package org.opentrafficsim.core.geometry;
   
   import org.djunits.value.vdouble.scalar.Angle;
   import org.djunits.value.vdouble.scalar.Direction;
   import org.djutils.base.AngleUtil;
   import org.djutils.draw.line.PolyLine2d;
   import org.djutils.draw.point.OrientedPoint2d;
   import org.djutils.draw.point.Point2d;
   import org.djutils.exceptions.Throw;
  import org.djutils.exceptions.Try;
  
  /**
   * Continuous definition of a clothoid. The following definitions are available:
   * <ul>
   * <li>A clothoid between two directed <i>points</i>.</li>
   * <li>A clothoid originating from a directed point with start curvature, end curvature, and <i>length</i> specified.</li>
   * <li>A clothoid originating from a directed point with start curvature, end curvature, and <i>A-value</i> specified.</li>
   * </ul>
   * This class is based on:
   * <ul>
   * <li>Dale Connor and Lilia Krivodonova (2014) "Interpolation of two-dimensional curves with Euler spirals", Journal of
   * Computational and Applied Mathematics, Volume 261, 1 May 2014, pp. 320-332.</li>
   * <li>D.J. Waltona and D.S. Meek (2009) "G<sup>1</sup> interpolation with a single Cornu spiral segment", Journal of
   * Computational and Applied Mathematics, Volume 223, Issue 1, 1 January 2009, pp. 86-96.</li>
   * </ul>
   * <p>
   * Copyright (c) 2023-2024 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://github.com/peter-knoppers">Peter Knoppers</a>
   * @author <a href="https://github.com/wjschakel">Wouter Schakel</a>
   * @see <a href="https://www.sciencedirect.com/science/article/pii/S0377042713006286">Connor and Krivodonova (2014)</a>
   * @see <a href="https://www.sciencedirect.com/science/article/pii/S0377042704000925">Waltona and Meek (2009)</a>
   */
  public class ContinuousClothoid implements ContinuousLine
  {
  
      /** Threshold to consider input to be a trivial straight or circle arc. The value is 1/10th of a degree. */
      private static final double ANGLE_TOLERANCE = 2.0 * Math.PI / 3600.0;
  
      /** Stopping tolerance for the Secant method to find optimal theta values. */
      private static final double SECANT_TOLERANCE = 1e-8;
  
      /** Start point with direction. */
      private final OrientedPoint2d startPoint;
  
      /** End point with direction. */
      private final OrientedPoint2d endPoint;
  
      /** Start curvature. */
      private final double startCurvature;
  
      /** End curvature. */
      private final double endCurvature;
  
      /** Length. */
      private final double length;
  
      /**
       * A-value; for scaling the Fresnel integral. The regular clothoid A-parameter is obtained by dividing by
       * {@code Math.sqrt(Math.PI)}.
       */
      private final double a;
  
      /** Minimum alpha value of line to draw. */
      private final double alphaMin;
  
      /** Maximum alpha value of line to draw. */
      private final double alphaMax;
  
      /** Unit vector from the origin of the clothoid, towards the positive side. */
      private final double[] t0;
  
      /** Normal unit vector to t0. */
      private final double[] n0;
  
      /** Whether the line needs to be flipped. */
      private final boolean opposite;
  
      /** Whether the line is reflected. */
      private final boolean reflected;
  
      /** Simplification to straight when valid. */
      private final ContinuousStraight straight;
  
      /** Simplification to arc when valid. */
      private final ContinuousArc arc;
  
      /** Whether the shift was determined. */
      private boolean shiftDetermined;
  
      /** Shift in x-coordinate of start point. */
      private double shiftX;
  
      /** Shift in y-coordinate of start point. */
      private double shiftY;
  
      /** Additional shift in x-coordinate towards end point. */
     private double dShiftX;
 
     /** Additional shift in y-coordinate towards end point. */
     private double dShiftY;
 
