double remainder = positionSI - this.lengthIndexedLine[index];
         double fraction = remainder / (this.lengthIndexedLine[index + 1] - this.lengthIndexedLine[index]);
         OtsPoint3d p1 = this.points[index];
         OtsPoint3d p2 = this.points[index + 1];
         return new DirectedPoint(p1.x + fraction * (p2.x - p1.x), p1.y + fraction * (p2.y - p1.y),
                 p1.z + fraction * (p2.z - p1.z), 0.0, 0.0, Math.atan2(p2.y - p1.y, p2.x - p1.x));
     }

     /**
      * Truncate a line at the given length (less than the length of the line, and larger than zero) and return a new line.
      * @param lengthSI double; the location where to truncate the line
      * @return a new OtsLine3d truncated at the exact position where line.getLength() == lengthSI
      * @throws OtsGeometryException when position less than 0.0 or more than line length.
      */
     public final synchronized OtsLine3d truncate(final double lengthSI) throws OtsGeometryException
     {
         makeLengthIndexedLine();
         if (lengthSI <= 0.0 || lengthSI > getLengthSI())
         {
             throw new OtsGeometryException("truncate for line: position <= 0.0 or > line length. Position = " + lengthSI
                     + " m. Length = " + getLengthSI() + " m.");
         }

         // handle special case: position == length
         if (lengthSI == getLengthSI())
         {
             return new OtsLine3d(getPoints());
         }

         // find the index of the line segment
         int index = find(lengthSI);
         double remainder = lengthSI - this.lengthIndexedLine[index];
         double fraction = remainder / (this.lengthIndexedLine[index + 1] - this.lengthIndexedLine[index]);
         OtsPoint3d p1 = this.points[index];
         OtsPoint3d lastPoint;
         if (0.0 == fraction)
         {
             index--;
             lastPoint = p1;
         }
         else
         {
             OtsPoint3d p2 = this.points[index + 1];
             lastPoint = new OtsPoint3d(p1.x + fraction * (p2.x - p1.x), p1.y + fraction * (p2.y - p1.y),
                     p1.z + fraction * (p2.z - p1.z));

         }
         OtsPoint3d[] coords = new OtsPoint3d[index + 2];
         for (int i = 0; i <= index; i++)
         {
             coords[i] = this.points[i];
         }
         coords[index + 1] = lastPoint;
         return new OtsLine3d(coords);
     }

     /*-
      * TODO finish this method if it is needed; remove otherwise.
      * Calculate the first point on this line that intersects somewhere with the provided line, or NaN if no intersection was
      * found.
      * @param line the line to test the intersection with
      * @return the fraction of the first intersection point
      *
     public final double firstIntersectionFraction(final OtsLine3d line)
     {
         List<Line2D.Double> segs = new ArrayList<>();
         for (int j = 1; j < line.getPoints().length; j++)
         {
             Line2D.Double seg =
                 new Line2D.Double(this.points[j - 1].x, this.points[j - 1].y, this.points[j].x, this.points[j].y);
             segs.add(seg);

         }
         for (int i = 1; i < this.points.length; i++)
         {
             Line2D.Double thisSeg =
                 new Line2D.Double(this.points[i - 1].x, this.points[i - 1].y, this.points[i].x, this.points[i].y);
             for (Line2D.Double seg : segs)
             {
                 if (thisSeg.intersectsLine(seg))
                 {
                     // Point2D.Double intersectionPoint = thisSeg.

