A class representing a quadratic path segment
/*
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
import java.awt.geom.Point2D;
import java.awt.geom.QuadCurve2D;
import java.awt.geom.Rectangle2D;
import java.util.Arrays;
/**
* A class representing a quadratic path segment.
*
* @version $Id: Quadradic.java 478249 2006-11-22 17:29:37Z dvholten $
*/
public class Quadradic extends AbstractSegment {
public Point2D.Double p1, p2, p3;
public Quadradic() {
p1 = new Point2D.Double();
p2 = new Point2D.Double();
p3 = new Point2D.Double();
}
public Quadradic(double x1, double y1,
double x2, double y2,
double x3, double y3) {
p1 = new Point2D.Double(x1, y1);
p2 = new Point2D.Double(x2, y2);
p3 = new Point2D.Double(x3, y3);
}
public Quadradic(Point2D.Double p1,
Point2D.Double p2,
Point2D.Double p3) {
this.p1 = p1;
this.p2 = p2;
this.p3 = p3;
}
public Object clone() {
return new Quadradic(new Point2D.Double(p1.x, p1.y),
new Point2D.Double(p2.x, p2.y),
new Point2D.Double(p3.x, p3.y));
}
public Segment reverse() {
return new Quadradic(new Point2D.Double(p3.x, p3.y),
new Point2D.Double(p2.x, p2.y),
new Point2D.Double(p1.x, p1.y));
}
private void getMinMax(double p1, double p2,
double p3, double [] minMax) {
if (p3 > p1){
minMax[0] = p1; minMax[1] = p3;
} else {
minMax[0] = p3; minMax[1] = p1;
}
double a = (p1-2*p2+p3);
double b = (p2-p1);
if (a == 0) return;
double tv = b/a;
if ((tv <= 0) || (tv >= 1)) return;
tv = ((p1-2*p2+p3)*tv+2*(p2-p1))*tv + p1;
if (tv < minMax[0]) minMax[0] = tv;
else if (tv > minMax[1]) minMax[1] = tv;
}
public double minX() {
double [] minMax = {0, 0};
getMinMax(p1.x, p2.x, p3.x, minMax);
return minMax[0];
}
public double maxX() {
double [] minMax = {0, 0};
getMinMax(p1.x, p2.x, p3.x, minMax);
return minMax[1];
}
public double minY() {
double [] minMax = {0, 0};
getMinMax(p1.y, p2.y, p3.y, minMax);
return minMax[0];
}
public double maxY() {
double [] minMax = {0, 0};
getMinMax(p1.y, p2.y, p3.y, minMax);
return minMax[1];
}
public Rectangle2D getBounds2D() {
double [] minMaxX = {0, 0};
getMinMax(p1.x, p2.x, p3.x, minMaxX);
double [] minMaxY = {0, 0};
getMinMax(p1.y, p2.y, p3.y, minMaxY);
return new Rectangle2D.Double
(minMaxX[0], minMaxY[0],
minMaxX[1]-minMaxX[0], minMaxY[1]-minMaxY[0]);
}
protected int findRoots(double y, double [] roots) {
double [] eqn = { p1.y-y, 2*(p2.y-p1.y), p1.y-2*p2.y+p3.y };
return QuadCurve2D.solveQuadratic(eqn, roots);
// return solveQuad(eqn[2], eqn[1], eqn[0], roots);
}
public Point2D.Double evalDt(double t) {
double x = 2*(p1.x-2*p2.x+p3.x)*t + 2*(p2.x-p1.x);
double y = 2*(p1.y-2*p2.y+p3.y)*t + 2*(p2.y-p1.y);
return new Point2D.Double(x, y);
}
public Point2D.Double eval(double t) {
double x = ((p1.x-2*p2.x+p3.x)*t+2*(p2.x-p1.x))*t + p1.x;
double y = ((p1.y-2*p2.y+p3.y)*t+2*(p2.y-p1.y))*t + p1.y;
return new Point2D.Double(x, y);
}
public Segment getSegment(double t0, double t1) {
double dt = t1-t0;
Point2D.Double np1 = eval(t0);
Point2D.Double dp1 = evalDt(t0);
Point2D.Double np2 = new Point2D.Double
(np1.x+.5*dt*dp1.x, np1.y+.5*dt*dp1.y);
Point2D.Double np3 = eval(t1);
return new Quadradic(np1, np2, np3);
}
/**
* Subdivides this Quadradic curve into two curves at t = 0.5.