     /**
      * Create clothoid between two directed points. This constructor is based on the procedure in:<br>
      * <br>
      * Dale Connor and Lilia Krivodonova (2014) "Interpolation of two-dimensional curves with Euler spirals", Journal of
      * Computational and Applied Mathematics, Volume 261, 1 May 2014, pp. 320-332.<br>
      * <br>
      * Which applies the theory proven in:<br>
      * <br>
      * D.J. Waltona and D.S. Meek (2009) "G<sup>1</sup> interpolation with a single Cornu spiral segment", Journal of
      * Computational and Applied Mathematics, Volume 223, Issue 1, 1 January 2009, pp. 86-96.<br>
      * <br>
      * This procedure guarantees that the resulting line has the minimal angle rotation that is required to connect the points.
      * If the points approximate a straight line or circle, with a tolerance of up 1/10th of a degree, those respective lines
      * are created. The numerical approximation of the underlying Fresnel integral is different from the paper. See
      * {@code Clothoid.fresnal()}.
      * @param startPoint start point.
      * @param endPoint end point.
      * @see <a href="https://www.sciencedirect.com/science/article/pii/S0377042713006286">Connor and Krivodonova (2014)</a>
      * @see <a href="https://www.sciencedirect.com/science/article/pii/S0377042704000925">Waltona and Meek (2009)</a>
      */
     public ContinuousClothoid(final OrientedPoint2d startPoint, final OrientedPoint2d endPoint)
     {
         Throw.whenNull(startPoint, "Start point may not be null.");
         Throw.whenNull(endPoint, "End point may not be null.");
         this.startPoint = startPoint;
         this.endPoint = endPoint;
 
         double dx = endPoint.x - startPoint.x;
         double dy = endPoint.y - startPoint.y;
         double d2 = Math.hypot(dx, dy); // length of straight line from start to end
         double d = Math.atan2(dy, dx); // angle of line through start and end points
 
         double phi1 = AngleUtil.normalizeAroundZero(d - startPoint.dirZ);
         double phi2 = AngleUtil.normalizeAroundZero(endPoint.dirZ - d);
         double phi1Abs = Math.abs(phi1);
         double phi2Abs = Math.abs(phi2);
 
         if (phi1Abs < ANGLE_TOLERANCE && phi2Abs < ANGLE_TOLERANCE)
         {
             // Straight
             this.length = Math.hypot(endPoint.x - startPoint.x, endPoint.y - startPoint.y);
             this.a = Double.POSITIVE_INFINITY;
             this.startCurvature = 0.0;
             this.endCurvature = 0.0;
             this.straight = new ContinuousStraight(startPoint, this.length);
             this.arc = null;
             this.alphaMin = 0.0;
             this.alphaMax = 0.0;
             this.t0 = null;
             this.n0 = null;
             this.opposite = false;
             this.reflected = false;
             return;
         }
         else if (Math.abs(phi2 - phi1) < ANGLE_TOLERANCE)
         {
             // Arc
             double r = .5 * d2 / Math.sin(phi1);
             double cosStartDirection = Math.cos(startPoint.dirZ);
             double sinStartDirection = Math.sin(startPoint.dirZ);
             double ang = Math.PI / 2.0;
             double cosAng = Math.cos(ang); // =0
             double sinAng = Math.sin(ang); // =1
             double x0 = startPoint.x - r * (cosStartDirection * cosAng + sinStartDirection * sinAng);
             double y0 = startPoint.y - r * (cosStartDirection * -sinAng + sinStartDirection * cosAng);
             double from = Math.atan2(startPoint.y - y0, startPoint.x - x0);
             double to = Math.atan2(endPoint.y - y0, endPoint.x - x0);
             if (r < 0 && to > from)
             {
                 to = to - 2.0 * Math.PI;
             }
             else if (r > 0 && to < from)
             {
                 to = to + 2.0 * Math.PI;
             }
             Angle angle = Angle.instantiateSI(Math.abs(to - from));
             this.length = angle.si * Math.abs(r);
             this.a = 0.0;
             this.startCurvature = 1.0 / r;
             this.endCurvature = 1.0 / r;
             this.straight = null;
             this.arc = new ContinuousArc(startPoint, Math.abs(r), r > 0.0, angle);
             this.alphaMin = 0.0;
             this.alphaMax = 0.0;
             this.t0 = null;
             this.n0 = null;
             this.opposite = false;
             this.reflected = false;
             return;
         }
         this.straight = null;
         this.arc = null;
 
         // The algorithm assumes |phi2| to be larger than |phi1|. If this is not the case, the clothoid is created in the
         // opposite direction.
         if (phi2Abs < phi1Abs)
         {
             this.opposite = true;
             double phi3 = phi1;
             phi1 = -phi2;
             phi2 = -phi3;
             dx = -dx;
             dy = -dy;
         }
         else
         {
             this.opposite = false;
         }
 
