Clothoid2d.java
package org.djutils.draw.curve;
import org.djutils.draw.function.ContinuousPiecewiseLinearFunction;
import org.djutils.draw.line.PolyLine2d;
import org.djutils.draw.point.DirectedPoint2d;
import org.djutils.draw.point.Point2d;
import org.djutils.exceptions.Throw;
import org.djutils.exceptions.Try;
import org.djutils.math.AngleUtil;
/**
* Continuous definition of a clothoid in 2d. The following definitions are available:
* <ul>
* <li>A clothoid between two <code>DirectedPoint2d</code>s.</li>
* <li>A clothoid originating from a <code>DirectedPoint2d</code> with start curvature, end curvature, and <code>length</code>
* specified.</li>
* <li>A clothoid originating from a <code>DirectedPoint2d</code> with start curvature, end curvature, and <code>A-value</code>
* 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) 2023-2025 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://www.tudelft.nl/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 Clothoid2d implements Curve2d, OffsetCurve2d
{
/** 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 DirectedPoint2d startPoint;
/** End point with direction. */
private final DirectedPoint2d 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 Straight2d straight;
/** Simplification to arc when valid. */
private final Arc2d 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>
* 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>
* Which applies the theory proven in:<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>
* 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.fresnel()}.
* @param startPoint start point
* @param endPoint end point
* @throws NullPointerException when <code>startPoint</code>, or <code>endPoint</code> is <code>null</code>
* @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 Clothoid2d(final DirectedPoint2d startPoint, final DirectedPoint2d endPoint)
{
Throw.whenNull(startPoint, "startPoint");
Throw.whenNull(endPoint, "endPoint");
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 Straight2d(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;
}
double angle = Math.abs(to - from);
this.length = angle * Math.abs(r);
this.a = 0.0;
this.startCurvature = 1.0 / r;
this.endCurvature = 1.0 / r;
this.straight = null;
this.arc = new Arc2d(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.
double[] cs = Fresnel.fresnel(alphaToT(phi1 + phi2));
double h = cs[1] * Math.cos(phi1) - cs[0] * 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;
double[] cs0 = Fresnel.fresnel(alphaToT(theta));
double[] cs1 = Fresnel.fresnel(alphaToT(v1));
this.a = d2 / ((cs1[1] + aSign * cs0[1]) * Math.sin(v2) + (cs1[0] + aSign * cs0[0]) * 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
* @throws NullPointerException when <code>startPoint</code> is <code>null</code>
* @throws IllegalArgumentException when <code>a ≤ 0.0</code>
*/
public Clothoid2d(final DirectedPoint2d startPoint, final double a, final double startCurvature, final double endCurvature)
{
Throw.whenNull(startPoint, "startPoint");
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]};
double endDirection = 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 = ang2 - Math.abs(this.alphaMax) + Math.PI;
}
PolyLine2d line = toPolyLine(new Flattener2d.NumSegments(1));
Point2d end = Try.assign(() -> line.get(line.size() - 1), "Line does not have an end point.");
this.endPoint = new DirectedPoint2d(end.x, end.y, endDirection);
// 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.
* @throws NullPointerException when <code>startPoint</code> is <code>null</code>
* @throws IllegalArgumentException when <code>length ≤ 0.0</code>
*/
public static Clothoid2d withLength(final DirectedPoint2d 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 Clothoid2d(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 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 IllegalArgumentException(
"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;
double[] cs0 = Fresnel.fresnel(alphaToT(theta));
double[] cs1 = Fresnel.fresnel(alphaToT(thetaPhi1 + phi2));
return (cs1[1] + sign * cs0[1]) * Math.cos(thetaPhi1) - (cs1[0] + sign * cs0[0]) * Math.sin(thetaPhi1);
}
@Override
public DirectedPoint2d getStartPoint()
{
return this.startPoint;
}
@Override
public DirectedPoint2d getEndPoint()
{
return this.endPoint;
}
/**
* Start curvature of this Clothoid.
* @return start curvature of this Clothoid
*/
public double getStartCurvature()
{
return this.startCurvature;
}
/**
* End curvature of this Clothoid.
* @return end curvature of this Clothoid
*/
public double getEndCurvature()
{
return this.endCurvature;
}
/**
* Start radius of this Clothoid.
* @return start radius of this Clothoid
*/
public double getStartRadius()
{
return 1.0 / this.startCurvature;
}
/**
* End radius of this Clothoid.