* can be done with getSegment but this is more efficent.
* @param q0 if non-null contains portion of curve from 0->.5
* @param q1 if non-null contains portion of curve from .5->1
*/
public void subdivide(Quadradic q0, Quadradic q1) {
if ((q0 == null) && (q1 == null)) return;
double x = (p1.x-2*p2.x+p3.x)*.25+(p2.x-p1.x) + p1.x;
double y = (p1.y-2*p2.y+p3.y)*.25+(p2.y-p1.y) + p1.y;
double dx = (p1.x-2*p2.x+p3.x)*.25 + (p2.x-p1.x)*.5;
double dy = (p1.y-2*p2.y+p3.y)*.25 + (p2.y-p1.y)*.5;
if (q0 != null) {
q0.p1.x = p1.x;
q0.p1.y = p1.y;
q0.p2.x = x-dx;
q0.p2.y = y-dy;
q0.p3.x = x;
q0.p3.y = y;
}
if (q1 != null) {
q1.p1.x = x;
q1.p1.y = y;
q1.p2.x = x+dx;
q1.p2.y = y+dy;
q1.p3.x = p3.x;
q1.p3.y = p3.y;
}
}
/**
* Subdivides this Quadradic curve into two curves at given t.
* @param q0 if non-null contains portion of curve from 0->t.
* @param q1 if non-null contains portion of curve from t->1.
*/
public void subdivide(double t, Quadradic q0, Quadradic q1) {
Point2D.Double np = eval(t);
Point2D.Double npd = evalDt(t);
if (q0 != null) {
q0.p1.x = p1.x;
q0.p1.y = p1.y;
q0.p2.x = np.x-(npd.x*t*.5);
q0.p2.y = np.y-(npd.y*t*.5);
q0.p3.x = np.x;
q0.p3.y = np.y;
}
if (q1 != null) {
q1.p1.x = np.x;
q1.p1.y = np.y;
q1.p2.x = np.x+(npd.x*(1-t)*.5);
q1.p2.y = np.y+(npd.y*(1-t)*.5);
q1.p3.x = p3.x;
q1.p3.y = p3.y;
}
}
/**
* Subdivides this Quadradic curve into two curves at t = 0.5.
* can be done with getSegment but this is more efficent.
* @param s0 if non-null contains portion of curve from 0->.5
* @param s1 if non-null contains portion of curve from .5->1
*/
public void subdivide(Segment s0, Segment s1) {
Quadradic q0=null, q1=null;
if (s0 instanceof Quadradic) q0 = (Quadradic)s0;
if (s1 instanceof Quadradic) q1 = (Quadradic)s1;
subdivide(q0, q1);
}
/**
* Subdivides this Quadradic curve into two curves at t.
* can be done with getSegment but this is more efficent.
