Bezier.java
package org.opentrafficsim.core.geometry;
import java.awt.geom.Line2D;
import org.djutils.exceptions.Throw;
import nl.tudelft.simulation.language.d3.DirectedPoint;
/**
* Generation of Bézier curves. <br>
* The class implements the cubic(...) method to generate a cubic Bézier curve using the following formula: B(t) = (1 -
* t)<sup>3</sup>P<sub>0</sub> + 3t(1 - t)<sup>2</sup> P<sub>1</sub> + 3t<sup>2</sup> (1 - t) P<sub>2</sub> + t<sup>3</sup>
* P<sub>3</sub> where P<sub>0</sub> and P<sub>3</sub> are the end points, and P<sub>1</sub> and P<sub>2</sub> the control
* points. <br>
* For a smooth movement, one of the standard implementations if the cubic(...) function offered is the case where P<sub>1</sub>
* is positioned halfway between P<sub>0</sub> and P<sub>3</sub> starting from P<sub>0</sub> in the direction of P<sub>3</sub>,
* and P<sub>2</sub> is positioned halfway between P<sub>3</sub> and P<sub>0</sub> starting from P<sub>3</sub> in the direction
* of P<sub>0</sub>.<br>
* Finally, an n-point generalization of the Bézier curve is implemented with the bezier(...) function.
* <p>
* Copyright (c) 2013-2019 Delft University of Technology, PO Box 5, 2600 AA, Delft, the Netherlands. All rights reserved. <br>
* BSD-style license. See <a href="http://opentrafficsim.org/docs/license.html">OpenTrafficSim License</a>.
* </p>
* $LastChangedDate: 2015-07-24 02:58:59 +0200 (Fri, 24 Jul 2015) $, @version $Revision: 1147 $, by $Author: averbraeck $,
* initial version Nov 14, 2015 <br>
* @author <a href="http://www.tbm.tudelft.nl/averbraeck">Alexander Verbraeck</a>
* @author <a href="http://www.tudelft.nl/pknoppers">Peter Knoppers</a>
*/
public final class Bezier
{
/** The default number of points to use to construct a Bézier curve. */
private static final int DEFAULT_NUM_POINTS = 64;
/** Cached factorial values. */
private static long[] fact = new long[] {1L, 1L, 2L, 6L, 24L, 120L, 720L, 5040L, 40320L, 362880L, 3628800L, 39916800L,
479001600L, 6227020800L, 87178291200L, 1307674368000L, 20922789888000L, 355687428096000L, 6402373705728000L,
121645100408832000L, 2432902008176640000L};
/** Utility class. */
private Bezier()
{
// do not instantiate
}
/**
* Construct a cubic Bézier curve from start to end with two control points.
* @param numPoints int; the number of points for the Bézier curve
* @param start OTSPoint3D; the start point of the Bézier curve
* @param control1 OTSPoint3D; the first control point
* @param control2 OTSPoint3D; the second control point
* @param end OTSPoint3D; the end point of the Bézier curve
* @return a cubic Bézier curve between start and end, with the two provided control points
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D cubic(final int numPoints, final OTSPoint3D start, final OTSPoint3D control1,
final OTSPoint3D control2, final OTSPoint3D end) throws OTSGeometryException
{
Throw.when(numPoints < 2, OTSGeometryException.class, "Number of points too small (got %d; minimum value is 2)",
numPoints);
OTSPoint3D[] points = new OTSPoint3D[numPoints];
for (int n = 0; n < numPoints; n++)
{
double t = n / (numPoints - 1.0);
double x = B3(t, start.x, control1.x, control2.x, end.x);
double y = B3(t, start.y, control1.y, control2.y, end.y);
double z = B3(t, start.z, control1.z, control2.z, end.z);
points[n] = new OTSPoint3D(x, y, z);
}
return new OTSLine3D(points);
}
/**
* Construct a cubic Bézier curve from start to end with two generated control points at half the distance between
* start and end. The z-value is interpolated in a linear way.
* @param numPoints int; the number of points for the Bézier curve
* @param start DirectedPoint; the directed start point of the Bézier curve
* @param end DirectedPoint; the directed end point of the Bézier curve
* @return a cubic Bézier curve between start and end, with the two provided control points
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D cubic(final int numPoints, final DirectedPoint start, final DirectedPoint end)
throws OTSGeometryException
{
return cubic(numPoints, start, end, 1.0);
}
/**
* Construct a cubic Bézier curve from start to end with two generated control points at half the distance between
* start and end. The z-value is interpolated in a linear way.