                 }
             }
         }
         return Double.NaN;
     }
      */

     /**
      * Returns the fractional position along this line of the orthogonal projection of point (x, y) on this line. If the point
      * is not orthogonal to the closest line segment, the nearest point is selected.
      * @param x double; x-coordinate of point to project
      * @param y double; y-coordinate of point to project
      * @return fractional position along this line of the orthogonal projection on this line of a point
      */
     public final synchronized double projectOrthogonal(final double x, final double y)
     {

         // prepare
         makeLengthIndexedLine();
         double minDistance = Double.POSITIVE_INFINITY;
         double minSegmentFraction = 0;
         int minSegment = -1;

         // code based on Line2D.ptSegDistSq(...)
         for (int i = 0; i < size() - 1; i++)
         {
             double dx = this.points[i + 1].x - this.points[i].x;
             double dy = this.points[i + 1].y - this.points[i].y;
             // vector relative to (x(i), y(i))
             double px = x - this.points[i].x;
             double py = y - this.points[i].y;
             // dot product
             double dot1 = px * dx + py * dy;
             double f;
             double distance;
             if (dot1 > 0)
             {
                 // vector relative to (x(i+1), y(i+1))
                 px = dx - px;
                 py = dy - py;
                 // dot product
                 double dot2 = px * dx + py * dy;
                 if (dot2 > 0)
                 {
                     // projection on line segment
                     double len2 = dx * dx + dy * dy;
                     double proj = dot2 * dot2 / len2;
                     f = dot1 / len2;
                     distance = px * px + py * py - proj;
                 }
                 else
                 {
                     // dot<=0 projection 'after' line segment
                     f = 1;
                     distance = px * px + py * py;
                 }
             }
             else
             {
                 // dot<=0 projection 'before' line segment
                 f = 0;
                 distance = px * px + py * py;
             }
             // check if closer than previous
             if (distance < minDistance)
             {
                 minDistance = distance;
                 minSegmentFraction = f;
                 minSegment = i;
             }
         }

         // return
         double segLen = this.lengthIndexedLine[minSegment + 1] - this.lengthIndexedLine[minSegment];
         return (this.lengthIndexedLine[minSegment] + segLen * minSegmentFraction) / getLengthSI();

     }

     /**
      * Returns the fractional projection of a point to a line. The projection works by taking slices in space per line segment
      * as shown below. A point is always projected to the nearest segment, but not necessarily to the closest point on that
      * segment. The slices in space are analogous to a Voronoi diagram, but for the line segments instead of points. If
      * fractional projection fails, the orthogonal projection is returned.<br>
      * <br>
      * The point 'A' is projected to point 'B' on the 3rd segment of line 'C-D'. The line from 'A' to 'B' extends towards point
      * 'E', which is the intersection of lines 'E-F' and 'E-G'. Line 'E-F' cuts the first bend of the 3rd segment (at point 'H')
      * in half, while the line 'E-G' cuts the second bend of the 3rd segment (at point 'I') in half.
      *
      * <pre>
      *            ____________________________     G                   .
      * .         |                            |    .                 .
      *   .       |  . . . .  helper lines     |    .               .
      *     .     |  _.._.._  projection line  |   I.             .
      *       .   |____________________________|  _.'._         .       L
      *        F.                              _.'  .  '-.    .
      *          ..                       B _.'     .     '-.
      *           . .                    _.\        .     .  D
      *            .  .               _.'   :       .   .
      *     J       .   .          _.'      \       . .
      *             ..    .     _.'          :      .                M
      *            .  .     ..-'             \      .
      *           .    .    /H.               A     .
      *          .      .  /    .                   .
      *        C _________/       .                 .
      *        .          .         .               .
      *   K   .            .          .             .
      *      .              .           .           .
      *     .                .            .         .           N
      *    .                  .             .       .
      *   .                    .              .     .
      *  .                      .               .   .
      * .                        .                . .
      *                           .                 .E
      *                            .                  .
      *                             .                   .
      *                              .                    .
      * </pre>
      *
      * Fractional projection may fail in three cases.
      * <ol>
      * <li>Numerical difficulties at slight bend, orthogonal projection returns the correct point.</li>
      * <li>Fractional projection is possible only to segments that aren't the nearest segment(s).</li>
      * <li>Fractional projection is possible for no segment.</li>
      * </ol>
      * In the latter two cases the projection is undefined and a orthogonal projection is returned if
      * {@code orthoFallback = true}, or {@code NaN} if {@code orthoFallback = false}.
      * @param start Direction; direction in first point
      * @param end Direction; direction in last point
      * @param x double; x-coordinate of point to project
      * @param y double; y-coordinate of point to project
      * @param fallback FractionalFallback; fallback method for when fractional projection fails
      * @return fractional position along this line of the fractional projection on that line of a point
      */
     public final synchronized double projectFractional(final Direction start, final Direction end, final double x,
             final double y, final FractionalFallback fallback) // XXX: DJ
     {