* @param s0 if non-null contains portion of curve from 0->.5
* @param s1 if non-null contains portion of curve from .5->1
*/
public void subdivide(double t, Segment s0, Segment s1) {
Quadradic q0=null, q1=null;
if (s0 instanceof Quadradic) q0 = (Quadradic)s0;
if (s1 instanceof Quadradic) q1 = (Quadradic)s1;
subdivide(t, q0, q1);
}
static int count = 0;
protected double subLength(double leftLegLen, double rightLegLen,
double maxErr) {
count++;
double dx, dy;
dx = p3.x-p1.x;
dy = p3.y-p1.y;
double cordLen = Math.sqrt(dx*dx+dy*dy);
double hullLen = leftLegLen+rightLegLen;
if (hullLen < maxErr) return (hullLen+cordLen)*.5;
double err = (hullLen-cordLen);
if (err < maxErr)
return (hullLen+cordLen)*.5;
Quadradic q = new Quadradic();
double x = (p1.x+2*p2.x+p3.x)*.25;
double y = (p1.y+2*p2.y+p3.y)*.25;
dx = .25*dx;
dy = .25*dy;
q.p1.x = p1.x;
q.p1.y = p1.y;
q.p2.x = x-dx;
q.p2.y = y-dy;
q.p3.x = x;
q.p3.y = y;
double midLen = .25*cordLen;
double len = q.subLength(leftLegLen*.5, midLen, maxErr*.5);
q.p1.x = x;
q.p1.y = y;
q.p2.x = x+dx;
q.p2.y = y+dy;
q.p3.x = p3.x;
q.p3.y = p3.y;
len += q.subLength(midLen, rightLegLen*.5, maxErr*.5);
return len;
}
public double getLength() {
return getLength(0.000001);
}
public double getLength(double maxErr) {
double dx, dy;
dx = p2.x-p1.x;
dy = p2.y-p1.y;
double leftLegLen = Math.sqrt(dx*dx+dy*dy);
dx = p3.x-p2.x;
dy = p3.y-p2.y;
double rightLegLen = Math.sqrt(dx*dx+dy*dy);
double eps = maxErr*(leftLegLen+rightLegLen);
return subLength(leftLegLen, rightLegLen, eps);
}
public String toString() {
return "M" + p1.x + ',' + p1.y +
'Q' + p2.x + ',' + p2.y + ' ' +
p3.x + ',' + p3.y;
}
/*
public static boolean epsEq(double a, double b) {
final double eps = 0.00001;
return (((a + eps) > b) && ((a-eps) < b));
}
public static void sub(Quadradic orig, Quadradic curr,
double t, double inc, int lev) {
Quadradic left=new Quadradic();
Quadradic right=new Quadradic();
curr.subdivide(left, right);
Point2D.Double ptl = left.eval(.5);
Point2D.Double ptr = right.eval(.5);
Point2D.Double pt1 = orig.eval(t-inc);
Point2D.Double pt2 = orig.eval(t+inc);
int steps = 100;
Point2D.Double l, r, o;
for (int i=0; i<=steps; i++) {
l = left.eval(i/(double)steps);
o = orig.eval(t-(2*inc)*(1-i/(double)steps));
if (!epsEq(l.x, o.x) || !epsEq(l.y, o.y))
System.err.println("Lf Pt: [" + l.x + "," + l.y +
"] Orig: [" + o.x + "," + o.y +"]");
r = right.eval(i/(double)steps);
o = orig.eval(t+(2*inc*i/(double)steps));
if (!epsEq(r.x, o.x) || !epsEq(r.y, o.y))
System.err.println("Rt Pt: [" + r.x + "," + r.y +
"] Orig: [" + o.x + "," + o.y +"]");
}
if (lev != 0) {
sub(orig, left, t-inc, inc/2, lev-1);
sub(orig, right, t+inc, inc/2, lev-1);
}
}
public static void evalQuad(Quadradic q) {
int steps = 1000000;
Point2D.Double oldP = q.eval(0);
Point2D.Double newP;
double len = 0;
for (int i=1; i<=steps; i++) {
newP = q.eval(i/(double)steps);
double dx = newP.x-oldP.x;
double dy = newP.y-oldP.y;
len += Math.sqrt(dx*dx + dy*dy);
oldP = newP;
}
System.err.println("Length(.1): " + q.getLength(.001) +
" x " + count); count = 0;
System.err.println("Length : " + q.getLength() +
" x " + count); count = 0;
System.err.println("D Len : " + len);
}
public static void main(String args[]) {
Quadradic q;
q = new Quadradic(0,0, 10,10, 30,0);
sub(q, q, .5, .25, 3);
evalQuad(q);
q = new Quadradic(0,0, 1,2, 3,0);
sub(q, q, .5, .25, 3);
evalQuad(q);
}
*/
}
/*
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
/**
* An interface that path segments must implement.