* @return end radius of this Clothoid
*/
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;
}
DirectedPoint2d p1 = this.opposite ? this.endPoint : this.startPoint;
DirectedPoint2d p2 = this.opposite ? this.startPoint : this.endPoint;
// Create first point to figure out the required overall shift
double[] csMin = Fresnel.fresnel(alphaToT(this.alphaMin));
double xMin = this.a * (csMin[0] * this.t0[0] - csMin[1] * this.n0[0]);
double yMin = this.a * (csMin[0] * this.t0[1] - csMin[1] * 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)
{
double[] csMax = Fresnel.fresnel(alphaToT(this.alphaMax));
double xMax = this.a * (csMax[0] * this.t0[0] - csMax[1] * this.n0[0]);
double yMax = this.a * (csMax[0] * this.t0[1] - csMax[1] * 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);
double[] cs = Fresnel.fresnel(alphaToT(alpha));
double x = this.shiftX + this.a * (cs[0] * this.t0[0] - cs[1] * this.n0[0]) + f * this.dShiftX;
double y = this.shiftY + this.a * (cs[0] * this.t0[1] - cs[1] * this.n0[1]) + f * this.dShiftY;
double d = getDirectionForAlpha(alpha) + Math.PI / 2;
return new Point2d(x + Math.cos(d) * offset, y + Math.sin(d) * offset);
}
@Override
public Point2d getPoint(final double fraction)
{
if (this.arc != null)
{
return this.arc.getPoint(fraction);
}
else if (this.straight != null)
{
return this.straight.getPoint(fraction);
}
return getPoint(fraction, 0);
}
@Override
public Point2d getPoint(final double fraction, final ContinuousPiecewiseLinearFunction of)
{
if (this.arc != null)
{
return this.arc.getPoint(fraction, of);
}
else if (this.straight != null)
{
return this.straight.getPoint(fraction, of);
}
return getPoint(fraction, of.get(fraction));
}
@Override
public Double getDirection(final double fraction)
{
if (this.arc != null)
{
return this.arc.getDirection(fraction);
}
else if (this.straight != null)
{
return this.straight.getDirection(fraction);
}
return getDirectionForAlpha(this.alphaMin + fraction * (this.alphaMax - this.alphaMin));
}
/**
* Returns the direction at given alpha.
* @param alpha alpha
* @return direction at given alpha
*/
private double getDirectionForAlpha(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 toPolyLine(final Flattener2d flattener)
{
if (this.straight != null)
{
return this.straight.toPolyLine(flattener);
}
if (this.arc != null)
{
return this.arc.toPolyLine(flattener);
}
assureShift();
return flattener.flatten(this);
}
@Override
public PolyLine2d toPolyLine(final OffsetFlattener2d flattener, final ContinuousPiecewiseLinearFunction offsets)
{
Throw.whenNull(offsets, "offsets");
if (this.straight != null)
{
return this.straight.toPolyLine(flattener, offsets);
}
if (this.arc != null)
{
return this.arc.toPolyLine(flattener, offsets);
}
assureShift();
return flattener.flatten(this, offsets);
}
@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 "Clothoid [startPoint=" + this.startPoint + ", endPoint=" + this.endPoint + ", startCurvature="
+ this.startCurvature + ", endCurvature=" + this.endCurvature + ", length=" + this.length + "]";
}
}
/**
* Utility class to create clothoid lines, in particular the Fresnel integral based on:
* <ul>
* <li>W.J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Mathematics of Computation, Vol. 22, Issue 102, pp.