         // prepare
         makeLengthIndexedLine();
         double minDistance = Double.POSITIVE_INFINITY;
         double minSegmentFraction = 0;
         int minSegment = -1;
         OtsPoint3d point = new OtsPoint3d(x, y);

         // determine helpers (centers and directions)
         determineFractionalHelpers(start, end);

         // get distance of point to each segment
         double[] d = new double[this.points.length - 1];
         double minD = Double.POSITIVE_INFINITY;
         for (int i = 0; i < this.points.length - 1; i++)
         {
             d[i] = Line2D.ptSegDist(this.points[i].x, this.points[i].y, this.points[i + 1].x, this.points[i + 1].y, x, y);
             minD = d[i] < minD ? d[i] : minD;
         }

         // loop over segments for projection
         double distance;
         for (int i = 0; i < this.points.length - 1; i++)
         {
             // skip if not the closest segment, note that often two segments are equally close in their shared end point
             if (d[i] > minD + FRAC_PROJ_PRECISION)
             {
                 continue;
             }
             OtsPoint3d center = this.fractionalHelperCenters[i];
             OtsPoint3d p;
             if (center != null)
             {
                 // get intersection of line "center - (x, y)" and the segment
                 p = OtsPoint3d.intersectionOfLines(center, point, this.points[i], this.points[i + 1]);
                 if (p == null || (x < center.x + FRAC_PROJ_PRECISION && center.x + FRAC_PROJ_PRECISION < p.x)
                         || (x > center.x - FRAC_PROJ_PRECISION && center.x - FRAC_PROJ_PRECISION > p.x)
                         || (y < center.y + FRAC_PROJ_PRECISION && center.y + FRAC_PROJ_PRECISION < p.y)
                         || (y > center.y - FRAC_PROJ_PRECISION && center.y - FRAC_PROJ_PRECISION > p.y))
                 {
                     // projected point may not be 'beyond' segment center (i.e. center may not be between (x, y) and (p.x, p.y)
                     continue;
                 }
             }
             else
             {
                 // parallel helper lines, project along direction
                 OtsPoint3d offsetPoint =
                         new OtsPoint3d(x + this.fractionalHelperDirections[i].x, y + this.fractionalHelperDirections[i].y);
                 p = OtsPoint3d.intersectionOfLines(point, offsetPoint, this.points[i], this.points[i + 1]);
             }
             double segLength = this.points[i].distance(this.points[i + 1]).si + FRAC_PROJ_PRECISION;
             if (p == null || this.points[i].distance(p).si > segLength || this.points[i + 1].distance(p).si > segLength)
             {
                 // intersection must be on the segment
                 // in case of p == null, the length of the fractional helper direction falls away due to precision
                 continue;
             }
             // distance from (x, y) to intersection on segment
             double dx = x - p.x;
             double dy = y - p.y;
             distance = Math.sqrt(dx * dx + dy * dy);
             // distance from start of segment to point on segment
             if (distance < minDistance)
             {
                 dx = p.x - this.points[i].x;
                 dy = p.y - this.points[i].y;
                 double dFrac = Math.sqrt(dx * dx + dy * dy);
                 // fraction to point on segment
                 minDistance = distance;
                 minSegmentFraction = dFrac / (this.lengthIndexedLine[i + 1] - this.lengthIndexedLine[i]);
                 minSegment = i;
             }
         }

         // return
         if (minSegment == -1)