* @param numPoints int; the number of points for the Bézier curve
* @param start DirectedPoint; the directed start point of the Bézier curve
* @param end DirectedPoint; the directed end point of the Bézier curve
* @param shape shape factor; 1 = control points at half the distance between start and end, > 1 results in a pointier
* shape, < 1 results in a flatter shape, value should be above 0
* @return a cubic Bézier curve between start and end, with the two determined control points
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D cubic(final int numPoints, final DirectedPoint start, final DirectedPoint end, final double shape)
throws OTSGeometryException
{
return cubic(numPoints, start, end, shape, false);
}
/**
* Construct a cubic Bézier curve from start to end with two generated control points at half the distance between
* start and end. The z-value is interpolated in a linear way.
* @param numPoints int; the number of points for the Bézier curve
* @param start DirectedPoint; the directed start point of the Bézier curve
* @param end DirectedPoint; the directed end point of the Bézier curve
* @param shape shape factor; 1 = control points at half the distance between start and end, > 1 results in a pointier
* shape, < 1 results in a flatter shape, value should be above 0
* @param weighted boolean; control point distance relates to distance to projected point on extended line from other end
* @return a cubic Bézier curve between start and end, with the two determined control points
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D cubic(final int numPoints, final DirectedPoint start, final DirectedPoint end, final double shape,
final boolean weighted) throws OTSGeometryException
{
OTSPoint3D control1;
OTSPoint3D control2;
if (weighted)
{
// each control point is 'w' * the distance between the end-points away from the respective end point
// 'w' is a weight given by the distance from the end point to the extended line of the other end point
double distance = shape * Math.sqrt((end.x - start.x) * (end.x - start.x) + (end.y - start.y) * (end.y - start.y));
double cosEnd = Math.cos(end.getRotZ());
double sinEnd = Math.sin(end.getRotZ());
double dStart = Line2D.ptLineDist(end.x, end.y, end.x + cosEnd, end.y + sinEnd, start.x, start.y);
double cosStart = Math.cos(start.getRotZ());
double sinStart = Math.sin(start.getRotZ());
double dEnd = Line2D.ptLineDist(start.x, start.y, start.x + cosStart, start.y + sinStart, end.x, end.y);
double wStart = dStart / (dStart + dEnd);
double wEnd = dEnd / (dStart + dEnd);
double wStartDistance = wStart * distance;
double wEndDistance = wEnd * distance;
control1 = new OTSPoint3D(start.x + wStartDistance * cosStart, start.y + wStartDistance * sinStart);
// - (minus) as the angle is where the line leaves, i.e. from shape point to end
control2 = new OTSPoint3D(end.x - wEndDistance * cosEnd, end.y - wEndDistance * sinEnd);
}
else
{
// each control point is half the distance between the end-points away from the respective end point
double distance2 =
shape * Math.sqrt((end.x - start.x) * (end.x - start.x) + (end.y - start.y) * (end.y - start.y)) / 2.0;
control1 = new OTSPoint3D(start.x + distance2 * Math.cos(start.getRotZ()),
start.y + distance2 * Math.sin(start.getRotZ()), start.z);
control2 = new OTSPoint3D(end.x - distance2 * Math.cos(end.getRotZ()), end.y - distance2 * Math.sin(end.getRotZ()),
end.z);
}
// Limit control points to not intersect with the other (infinite) line
OTSPoint3D s = new OTSPoint3D(start);
OTSPoint3D e = new OTSPoint3D(end);
// return cubic(numPoints, new OTSPoint3D(start), control1, control2, new OTSPoint3D(end));
return bezier(numPoints, s, control1, control2, e);
}
/**
* Construct a cubic Bézier curve from start to end with two generated control points at half the distance between
* start and end. The z-value is interpolated in a linear way.
* @param start DirectedPoint; the directed start point of the Bézier curve
* @param end DirectedPoint; the directed end point of the Bézier curve
* @return a cubic Bézier curve between start and end, with the two provided control points
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D cubic(final DirectedPoint start, final DirectedPoint end) throws OTSGeometryException
{
return cubic(DEFAULT_NUM_POINTS, start, end);
}
/**
* Calculate the cubic Bézier point with B(t) = (1 - t)<sup>3</sup>P<sub>0</sub> + 3t(1 - t)<sup>2</sup>
* P<sub>1</sub> + 3t<sup>2</sup> (1 - t) P<sub>2</sub> + t<sup>3</sup> P<sub>3</sub>.