* 450–453.</li>
* </ul>
* <p>
* Copyright (c) 2023-2025 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.ams.org/journals/mcom/1985-44-170/S0025-5718-1985-0777277-6/S0025-5718-1985-0777277-6.pdf">Cody
* (1968)</a>
*/
final class Fresnel
{
// {@formatter:off}
/** Numerator coefficients to calculate C(t) in region 1. */
private static final double[] CN1 = new double[] {
9.999999999999999421E-01,
-1.994608988261842706E-01,
1.761939525434914045E-02,
-5.280796513726226960E-04,
5.477113856826871660E-06
};
/** Denominator coefficients to calculate C(t) in region 1. */
private static final double[] CD1 = new double[] {
1.000000000000000000E+00,
4.727921120104532689E-02,
1.099572150256418851E-03,
1.552378852769941331E-05,
1.189389014228757184E-07
};
/** Numerator coefficients to calculate C(t) in region 2. */
private static final double[] CN2 = new double[] {
1.00000000000111043640E+00,
-2.07073360335323894245E-01,
1.91870279431746926505E-02,
-6.71376034694922109230E-04,
1.02365435056105864908E-05,
-5.68293310121870728343E-08
};
/** Denominator coefficients to calculate C(t) in region 3. */
private static final double[] CD2 = new double[] {
1.00000000000000000000E+00,
3.96667496952323433510E-02,
7.88905245052359907842E-04,
1.01344630866749406081E-05,
8.77945377892369265356E-08,
4.41701374065009620393E-10
};
/** Numerator coefficients to calculate S(t) in region 1. */
private static final double[] SN1 = new double[] {
5.2359877559829887021E-01,
-7.0748991514452302596E-02,
3.8778212346368287939E-03,
-8.4555728435277680591E-05,
6.7174846662514086196E-07
};
/** Denominator coefficients to calculate S(t) in region 1. */
private static final double[] SD1 = new double[] {
1.0000000000000000000E+00,
4.1122315114238422205E-02,
8.1709194215213447204E-04,
9.6269087593903403370E-06,
5.9528122767840998345E-08
};
/** Numerator coefficients to calculate S(t) in region 2. */
private static final double[] SN2 = new double[] {
5.23598775598344165913E-01,
-7.37766914010191323867E-02,
4.30730526504366510217E-03,
-1.09540023911434994566E-04,
1.28531043742724820610E-06,
-5.76765815593088804567E-09
};
/** Denominator coefficients to calculate S(t) in region 2. */
private static final double[] SD2 = new double[] {
1.00000000000000000000E+00,
3.53398342167472162540E-02,
6.18224620195473216538E-04,
6.87086265718620117905E-06,
5.03090581246612375866E-08,
2.05539124458579596075E-10
};
/** Numerator coefficients to calculate f(t) in region 3. */
private static final double[] FN3 = new double[] {
3.1830975293580985290E-01,
1.2226000551672961219E+01,
1.2924886131901657025E+02,
4.3886367156695547655E+02,
4.1466722177958961672E+02,
5.6771463664185116454E+01
};
/** Denominator coefficients to calculate f(t) in region 3. */
private static final double[] FD3 = new double[] {
1.0000000000000000000E+00,
3.8713003365583442831E+01,
4.1674359830705629745E+02,
1.4740030733966610568E+03,
1.5371675584895759916E+03,
2.9113088788847831515E+02
};
/** Numerator coefficients to calculate f(t) in region 4. */
private static final double[] FN4 = new double[] {
3.183098818220169217E-01,
1.958839410219691002E+01,
3.398371349269842400E+02,
1.930076407867157531E+03,
3.091451615744296552E+03,
7.177032493651399590E+02
};
/** Denominator coefficients to calculate f(t) in region 4. */
private static final double[] FD4 = new double[] {
1.000000000000000000E+00,
6.184271381728873709E+01,
1.085350675006501251E+03,
6.337471558511437898E+03,
1.093342489888087888E+04,
3.361216991805511494E+03
};
/** Numerator coefficients to calculate f(t) in region 5. */
private static final double[] FN5 = new double[] {
-9.675460329952532343E-02,
-2.431275407194161683E+01,
-1.947621998306889176E+03,
-6.059852197160773639E+04,
-7.076806952837779823E+05,
-2.417656749061154155E+06,
-7.834914590078311336E+05
};
/** Denominator coefficients to calculate f(t) in region 5. */
private static final double[] FD5 = new double[] {
1.000000000000000000E+00,
2.548289012949732752E+02,
2.099761536857815105E+04,
6.924122509827708985E+05,
9.178823229918143780E+06,
4.292733255630186679E+07,
4.803294184260528342E+07
};
/** Numerator coefficients to calculate g(t) in region 3. */
private static final double[] GN3 = new double[] {
1.013206188102747985E-01,
4.445338275505123778E+00,
5.311228134809894481E+01,
1.991828186789025318E+02,
1.962320379716626191E+02,
2.054214324985006303E+01
};
/** Denominator coefficients to calculate g(t) in region 3. */
private static final double[] GD3 = new double[] {
1.000000000000000000E+00,
4.539250196736893605E+01,
5.835905757164290666E+02,
2.544731331818221034E+03,
3.481121478565452837E+03,
1.013794833960028555E+03
};
/** Numerator coefficients to calculate g(t) in region 4. */
private static final double[] GN4 = new double[] {
1.01321161761804586E-01,
7.11205001789782823E+00,
1.40959617911315524E+02,
9.08311749529593938E+02,
1.59268006085353864E+03,
3.13330163068755950E+02
};
/** Denominator coefficients to calculate g(t) in region 4. */
private static final double[] GD4 = new double[] {
1.00000000000000000E+00,
7.17128596939302198E+01,
1.49051922797329229E+03,
1.06729678030583897E+04,
2.41315567213369742E+04,
1.15149832376260604E+04
};
/** Numerator coefficients to calculate g(t) in region 5. */
private static final double[] GN5 = new double[] {
-1.53989733819769316E-01,
-4.31710157823357568E+01,
-3.87754141746378493E+03,
-1.35678867813756347E+05,
-1.77758950838029676E+06,
-6.66907061668636416E+06,
-1.72590224654836845E+06
};
/** Denominator coefficients to calculate g(t) in region 5. */
private static final double[] GD5 = new double[] {
1.00000000000000000E+00,
2.86733194975899483E+02,
2.69183180396242536E+04,
1.02878693056687506E+06,
1.62095600500231646E+07,
9.38695862531635179E+07,
1.40622441123580005E+08
};
// {@formatter:on}
/** Utility class. */
private Fresnel()
{
// do not instantiate
}
/**
* Approximate the Fresnel integral. The method used is based on Cody (1968). This method applies rational approximation to
* approximate the clothoid. For clothoid rotation beyond 1.6 rad, this occurs in polar form. The polar form is robust for
* arbitrary large numbers, unlike polynomial expansion, and will at a large threshold converge to (0.5, 0.5). There are 5
* regions with different fitted values for the rational approximations, in Cartesian or polar form.<br>
* <br>
* W.J. Cody (1968) Chebyshev approximations for the Fresnel integrals. Mathematics of Computation, Vol. 22, Issue 102, pp.