         {
             /*
              * If fractional projection fails (x, y) is either outside of the applicable area for fractional projection, or is
              * inside an area where numerical difficulties arise (i.e. far away outside of very slight bend which is considered
              * parallel).
              */
             // CategoryLogger.info(Cat.CORE, "projectFractional failed to project " + point + " on " + this
             // + "; using fallback approach");
             return fallback.getFraction(this, x, y);
         }

         double segLen = this.lengthIndexedLine[minSegment + 1] - this.lengthIndexedLine[minSegment];
         return (this.lengthIndexedLine[minSegment] + segLen * minSegmentFraction) / getLengthSI();

     }

     /**
      * Fallback method for when fractional projection fails as the point is beyond the line or from numerical limitations.
      * <p>
      * Copyright (c) 2013-2023 Delft University of Technology, PO Box 5, 2600 AA, Delft, the Netherlands. All rights reserved.
      * <br>
      * BSD-style license. See <a href="https://opentrafficsim.org/docs/license.html">OpenTrafficSim License</a>.
      * </p>
      * @author <a href="https://github.com/averbraeck">Alexander Verbraeck</a>
      * @author <a href="https://tudelft.nl/staff/p.knoppers-1">Peter Knoppers</a>
      * @author <a href="https://dittlab.tudelft.nl">Wouter Schakel</a>
      */
     public enum FractionalFallback // XXX: DJ
     {
         /** Orthogonal projection. */
         ORTHOGONAL
         {
             @Override
             double getFraction(final OtsLine3d line, final double x, final double y)
             {
                 return line.projectOrthogonal(x, y);
             }
         },

         /** Distance to nearest end point. */
         ENDPOINT
         {
             @Override
             double getFraction(final OtsLine3d line, final double x, final double y)
             {
                 OtsPoint3d point = new OtsPoint3d(x, y);
                 double dStart = point.distanceSI(line.getFirst());
                 double dEnd = point.distanceSI(line.getLast());
                 if (dStart < dEnd)
                 {
                     return -dStart / line.getLengthSI();
                 }
                 else
                 {
                     return (dEnd + line.getLengthSI()) / line.getLengthSI();
                 }
             }
         },

         /** NaN value. */
         NaN
         {
             @Override
             double getFraction(final OtsLine3d line, final double x, final double y)
             {
                 return Double.NaN;
             }
         };

         /**
          * Returns fraction for when fractional projection fails as the point is beyond the line or from numerical limitations.
          * @param line OtsLine3d; line
          * @param x double; x coordinate of point
          * @param y double; y coordinate of point
          * @return double; fraction for when fractional projection fails
          */
         abstract double getFraction(OtsLine3d line, double x, double y);

     }

     /**
      * Determines all helpers (points and/or directions) for fractional projection and stores fixed information in properties
      * while returning the first and last center points (.
      * @param start Direction; direction in first point
      * @param end Direction; direction in last point
      */
     private synchronized void determineFractionalHelpers(final Direction start, final Direction end) // XXX: DJ
     {

         final int n = this.points.length - 1;