*
* @version $Id: Segment.java 478249 2006-11-22 17:29:37Z dvholten $
*/
interface Segment extends Cloneable {
double minX();
double maxX();
double minY();
double maxY();
Rectangle2D getBounds2D();
Point2D.Double evalDt(double t);
Point2D.Double eval(double t);
Segment getSegment(double t0, double t1);
Segment splitBefore(double t);
Segment splitAfter(double t);
void subdivide(Segment s0, Segment s1);
void subdivide(double t, Segment s0, Segment s1);
double getLength();
double getLength(double maxErr);
SplitResults split(double y);
class SplitResults {
Segment [] above;
Segment [] below;
SplitResults(Segment []below, Segment []above) {
this.below = below;
this.above = above;
}
Segment [] getBelow() {
return below;
}
Segment [] getAbove() {
return above;
}
}
}
/*
Licensed to the Apache Software Foundation (ASF) under one or more
contributor license agreements. See the NOTICE file distributed with
this work for additional information regarding copyright ownership.
The ASF licenses this file to You under the Apache License, Version 2.0
(the "License"); you may not use this file except in compliance with
the License. You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
/**
* An abstract class for path segments.
*
* @version $Id: AbstractSegment.java 478249 2006-11-22 17:29:37Z dvholten $
*/
abstract class AbstractSegment implements Segment {
protected abstract int findRoots(double y, double [] roots);
public Segment.SplitResults split(double y) {
double [] roots = { 0, 0, 0 };
int numSol = findRoots(y, roots);
if (numSol == 0) return null; // No split
Arrays.sort(roots, 0, numSol);
double [] segs = new double[numSol+2];
int numSegments=0;
segs[numSegments++] = 0;
for (int i=0; i<numSol; i++) {
double r = roots[i];
if (r <= 0.0) continue;
if (r >= 1.0) break;
if (segs[numSegments-1] != r)
segs[numSegments++] = r;
}
segs[numSegments++] = 1.0;
if (numSegments == 2) return null;
// System.err.println("Y: " + y + "#Seg: " + numSegments +
// " Seg: " + this);
Segment [] parts = new Segment[numSegments];
double pT = 0.0;
int pIdx = 0;
boolean firstAbove=false, prevAbove=false;
for (int i=1; i<numSegments; i++) {
// System.err.println("Segs: " + segs[i-1]+", "+segs[i]);
parts[pIdx] = getSegment(segs[i-1], segs[i]);
Point2D.Double pt = parts[pIdx].eval(0.5);
// System.err.println("Pt: " + pt);
if (pIdx == 0) {
pIdx++;
firstAbove = prevAbove = (pt.y < y);
continue;
}
boolean above = (pt.y < y);
if (prevAbove == above) {
// Merge segments
parts[pIdx-1] = getSegment(pT, segs[i]);
} else {
pIdx++;
pT=segs[i-1];
prevAbove = above;
}
}
if (pIdx == 1) return null;
Segment [] below, above;
if (firstAbove) {
above = new Segment[(pIdx+1)/2];
below = new Segment[pIdx/2];
} else {
above = new Segment[pIdx/2];
below = new Segment[(pIdx+1)/2];
}
int ai=0, bi=0;
for (int i=0; i<pIdx; i++) {
if (firstAbove) above[ai++] = parts[i];
else below[bi++] = parts[i];
firstAbove = !firstAbove;
}
return new SplitResults(below, above);
}
public Segment splitBefore(double t) {
return getSegment(0.0, t);
}
public Segment splitAfter(double t) {
return getSegment(t, 1.0);
}
// Doubles have 48bit precision
static final double eps = 1/(double)(1L<<48);
static final double tol = 4.0*eps;
public static int solveLine(double a, double b,
double [] roots) {
if (a == 0) {
if (b != 0)
// No intersection.