         // The algorithm assumes 0 < phi2 < pi. If this is not the case, the input and output are reflected on 'd'.
         this.reflected = phi2 < 0 || phi2 > Math.PI;
         if (this.reflected)
         {
             phi1 = -phi1;
             phi2 = -phi2;
         }
 
         // h(phi1, phi2) guarantees for negative values along with 0 < phi1 < phi2 < pi, that a C-shaped clothoid exists.
         Fresnel cs = Fresnel.integral(alphaToT(phi1 + phi2));
         double h = cs.s() * Math.cos(phi1) - cs.c() * Math.sin(phi1);
         boolean cShape = 0 < phi1 && phi1 < phi2 && phi2 < Math.PI && h < 0; // otherwise, S-shape
         double theta = getTheta(phi1, phi2, cShape);
         double aSign = cShape ? -1.0 : 1.0;
         double thetaSign = -aSign;
 
         double v1 = theta + phi1 + phi2;
         double v2 = theta + phi1;
         Fresnel cs0 = Fresnel.integral(alphaToT(theta));
         Fresnel cs1 = Fresnel.integral(alphaToT(v1));
         this.a = d2 / ((cs1.s() + aSign * cs0.s()) * Math.sin(v2) + (cs1.c() + aSign * cs0.c()) * Math.cos(v2));
 
         dx /= d2; // normalized
         dy /= d2;
         if (this.reflected)
         {
             // reflect t0 and n0 on 'd' so that the created output clothoid is reflected back after input was reflected
             this.t0 = new double[] {Math.cos(-v2) * dx + Math.sin(-v2) * dy, -Math.sin(-v2) * dx + Math.cos(-v2) * dy};
             this.n0 = new double[] {-this.t0[1], this.t0[0]};
         }
         else
         {
             this.t0 = new double[] {Math.cos(v2) * dx + Math.sin(v2) * dy, -Math.sin(v2) * dx + Math.cos(v2) * dy};
             this.n0 = new double[] {this.t0[1], -this.t0[0]};
         }
 
         this.alphaMin = thetaSign * theta;
         this.alphaMax = v1; // alphaMax = theta + phi1 + phi2, which is v1
         double sign = (this.reflected ? -1.0 : 1.0);
         double curveMin = Math.PI * alphaToT(this.alphaMin) / this.a;
         double curveMax = Math.PI * alphaToT(v1) / this.a;
         this.startCurvature = sign * (this.opposite ? -curveMax : curveMin);
         this.endCurvature = sign * (this.opposite ? -curveMin : curveMax);
         this.length = this.a * (alphaToT(v1) - alphaToT(this.alphaMin));
     }
 
     /**
      * Create clothoid from one point based on curvature and A-value.
      * @param startPoint start point.
      * @param a A-value.
      * @param startCurvature start curvature.
      * @param endCurvature end curvature;
      */
     public ContinuousClothoid(final OrientedPoint2d startPoint, final double a, final double startCurvature,
             final double endCurvature)
     {
         Throw.whenNull(startPoint, "Start point may not be null.");
         Throw.when(a <= 0.0, IllegalArgumentException.class, "A value must be above 0.");
         this.startPoint = startPoint;
         // Scale 'a', due to parameter conversion between C(alpha)/S(alpha) and C(t)/S(t); t = sqrt(2*alpha/pi).
         this.a = a * Math.sqrt(Math.PI);
         this.length = a * a * Math.abs(endCurvature - startCurvature);
         this.startCurvature = startCurvature;
         this.endCurvature = endCurvature;
 
         double l1 = a * a * startCurvature;
         double l2 = a * a * endCurvature;
         this.alphaMin = Math.abs(l1) * startCurvature / 2.0;
         this.alphaMax = Math.abs(l2) * endCurvature / 2.0;
 
         double ang = AngleUtil.normalizeAroundZero(startPoint.dirZ) - Math.abs(this.alphaMin);
         this.t0 = new double[] {Math.cos(ang), Math.sin(ang)};
         this.n0 = new double[] {this.t0[1], -this.t0[0]};
         Direction endDirection = Direction.instantiateSI(ang + Math.abs(this.alphaMax));
         if (startCurvature > endCurvature)
         {
             // In these cases the algorithm works in the negative direction. We need to flip over the line through the start
             // point that runs perpendicular to the start direction.
             double m = Math.tan(startPoint.dirZ + Math.PI / 2.0);
 