         // calculate fixed helpers if not done yet
         if (this.fractionalHelperCenters == null)
         {
             this.fractionalHelperCenters = new OtsPoint3d[n];
             this.fractionalHelperDirections = new Point2D.Double[n];
             if (this.points.length > 2)
             {
                 // intersection of parallel lines of first and second segment
                 OtsLine3d prevOfsSeg = unitOffsetSegment(0);
                 OtsLine3d nextOfsSeg = unitOffsetSegment(1);
                 OtsPoint3d parStartPoint;
                 try
                 {
                     parStartPoint = OtsPoint3d.intersectionOfLines(prevOfsSeg.get(0), prevOfsSeg.get(1), nextOfsSeg.get(0),
                             nextOfsSeg.get(1));
                     if (parStartPoint == null || prevOfsSeg.get(1).distanceSI(nextOfsSeg.get(0)) < Math
                             .min(prevOfsSeg.get(1).distanceSI(parStartPoint), nextOfsSeg.get(0).distanceSI(parStartPoint)))
                     {
                         parStartPoint = new OtsPoint3d((prevOfsSeg.get(1).x + nextOfsSeg.get(0).x) / 2,
                                 (prevOfsSeg.get(1).y + nextOfsSeg.get(0).y) / 2);
                     }
                 }
                 catch (OtsGeometryException oge)
                 {
                     // cannot happen as only the first and second point (which are always present) are requested
                     throw new RuntimeException(oge);
                 }
                 // remember the intersection of the first two unit offset segments
                 this.firstOffsetIntersection = parStartPoint;
                 // loop segments
                 for (int i = 1; i < this.points.length - 2; i++)
                 {
                     prevOfsSeg = nextOfsSeg;
                     nextOfsSeg = unitOffsetSegment(i + 1);
                     OtsPoint3d parEndPoint;
                     try
                     {
                         parEndPoint = OtsPoint3d.intersectionOfLines(prevOfsSeg.get(0), prevOfsSeg.get(1), nextOfsSeg.get(0),
                                 nextOfsSeg.get(1));
                         if (parEndPoint == null || prevOfsSeg.get(1).distanceSI(nextOfsSeg.get(0)) < Math
                                 .min(prevOfsSeg.get(1).distanceSI(parEndPoint), nextOfsSeg.get(0).distanceSI(parEndPoint)))
                         {
                             parEndPoint = new OtsPoint3d((prevOfsSeg.get(1).x + nextOfsSeg.get(0).x) / 2,
                                     (prevOfsSeg.get(1).y + nextOfsSeg.get(0).y) / 2);
                         }
                     }
                     catch (OtsGeometryException oge)
                     {
                         // cannot happen as only the first and second point (which are always present) are requested
                         throw new RuntimeException(oge);
                     }
                     // center = intersections of helper lines
                     this.fractionalHelperCenters[i] =
                             OtsPoint3d.intersectionOfLines(this.points[i], parStartPoint, this.points[i + 1], parEndPoint);
                     if (this.fractionalHelperCenters[i] == null)
                     {
                         // parallel helper lines, parallel segments or /\/ cause parallel helper lines, use direction
                         this.fractionalHelperDirections[i] =
                                 new Point2D.Double(parStartPoint.x - this.points[i].x, parStartPoint.y - this.points[i].y);
                     }
                     parStartPoint = parEndPoint;
                 }
                 // remember the intersection of the last two unit offset segments
                 this.lastOffsetIntersection = parStartPoint;
             }
         }

         // use directions at start and end to get unit offset points to the left at a distance of 1
         double ang = (start == null ? Math.atan2(this.points[1].y - this.points[0].y, this.points[1].x - this.points[0].x)
                 : start.getInUnit(DirectionUnit.DEFAULT)) + Math.PI / 2; // start.si + Math.PI / 2;
         OtsPoint3d p1 = new OtsPoint3d(this.points[0].x + Math.cos(ang), this.points[0].y + Math.sin(ang));
         ang = (end == null ? Math.atan2(this.points[n].y - this.points[n - 1].y, this.points[n].x - this.points[n - 1].x)
                 : end.getInUnit(DirectionUnit.DEFAULT)) + Math.PI / 2; // end.si + Math.PI / 2;
         OtsPoint3d p2 = new OtsPoint3d(this.points[n].x + Math.cos(ang), this.points[n].y + Math.sin(ang));

         // calculate first and last center (i.e. intersection of unit offset segments), which depend on inputs 'start' and 'end'
         if (this.points.length > 2)
         {
             this.fractionalHelperCenters[0] =
                     OtsPoint3d.intersectionOfLines(this.points[0], p1, this.points[1], this.firstOffsetIntersection);
             this.fractionalHelperCenters[n - 1] =
                     OtsPoint3d.intersectionOfLines(this.points[n - 1], this.lastOffsetIntersection, this.points[n], p2);
             if (this.fractionalHelperCenters[n - 1] == null)
             {
                 // parallel helper lines, use direction for projection
                 this.fractionalHelperDirections[n - 1] = new Point2D.Double(p2.x - this.points[n].x, p2.y - this.points[n].y);
             }
         }
         else
         {
             // only a single segment
             this.fractionalHelperCenters[0] = OtsPoint3d.intersectionOfLines(this.points[0], p1, this.points[1], p2);
         }
         if (this.fractionalHelperCenters[0] == null)
         {
             // parallel helper lines, use direction for projection
             this.fractionalHelperDirections[0] = new Point2D.Double(p1.x - this.points[0].x, p1.y - this.points[0].y);
         }