return 0;
// All pts intersect just return 0.
roots[0] = 0;
return 1;
}
roots[0] = -b/a;
return 1;
}
public static int solveQuad(double a, double b, double c,
double [] roots) {
// System.err.println("Quad: " + a +"t^2 + " + b +"t + " + c);
if (a == 0) {
// no square term.
return solveLine(b, c, roots);
}
double det = b*b-4*a*c;
// System.err.println("Det: " + det);
if (Math.abs(det) <= tol*b*b) {
// one real root (det doesn't contain any useful info)
roots[0] = -b/(2*a);
return 1;
}
if (det < 0)
return 0; // No real roots
// Two real roots
det = Math.sqrt(det);
double w = -(b + matchSign(det, b));
roots[0] = (2*c)/w;
roots[1] = w/(2*a);
return 2;
}
public static double matchSign(double a, double b) {
if (b < 0) return (a < 0)?a:-a;
return (a > 0)?a:-a;
}
public static int solveCubic(double a3, double a2,
double a1, double a0,
double [] roots) {
// System.err.println("Cubic: " + a3 + "t^3 + " +
// a2 +"t^2 + " +
// a1 +"t + " + a0);
double [] dRoots = { 0, 0};
int dCnt = solveQuad(3*a3, 2*a2, a1, dRoots);
double [] yVals = {0, 0, 0, 0};
double [] tVals = {0, 0, 0, 0};
int yCnt=0;
yVals[yCnt] = a0;
tVals[yCnt++] = 0;
double r;
switch (dCnt) {
case 1:
r = dRoots[0];
if ((r > 0) && (r < 1)) {
yVals[yCnt] = ((a3*r+a2)*r+a1)*r+a0;
tVals[yCnt++] = r;
}
break;
case 2:
if (dRoots[0] > dRoots[1]) {
double t = dRoots[0];
dRoots[0] = dRoots[1];
dRoots[1] = t;
}
r = dRoots[0];
if ((r > 0) && (r < 1)) {
yVals[yCnt] = ((a3*r+a2)*r+a1)*r+a0;
tVals[yCnt++] = r;
}
r = dRoots[1];
if ((r > 0) && (r < 1)) {
yVals[yCnt] = ((a3*r+a2)*r+a1)*r+a0;
tVals[yCnt++] = r;
}
break;
default: break;
}
yVals[yCnt] = a3+a2+a1+a0;
tVals[yCnt++] = 1.0;
int ret=0;
for (int i=0; i<yCnt-1; i++) {
double y0 = yVals[i], t0 = tVals[i];
double y1 = yVals[i+1], t1 = tVals[i+1];
if ((y0 < 0) && (y1 < 0)) continue;
if ((y0 > 0) && (y1 > 0)) continue;
if (y0 > y1) { // swap so y0 < 0 and y1 > 0
double t;
t = y0; y0=y1; y1=t;
t = t0; t0=t1; t1=t;
}
if (-y0 < tol*y1) { roots[ret++] = t0; continue; }
if (y1 < -tol*y0) { roots[ret++] = t1; i++; continue; }
double epsZero = tol*(y1-y0);
int cnt;
for (cnt=0; cnt<20; cnt++) {
double dt = t1-t0;
double dy = y1-y0;
// double t = (t0+t1)/2;
// double t= t0+Math.abs(y0/dy)*dt;
// This tends to make sure that we come up
// a little short each time this generaly allows
// you to eliminate as much of the range as possible
// without overshooting (in which case you may eliminate
// almost nothing).