             // Linear algebra flipping, see: https://math.stackexchange.com/questions/525082/reflection-across-a-line
             double onePlusMm = 1.0 + m * m;
             double oneMinusMm = 1.0 - m * m;
             double mmMinusOne = m * m - 1.0;
             double twoM = 2.0 * m;
             double t00 = this.t0[0];
             double t01 = this.t0[1];
             double n00 = this.n0[0];
             double n01 = this.n0[1];
             this.t0[0] = (oneMinusMm * t00 + 2 * m * t01) / onePlusMm;
             this.t0[1] = (twoM * t00 + mmMinusOne * t01) / onePlusMm;
             this.n0[0] = (oneMinusMm * n00 + 2 * m * n01) / onePlusMm;
             this.n0[1] = (twoM * n00 + mmMinusOne * n01) / onePlusMm;
 
             double ang2 = Math.atan2(this.t0[1], this.t0[0]);
             endDirection = Direction.instantiateSI(ang2 - Math.abs(this.alphaMax) + Math.PI);
         }
         PolyLine2d line = flatten(new Flattener.NumSegments(1));
         Point2d end = Try.assign(() -> line.get(line.size() - 1), "Line does not have an end point.");
         this.endPoint = new OrientedPoint2d(end.x, end.y, endDirection.si);
 
         // Fields not relevant for definition with curvatures
         this.straight = null;
         this.arc = null;
         this.opposite = false;
         this.reflected = false;
     }
 
     /**
      * Create clothoid from one point based on curvature and length. This method calculates the A-value as
      * <i>sqrt(L/|k2-k1|)</i>, where <i>L</i> is the length of the resulting clothoid, and <i>k2</i> and <i>k1</i> are the end
      * and start curvature.
      * @param startPoint start point.
      * @param length Length of the resulting clothoid.
      * @param startCurvature start curvature.
      * @param endCurvature end curvature;
      * @return clothoid based on curvature and length.
      */
     public static ContinuousClothoid withLength(final OrientedPoint2d startPoint, final double length,
             final double startCurvature, final double endCurvature)
     {
         Throw.when(length <= 0.0, IllegalArgumentException.class, "Length must be above 0.");
         double a = Math.sqrt(length / Math.abs(endCurvature - startCurvature));
         return new ContinuousClothoid(startPoint, a, startCurvature, endCurvature);
     }
 
     /**
      * Performs alpha to t variable change.
      * @param alpha alpha value, must be positive.
      * @return t value (length along the Fresnel integral, also known as x).
      */
     private static double alphaToT(final double alpha)
     {
         return alpha >= 0 ? Math.sqrt(alpha * 2.0 / Math.PI) : -Math.sqrt(-alpha * 2.0 / Math.PI);
     }
 
     /**
      * Returns theta value given shape to use. If no such value is found, the other shape may be attempted.
      * @param phi1 phi1.
      * @param phi2 phi2.
      * @param cShape C-shaped, or S-shaped otherwise.
      * @return theta value; the number of radians that is moved on to a side of the full clothoid.
      */
     private static double getTheta(final double phi1, final double phi2, final boolean cShape)
     {
         double sign, phiMin, phiMax;
         if (cShape)
         {
             double lambda = (1 - Math.cos(phi1)) / (1 - Math.cos(phi2));
             phiMin = 0.0;
             phiMax = (lambda * lambda * (phi1 + phi2)) / (1 - (lambda * lambda));
             sign = -1.0;
         }
         else
         {
             phiMin = Math.max(0, -phi1);
             phiMax = Math.PI / 2 - phi1;
             sign = 1;
         }
 
         double fMin = fTheta(phiMin, phi1, phi2, sign);
         double fMax = fTheta(phiMax, phi1, phi2, sign);
         if (fMin * fMax > 0)
         {
             throw new RuntimeException("f(phiMin) and f(phiMax) have the same sign, we cant find f(theta) = 0 between them.");
         }
 