     }

     /**
      * Helper method for fractional projection which returns an offset line to the left of a segment at a distance of 1.
      * @param segment int; segment number
      * @return parallel line to the left of a segment at a distance of 1
      */
     private synchronized OtsLine3d unitOffsetSegment(final int segment) // XXX: DJ
     {

         // double angle = Math.atan2(this.points[segment + 1].y - this.points[segment].y,
         // this.points[segment + 1].x - this.points[segment].x) + Math.PI / 2;
         // while (angle > Math.PI)
         // {
         // angle -= Math.PI;
         // }
         // while (angle < -Math.PI)
         // {
         // angle += Math.PI;
         // }
         // OtsPoint3d from = new OtsPoint3d(this.points[segment].x + Math.cos(angle), this.points[segment].y + Math.sin(angle));
         // OtsPoint3d to =
         // new OtsPoint3d(this.points[segment + 1].x + Math.cos(angle), this.points[segment + 1].y + Math.sin(angle));
         // try
         // {
         // return new OtsLine3d(from, to);
         // }
         // catch (OTSGeometryException oge)
         // {
         // // cannot happen as points are from this OtsLine3d which performed the same checks and 2 points are given
         // throw new RuntimeException(oge);
         // }
         OtsPoint3d from = new OtsPoint3d(this.points[segment].x, this.points[segment].y);
         OtsPoint3d to = new OtsPoint3d(this.points[segment + 1].x, this.points[segment + 1].y);
         try
         {
             OtsLine3d line = new OtsLine3d(from, to);
             return line.offsetLine(1.0);
         }
         catch (OtsGeometryException oge)
         {
             // Cannot happen as points are from this OtsLine3d which performed the same checks and 2 points are given
             throw new RuntimeException(oge);
         }
     }

     /**
      * Returns the projected directional radius of the line at a given fraction. Negative values reflect right-hand curvature in
      * the design-line direction. The radius is taken as the minimum of the radii at the vertices before and after the given
      * fraction. The radius at a vertex is calculated as the radius of a circle that is equidistant from both edges connected to
      * the vertex. The circle center is on a line perpendicular to the shortest edge, crossing through the middle of the
      * shortest edge. This method ignores Z components.
      * @param fraction double; fraction along the line, between 0.0 and 1.0 (both inclusive)
      * @return Length; radius; the local radius; or si field set to Double.NaN if line is totally straight
      * @throws OtsGeometryException fraction out of bounds
      */
     public synchronized Length getProjectedRadius(final double fraction) throws OtsGeometryException // XXX: DJ
     {
         Throw.when(fraction < 0.0 || fraction > 1.0, OtsGeometryException.class, "Fraction %f is out of bounds [0.0 ... 1.0]",
                 fraction);
         if (this.vertexRadii == null)
         {
             this.vertexRadii = new Length[size() - 1];
         }
         int index = find(fraction * getLength().si);
         if (index > 0 && this.vertexRadii[index] == null)
         {
             this.vertexRadii[index] = getProjectedVertexRadius(index);
         }
         if (index < size() - 2 && this.vertexRadii[index + 1] == null)
         {
             this.vertexRadii[index + 1] = getProjectedVertexRadius(index + 1);
         }
         if (index == 0)
         {
             if (this.vertexRadii.length < 2)
             {
                 return Length.instantiateSI(Double.NaN);
             }
             return this.vertexRadii[1];
         }
         if (index == size() - 2)
         {
             return this.vertexRadii[size() - 2];
         }
         return Math.abs(this.vertexRadii[index].si) < Math.abs(this.vertexRadii[index + 1].si) ? this.vertexRadii[index]
                 : this.vertexRadii[index + 1];
     }