* 450–453.
* @param x length along the standard Fresnel integral (no scaling).
* @return array with two double values c and s
* @see <a href="https://www.ams.org/journals/mcom/1968-22-102/S0025-5718-68-99871-2/S0025-5718-68-99871-2.pdf">Cody
* (1968)</a>
*/
public static double[] fresnel(final double x)
{
final double t = Math.abs(x);
double cc, ss;
if (t < 1.2)
{
cc = t * ratioEval(t, CN1, +1) / ratioEval(t, CD1, +1);
ss = t * t * t * ratioEval(t, SN1, +1) / ratioEval(t, SD1, +1);
}
else if (t < 1.6)
{
cc = t * ratioEval(t, CN2, +1) / ratioEval(t, CD2, +1);
ss = t * t * t * ratioEval(t, SN2, +1) / ratioEval(t, SD2, +1);
}
else if (t < 1.9)
{
double pitt2 = Math.PI * t * t / 2;
double sinpitt2 = Math.sin(pitt2);
double cospitt2 = Math.cos(pitt2);
double ft = (1 / t) * ratioEval(t, FN3, -1) / ratioEval(t, FD3, -1);
double gt = (1 / (t * t * t)) * ratioEval(t, GN3, -1) / ratioEval(t, GD3, -1);
cc = .5 + ft * sinpitt2 - gt * cospitt2;
ss = .5 - ft * cospitt2 - gt * sinpitt2;
}
else if (t < 2.4)
{
double pitt2 = Math.PI * t * t / 2;
double sinpitt2 = Math.sin(pitt2);
double cospitt2 = Math.cos(pitt2);
double tinv = 1 / t;
double tttinv = tinv * tinv * tinv;
double ft = tinv * ratioEval(t, FN4, -1) / ratioEval(t, FD4, -1);
double gt = tttinv * ratioEval(t, GN4, -1) / ratioEval(t, GD4, -1);
cc = .5 + ft * sinpitt2 - gt * cospitt2;
ss = .5 - ft * cospitt2 - gt * sinpitt2;
}
else
{
double pitt2 = Math.PI * t * t / 2;
double sinpitt2 = Math.sin(pitt2);
double cospitt2 = Math.cos(pitt2);
double piinv = 1 / Math.PI;
double tinv = 1 / t;
double tttinv = tinv * tinv * tinv;
double ttttinv = tttinv * tinv;
double ft = tinv * (piinv + (ttttinv * ratioEval(t, FN5, -1) / ratioEval(t, FD5, -1)));
double gt = tttinv * ((piinv * piinv) + (ttttinv * ratioEval(t, GN5, -1) / ratioEval(t, GD5, -1)));
cc = .5 + ft * sinpitt2 - gt * cospitt2;
ss = .5 - ft * cospitt2 - gt * sinpitt2;
}
if (x < 0)
{
cc = -cc;
ss = -ss;
}
return new double[] {cc, ss};
}
/**
* Evaluate numerator or denominator of rational approximation.
* @param t value along the clothoid
* @param coef rational approximation coefficients
* @param sign sign of exponent, +1 for Cartesian rational approximation, -1 for polar approximation
* @return numerator or denominator of rational approximation
*/
private static double ratioEval(final double t, final double[] coef, final double sign)
{
double value = 0;
for (int s = 0; s < coef.length; s++)
{
value += coef[s] * Math.pow(t, sign * 4 * s);
}
return value;
}
}