         // Find optimum using Secant method, see https://en.wikipedia.org/wiki/Secant_method
         double x0 = phiMin;
         double x1 = phiMax;
         double x2 = 0;
         for (int i = 0; i < 100; i++) // max 100 iterations, otherwise use latest x2 value
         {
             double f1 = fTheta(x1, phi1, phi2, sign);
             x2 = x1 - f1 * (x1 - x0) / (f1 - fTheta(x0, phi1, phi2, sign));
             x2 = Math.max(Math.min(x2, phiMax), phiMin); // this line is an essential addition to keep the algorithm at bay
             x0 = x1;
             x1 = x2;
             if (Math.abs(x0 - x1) < SECANT_TOLERANCE || Math.abs(x0 / x1 - 1) < SECANT_TOLERANCE
                     || Math.abs(f1) < SECANT_TOLERANCE)
             {
                 return x2;
             }
         }
 
         return x2;
     }
 
     /**
      * Function who's solution <i>f</i>(<i>theta</i>) = 0 for the given value of <i>phi1</i> and <i>phi2</i> gives the angle
      * that solves fitting a C-shaped clothoid through two points. This assumes that <i>sign</i> = -1. If <i>sign</i> = 1, this
      * changes to <i>g</i>(<i>theta</i>) = 0 being a solution for an S-shaped clothoid.
      * @param theta angle defining the curvature of the resulting clothoid.
      * @param phi1 angle between the line through both end points, and the direction of the first point.
      * @param phi2 angle between the line through both end points, and the direction of the last point.
      * @param sign 1 for C-shaped, -1 for S-shaped.
      * @return <i>f</i>(<i>theta</i>) for <i>sign</i> = -1, or <i>g</i>(<i>theta</i>) for <i>sign</i> = 1.
      */
     private static double fTheta(final double theta, final double phi1, final double phi2, final double sign)
     {
         double thetaPhi1 = theta + phi1;
         Fresnel cs0 = Fresnel.integral(alphaToT(theta));
         Fresnel cs1 = Fresnel.integral(alphaToT(thetaPhi1 + phi2));
         return (cs1.s() + sign * cs0.s()) * Math.cos(thetaPhi1) - (cs1.c() + sign * cs0.c()) * Math.sin(thetaPhi1);
     }
 
     @Override
     public OrientedPoint2d getStartPoint()
     {
         return this.startPoint;
     }
 
     @Override
     public OrientedPoint2d getEndPoint()
     {
         return this.endPoint;
     }
 
     @Override
     public double getStartCurvature()
     {
         return this.startCurvature;
     }
 
     @Override
     public double getEndCurvature()
     {
         return this.endCurvature;
     }
 
     @Override
     public double getStartRadius()
     {
         return 1.0 / this.startCurvature;
     }
 
     @Override
     public double getEndRadius()
     {
         return 1.0 / this.endCurvature;
     }
 
     /**
      * Return A, the clothoid scaling parameter.
      * @return a, the clothoid scaling parameter.
      */
     public double getA()
     {
         // Scale 'a', due to parameter conversion between C(alpha)/S(alpha) and C(t)/S(t); t = sqrt(2*alpha/pi).
         // The value of 'this.a' is used when scaling the Fresnel integral, which is why this is stored.
         return this.a / Math.sqrt(Math.PI);
     }
 
     /**
      * Calculates shifts if these have not yet been calculated.
      */
     private void assureShift()
     {
         if (this.shiftDetermined)
         {
             return;
         }
 
         OrientedPoint2d p1 = this.opposite ? this.endPoint : this.startPoint;
         OrientedPoint2d p2 = this.opposite ? this.startPoint : this.endPoint;
 
         // Create first point to figure out the required overall shift
         Fresnel csMin = Fresnel.integral(alphaToT(this.alphaMin));
         double xMin = this.a * (csMin.c() * this.t0[0] - csMin.s() * this.n0[0]);
         double yMin = this.a * (csMin.c() * this.t0[1] - csMin.s() * this.n0[1]);
         this.shiftX = p1.x - xMin;
         this.shiftY = p1.y - yMin;
 