     /**
      * Calculates the directional radius at a vertex. Negative values reflect right-hand curvature in the design-line direction.
      * The radius at a vertex is calculated as the radius of a circle that is equidistant from both edges connected to the
      * vertex. The circle center is on a line perpendicular to the shortest edge, crossing through the middle of the shortest
      * edge. This function ignores Z components.
      * @param index int; index of the vertex in range [1 ... size() - 2]
      * @return Length; radius at the vertex
      * @throws OtsGeometryException if the index is out of bounds
      */
     public synchronized Length getProjectedVertexRadius(final int index) throws OtsGeometryException // XXX: DJ
     {
         Throw.when(index < 1 || index > size() - 2, OtsGeometryException.class, "Index %d is out of bounds [1 ... size() - 2].",
                 index);
         makeLengthIndexedLine();
         determineFractionalHelpers(null, null);
         double length1 = this.lengthIndexedLine[index] - this.lengthIndexedLine[index - 1];
         double length2 = this.lengthIndexedLine[index + 1] - this.lengthIndexedLine[index];
         int shortIndex = length1 < length2 ? index : index + 1;
         // center of shortest edge
         OtsPoint3d p1 = new OtsPoint3d(.5 * (this.points[shortIndex - 1].x + this.points[shortIndex].x),
                 .5 * (this.points[shortIndex - 1].y + this.points[shortIndex].y),
                 .5 * (this.points[shortIndex - 1].z + this.points[shortIndex].z));
         // perpendicular to shortest edge, line crossing p1
         OtsPoint3d p2 = new OtsPoint3d(p1.x + (this.points[shortIndex].y - this.points[shortIndex - 1].y),
                 p1.y - (this.points[shortIndex].x - this.points[shortIndex - 1].x), p1.z);
         // vertex
         OtsPoint3d p3 = this.points[index];
         // point on line that splits angle between edges at vertex 50-50
         OtsPoint3d p4 = this.fractionalHelperCenters[index];
         if (p4 == null)
         {
             // parallel helper lines
             p4 = new OtsPoint3d(p3.x + this.fractionalHelperDirections[index].x,
                     p3.y + this.fractionalHelperDirections[index].y);
         }
         OtsPoint3d intersection = OtsPoint3d.intersectionOfLines(p1, p2, p3, p4);
         if (null == intersection)
         {
             return Length.instantiateSI(Double.NaN);
         }
         // determine left or right
         double refLength = length1 < length2 ? length1 : length2;
         Length radius = intersection.horizontalDistance(p1);
         Length i2p2 = intersection.horizontalDistance(p2);
         if (radius.si < i2p2.si && i2p2.si > refLength)
         {
             // left as p1 is closer than p2 (which was placed to the right) and not on the perpendicular line
             return radius;
         }
         // right as not left
         return radius.neg();
     }

     /**
      * Returns the length fraction at the vertex.
      * @param index int; index of vertex [0 ... size() - 1]
      * @return double; length fraction at the vertex
      * @throws OtsGeometryException if the index is out of bounds
      */
     public synchronized double getVertexFraction(final int index) throws OtsGeometryException // XXX: DJ
     {
         Throw.when(index < 0 || index > size() - 1, OtsGeometryException.class, "Index %d is out of bounds [0 %d].", index,
                 size() - 1);
         makeLengthIndexedLine();
         return this.lengthIndexedLine[index] / getLengthSI();
     }

