/* Copyright (C) 2005-2011 Fabio Riccardi */
package com.lightcrafts.ui.region.curves;
import java.awt.*;
import java.awt.geom.GeneralPath;
import java.awt.geom.Point2D;
import java.util.Iterator;
/** This class represents a closed sequence of cubic parametric curve segments
* that smoothly interpolate a list of points. It may be accessed as a
* Shape by <code>getShape()</code>, and it can paint itself by
* <code>paint()</code>
* <p>
* It is not a Component though, so that two CubicCurves may overlap.
* <p>
* The linear algebra at the core is copied from Tim Lambert,
* http://www.cse.unsw.edu.au/~lambert/splines/source.html,
* who transcribed the formulae from
* "Two Dimensional Spline Interpolation Algorithms" by Helmuth Spath.
*/
public class CubicBezierCurve extends AbstractCurve {
// Update "shape" and "segments" from the current value of "points":
void updateShape() {
if (points.size() < 2) {
shape = null;
segments.clear();
return;
}
GeneralPath path = new GeneralPath();
int[] xPts = new int[points.size()];
int[] yPts = new int[points.size()];
int count = 0;
for (Iterator i=points.iterator(); i.hasNext(); ) {
Point2D p = (Point2D) i.next();
xPts[count] = (int) p.getX();
yPts[count] = (int) p.getY();
count++;
}
Cubic[] xPoly = calcNaturalCubic(xPts);
Cubic[] yPoly = calcNaturalCubic(yPts);
segments.clear();
for (int n=0; n<count; n++) {
Shape curve = Cubic.createShape(xPoly[n], yPoly[n]);
segments.add(curve);
path.append(curve, true);
}
path.closePath();
shape = path;
}
/** Calculates the closed natural cubic spline that interpolates
* x[0], x[1], ... x[n]. The first segment is returned as
*
* C[0].a + C[0].b*u + C[0].c*u^2 + C[0].d*u^3
*
* on the interval u in [0,1). The other segments are in C[1],C[2],...C[n].
*/
private Cubic[] calcNaturalCubic(int[] x) {
int n = x.length - 1;
double[] w = new double[n+1];
double[] v = new double[n+1];
double[] y = new double[n+1];
double[] D = new double[n+1];
double z, F, G, H;
int k;
/* We solve the equation
[4 1 1] [D[0]] [3(x[1] - x[n]) ]
|1 4 1 | |D[1]| |3(x[2] - x[0]) |
| 1 4 1 | | . | = | . |
| ..... | | . | | . |
| 1 4 1| | . | |3(x[n] - x[n-2])|
[1 1 4] [D[n]] [3(x[0] - x[n-1])]
by decomposing the matrix into upper triangular and lower matrices
and then back sustitution. See Spath "Spline Algorithms for Curves
and Surfaces" pp 19--21. The D[i] are the derivatives at the knots.
*/
w[1] = v[1] = z = 1. / 4.;
y[0] = z * 3 * (x[1] - x[n]);
H = 4;
F = 3 * (x[0] - x[n-1]);
G = 1;
for ( k = 1; k < n; k++) {
v[k+1] = z = 1 / (4 - v[k]);
w[k+1] = - z * w[k];
y[k] = z * (3 * (x[k+1] - x[k-1]) - y[k-1]);
H = H - G * w[k];
F = F - G * y[k-1];
G = - v[k] * G;
}
H = H - (G + 1) * (v[n] + w[n]);
y[n] = F - (G + 1) * y[n-1];
D[n] = y[n] / H;
/* This equation is WRONG! in my copy of Spath: */
D[n-1] = y[n-1] - (v[n] + w[n]) * D[n];
for (k=n-2; k>=0; k--) {
D[k] = y[k] - v[k+1] * D[k+1] - w[k+1] * D[n];
}
/* now compute the coefficients of the cubics */
Cubic[] C = new Cubic[n+1];
for (k=0; k<n; k++) {
C[k] = new Cubic(
(double) x[k],
D[k],
3 * (x[k+1] - x[k]) - 2 * D[k] - D[k+1],
2 * (x[k] - x[k+1]) + D[k] + D[k+1]
);
}
C[n] = new Cubic(
(double) x[n],
D[n],
3 * (x[0] - x[n]) - 2*D[n] - D[0],
2 * (x[n] - x[0]) + D[n] + D[0]
);
return C;
}
}