         // Due to numerical precision, we linearly scale over alpha such that the final point is exactly on p2
         if (p2 != null)
         {
             Fresnel csMax = Fresnel.integral(alphaToT(this.alphaMax));
             double xMax = this.a * (csMax.c() * this.t0[0] - csMax.s() * this.n0[0]);
             double yMax = this.a * (csMax.c() * this.t0[1] - csMax.s() * this.n0[1]);
             this.dShiftX = p2.x - (xMax + this.shiftX);
             this.dShiftY = p2.y - (yMax + this.shiftY);
         }
         else
         {
             this.dShiftX = 0.0;
             this.dShiftY = 0.0;
         }
 
         this.shiftDetermined = true;
     }
 
     /**
      * Returns a point on the clothoid at a fraction of curvature along the clothoid.
      * @param fraction fraction of curvature along the clothoid.
      * @param offset offset relative to radius.
      * @return point on the clothoid at a fraction of curvature along the clothoid.
      */
     private Point2d getPoint(final double fraction, final double offset)
     {
         double f = this.opposite ? 1.0 - fraction : fraction;
         double alpha = this.alphaMin + f * (this.alphaMax - this.alphaMin);
         Fresnel cs = Fresnel.integral(alphaToT(alpha));
         double x = this.shiftX + this.a * (cs.c() * this.t0[0] - cs.s() * this.n0[0]) + f * this.dShiftX;
         double y = this.shiftY + this.a * (cs.c() * this.t0[1] - cs.s() * this.n0[1]) + f * this.dShiftY;
         double d = getDirection(alpha) + Math.PI / 2;
         return new Point2d(x + Math.cos(d) * offset, y + Math.sin(d) * offset);
     }
 
     /**
      * Returns the direction at given alpha.
      * @param alpha alpha.
      * @return direction at given alpha.
      */
     private double getDirection(final double alpha)
     {
         double rot = Math.atan2(this.t0[1], this.t0[0]);
         // abs because alpha = -3deg has the same direction as alpha = 3deg in an S-curve where alpha = 0 is the middle
         rot += this.reflected ? -Math.abs(alpha) : Math.abs(alpha);
         if (this.opposite)
         {
             rot += Math.PI;
         }
         return AngleUtil.normalizeAroundZero(rot);
     }
 
     @Override
     public PolyLine2d flatten(final Flattener flattener)
     {
         Throw.whenNull(flattener, "Flattener may not be null.");
         if (this.straight != null)
         {
             return this.straight.flatten(flattener);
         }
         if (this.arc != null)
         {
             return this.arc.flatten(flattener);
         }
         assureShift();
         return flattener.flatten(new FlattableLine()
         {
             @Override
             public Point2d get(final double fraction)
             {
                 return getPoint(fraction, 0.0);
             }
 
             @Override
             public double getDirection(final double fraction)
             {
                 return ContinuousClothoid.this.getDirection(ContinuousClothoid.this.alphaMin
                         + fraction * (ContinuousClothoid.this.alphaMax - ContinuousClothoid.this.alphaMin));
             }
         });
     }
 
     @Override
     public PolyLine2d flattenOffset(final ContinuousDoubleFunction offset, final Flattener flattener)
     {
         Throw.whenNull(offset, "Offsets may not be null.");
         Throw.whenNull(flattener, "Flattener may not be null.");
         if (this.straight != null)
         {
             return this.straight.flattenOffset(offset, flattener);
         }
         if (this.arc != null)
         {
             return this.arc.flattenOffset(offset, flattener);
         }
         assureShift();
         return flattener.flatten(new FlattableLine()
         {
             @Override
             public Point2d get(final double fraction)
             {
                 return getPoint(fraction, offset.apply(fraction));
             }
 
             @Override
             public double getDirection(final double fraction)
             {
                 double derivativeOffset = offset.getDerivative(fraction) / ContinuousClothoid.this.length;
                 return ContinuousClothoid.this.getDirection(ContinuousClothoid.this.alphaMin
                         + fraction * (ContinuousClothoid.this.alphaMax - ContinuousClothoid.this.alphaMin))
                         + Math.atan(derivativeOffset);
             }
         });
     }
 
     @Override
     public double getLength()
     {
         return this.length;
     }
 
     /**
      * Returns whether the shape was applied as a Clothoid, an Arc, or as a Straight, depending on start and end position and
      * direction.
      * @return "Clothoid", "Arc" or "Straight".
      */
     public String getAppliedShape()
     {
         return this.straight == null ? (this.arc == null ? "Clothoid" : "Arc") : "Straight";
     }
 
     @Override
     public String toString()
     {
         return "ContinuousClothoid [startPoint=" + this.startPoint + ", endPoint=" + this.endPoint + ", startCurvature="
                 + this.startCurvature + ", endCurvature=" + this.endCurvature + ", length=" + this.length + "]";
     }
 
 }