double t= t0+(Math.abs(y0/dy)*99+.5)*dt/100;
double v = ((a3*t+a2)*t+a1)*t+a0;
if (Math.abs(v) < epsZero) {
roots[ret++] = t; break;
}
if (v < 0) { t0 = t; y0=v;}
else { t1 = t; y1=v;}
}
if (cnt == 20)
roots[ret++] = (t0+t1)/2;
}
return ret;
}
/*
public static void check(Segment seg, float y, PrintStream ps) {
ps.println("<path fill=\"none\" stroke=\"black\" " +
" stroke-width=\"3\" d=\"" + seg + "\"/>");
ps.println("<line x1=\"-1000\" y1=\""+y+
"\" x2=\"1000\" y2=\""+y+"\" fill=\"none\" stroke=\"orange\"/>\n");
SplitResults sr = seg.split(y);
if (sr == null) return;
Segment [] above = sr.getAbove();
Segment [] below = sr.getBelow();
for (int i=0; i<above.length; i++) {
ps.println("<path fill=\"none\" stroke=\"blue\" " +
" stroke-width=\"2.5\" " +
" d=\"" + above[i] + "\"/>");
}
for (int i=0; i<below.length; i++) {
ps.println("<path fill=\"none\" stroke=\"red\" " +
" stroke-width=\"2\" " +
"d=\"" + below[i] + "\"/>");
}
}
public static void main(String [] args) {
PrintStream ps;
double [] roots = { 0, 0, 0 };
int n = solveCubic (-0.10000991821289062, 9.600013732910156,
-35.70000457763672, 58.0, roots);
for (int i=0; i<n; i++)
System.err.println("Root: " + roots[i]);
Cubic c;
c = new Cubic(new Point2D.Double(153.6999969482422,5.099999904632568),
new Point2D.Double(156.6999969482422,4.099999904632568),
new Point2D.Double(160.39999389648438,2.3999998569488525),
new Point2D.Double(164.6999969482422,0.0));
c.split(0);
c = new Cubic(new Point2D.Double(24.899999618530273,23.10000228881836),
new Point2D.Double(41.5,8.399999618530273),
new Point2D.Double(64.69999694824219,1.0),
new Point2D.Double(94.5999984741211,1.0));
c.split(0);
try {
ps = new PrintStream(new FileOutputStream(args[0]));
} catch(java.io.IOException ioe) {
ioe.printStackTrace();
return;
}
ps.println("<?xml version=\"1.0\" standalone=\"no\"?>\n" +
"<!DOCTYPE svg PUBLIC \"-//W3C//DTD SVG 1.0//EN\"\n" +
"\"http://www.w3.org/TR/2001/REC-SVG-20010904/DTD/svg10.dtd\">\n" +
"<svg width=\"450\" height=\"500\"\n" +
" viewBox=\"-100 -100 450 500\"\n" +
" xmlns=\"http://www.w3.org/2000/svg\"\n" +
" xmlns:xlink=\"http://www.w3.org/1999/xlink\">");
check(new Cubic(new Point2D.Double(0, 0),
new Point2D.Double(100, 100),
new Point2D.Double(-50, 100),
new Point2D.Double(50, 0)), 40, ps);
check(new Cubic(new Point2D.Double(100, 0),
new Point2D.Double(200, 100),
new Point2D.Double(50, -50),
new Point2D.Double(150, 30)), 20, ps);
check(new Cubic(new Point2D.Double(200, 0),
new Point2D.Double(300, 100),
new Point2D.Double(150, 100),
new Point2D.Double(250, 0)), 75, ps);
check(new Quadradic(new Point2D.Double(0, 100),
new Point2D.Double(50,150),
new Point2D.Double(10,100)), 115, ps);
check(new Linear(new Point2D.Double(100, 100),
new Point2D.Double(150,150)), 115, ps);
ps.println("</svg>");
}
*/
}
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