     /**
      * Calculate the centroid of this line, and the bounds, and cache for later use.
      */
     private synchronized void calcCentroidBounds()
     {
         double minX = Double.POSITIVE_INFINITY;
         double minY = Double.POSITIVE_INFINITY;
         double minZ = Double.POSITIVE_INFINITY;
         double maxX = Double.NEGATIVE_INFINITY;
         double maxY = Double.NEGATIVE_INFINITY;
         double maxZ = Double.NEGATIVE_INFINITY;
         for (OtsPoint3d p : this.points)
         {
             minX = Math.min(minX, p.x);
             minY = Math.min(minY, p.y);
             minZ = Math.min(minZ, p.z);
             maxX = Math.max(maxX, p.x);
             maxY = Math.max(maxY, p.y);
             maxZ = Math.max(maxZ, p.z);
         }
         this.centroid = new OtsPoint3d((maxX + minX) / 2, (maxY + minY) / 2, (maxZ + minZ) / 2);
         double deltaX = maxX - minX;
         double deltaY = maxY - minY;
         double deltaZ = maxZ - minZ;
         this.bounds = new Bounds(new Point3d(-deltaX / 2.0, -deltaY / 2.0, -deltaZ / 2.0),
                 new Point3d(deltaX / 2, deltaY / 2, deltaZ / 2));
         this.envelope = new Bounds(new Point3d(minX, minY, 0), new Point3d(maxX, maxY, 0));
     }

     /**
      * Retrieve the centroid of this OtsLine3d.
      * @return OtsPoint3d; the centroid of this OtsLine3d
      */
     public final synchronized OtsPoint3d getCentroid()
     {
         if (this.centroid == null)
         {
             calcCentroidBounds();
         }
         return this.centroid;
     }

     /**
      * Get the bounding rectangle of this OtsLine3d.
      * @return Rectangle2D; the bounding rectangle of this OtsLine3d
      */
     public final synchronized Bounds getEnvelope() // XXX: DJ uses BoundingRectangle
     {
         if (this.envelope == null)
         {
             calcCentroidBounds();
         }
         return this.envelope;
     }

     /** {@inheritDoc} */
     @Override
     @SuppressWarnings("checkstyle:designforextension")
     public synchronized DirectedPoint getLocation() // XXX: DJ
     {
         if (this.centroid == null)
         {
             calcCentroidBounds();
         }
         return this.centroid.getDirectedPoint();
     }

     /** {@inheritDoc} */
     @Override
     @SuppressWarnings("checkstyle:designforextension")
     public synchronized Bounds getBounds() // XXX: DJ
     {
         if (this.bounds == null)
         {
             calcCentroidBounds();
         }
         return this.bounds;
     }

     /** {@inheritDoc} */
     @Override
     @SuppressWarnings("checkstyle:designforextension")
     public String toString()
     {
         return Arrays.toString(this.points);
     }

     /** {@inheritDoc} */
     @Override
     @SuppressWarnings("checkstyle:designforextension")
     public int hashCode()
     {
         final int prime = 31;
         int result = 1;
         result = prime * result + Arrays.hashCode(this.points);
         return result;
     }

     /** {@inheritDoc} */
     @Override
     @SuppressWarnings({"checkstyle:designforextension", "checkstyle:needbraces"})
     public boolean equals(final Object obj)
     {
         if (this == obj)
             return true;
         if (obj == null)
             return false;
         if (getClass() != obj.getClass())
             return false;
         OtsLine3d other = (OtsLine3d) obj;
         if (!Arrays.equals(this.points, other.points))
             return false;
         return true;
     }

     /**
      * Convert the 2D projection of this OtsLine3d to something that MS-Excel can plot.
      * @return excel XY plottable output
      */
     public final String toExcel()
     {
         StringBuffer s = new StringBuffer();
         for (OtsPoint3d p : this.points)
         {
             s.append(p.x + "\t" + p.y + "\n");
         }
         return s.toString();
     }

     /**
      * Convert the 2D projection of this OtsLine3d to Peter's plot format.
      * @return Peter's format plot output
      */
     public final String toPlot() // XXX: DJ
     {
         StringBuffer result = new StringBuffer();
         for (OtsPoint3d p : this.points)
         {
             result.append(String.format(Locale.US, "%s%.3f,%.3f", 0 == result.length() ? "M" : " L", p.x, p.y));
         }
         result.append("\n");
         return result.toString();
     }

 }