* @param t double; the fraction
* @param p0 double; the first point of the curve
* @param p1 double; the first control point
* @param p2 double; the second control point
* @param p3 double; the end point of the curve
* @return the cubic bezier value B(t)
*/
@SuppressWarnings("checkstyle:methodname")
private static double B3(final double t, final double p0, final double p1, final double p2, final double p3)
{
double t2 = t * t;
double t3 = t2 * t;
double m = (1.0 - t);
double m2 = m * m;
double m3 = m2 * m;
return m3 * p0 + 3.0 * t * m2 * p1 + 3.0 * t2 * m * p2 + t3 * p3;
}
/**
* Construct a Bézier curve of degree n.
* @param numPoints int; the number of points for the Bézier curve to be constructed
* @param points OTSPoint3D...; the points of the curve, where the first and last are begin and end point, and the
* intermediate ones are control points. There should be at least two points.
* @return the Bézier value B(t) of degree n, where n is the number of points in the array
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D bezier(final int numPoints, final OTSPoint3D... points) throws OTSGeometryException
{
OTSPoint3D[] result = new OTSPoint3D[numPoints];
double[] px = new double[points.length];
double[] py = new double[points.length];
double[] pz = new double[points.length];
for (int i = 0; i < points.length; i++)
{
px[i] = points[i].x;
py[i] = points[i].y;
pz[i] = points[i].z;
}
for (int n = 0; n < numPoints; n++)
{
double t = n / (numPoints - 1.0);
double x = Bn(t, px);
double y = Bn(t, py);
double z = Bn(t, pz);
result[n] = new OTSPoint3D(x, y, z);
}
return new OTSLine3D(result);
}
/**
* Construct a Bézier curve of degree n.
* @param points OTSPoint3D...; the points of the curve, where the first and last are begin and end point, and the
* intermediate ones are control points. There should be at least two points.
* @return the Bézier value B(t) of degree n, where n is the number of points in the array
* @throws OTSGeometryException in case the number of points is less than 2 or the Bézier curve could not be
* constructed
*/
public static OTSLine3D bezier(final OTSPoint3D... points) throws OTSGeometryException
{
return bezier(DEFAULT_NUM_POINTS, points);
}
/**
* Calculate the Bézier point of degree n, with B(t) = Sum(i = 0..n) [C(n, i) * (1 - t)<sup>n-i</sup> t<sup>i</sup>
* P<sub>i</sub>], where C(n, k) is the binomial coefficient defined by n! / ( k! (n-k)! ), ! being the factorial operator.
* @param t double; the fraction
* @param p double...; the points of the curve, where the first and last are begin and end point, and the intermediate ones
* are control points
* @return the Bézier value B(t) of degree n, where n is the number of points in the array
*/
@SuppressWarnings("checkstyle:methodname")
private static double Bn(final double t, final double... p)
{
double b = 0.0;
double m = (1.0 - t);
int n = p.length - 1;
double fn = factorial(n);
for (int i = 0; i <= n; i++)
{
double c = fn / (factorial(i) * (factorial(n - i)));
b += c * Math.pow(m, n - i) * Math.pow(t, i) * p[i];
}
return b;
}
/**
* Calculate factorial(k), which is k * (k-1) * (k-2) * ... * 1. For factorials up to 20, a lookup table is used.
* @param k int; the parameter
* @return factorial(k)
*/
private static double factorial(final int k)
{
if (k < fact.length)
{
return fact[k];
}
double f = 1;
for (int i = 2; i <= k; i++)
{
f = f * i;
}
return f;
}
/**
* @param args String[]; args
* @throws OTSGeometryException ne
*/
public static void main(final String[] args) throws OTSGeometryException
{
// DirectedPoint s = new DirectedPoint(0, 0, 0, 0, 0, -Math.PI/2.0);
// DirectedPoint e = new DirectedPoint(10, 10, 20, 0, 0, Math.PI);
// OTSLine3D b1 = Bezier.cubic(s, e);
// for (OTSPoint3D p : b1.getPoints())
// {
// System.out.println(p.x + "\t" + p.y + "\t" + p.z);
// }
OTSPoint3D s = new OTSPoint3D(0, 0, 0);
OTSPoint3D s1 = new OTSPoint3D(10, 0, 0);
OTSPoint3D m1 = new OTSPoint3D(25, 5, 0);
OTSPoint3D m2 = new OTSPoint3D(-15, 5, 0);
OTSPoint3D e0 = new OTSPoint3D(0, 10, 20);
OTSPoint3D e = new OTSPoint3D(10, 10, 20);
OTSLine3D b1 = Bezier.bezier(s, s1, m1, m2, e0, e);
for (OTSPoint3D p : b1.getPoints())
{
System.out.println(p.x + "\t" + p.y + "\t" + p.z);
}
}
}