/* * Copyright (c) 2007, 2011, Oracle and/or its affiliates. All rights reserved. * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. * * This code is free software; you can redistribute it and/or modify it * under the terms of the GNU General Public License version 2 only, as * published by the Free Software Foundation. Oracle designates this * particular file as subject to the "Classpath" exception as provided * by Oracle in the LICENSE file that accompanied this code. * * This code is distributed in the hope that it will be useful, but WITHOUT * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License * version 2 for more details (a copy is included in the LICENSE file that * accompanied this code). * * You should have received a copy of the GNU General Public License version * 2 along with this work; if not, write to the Free Software Foundation, * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. * * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA * or visit www.oracle.com if you need additional information or have any * questions. */ package sun.java2d.pisces; import java.util.Iterator; final class Curve { float ax, ay, bx, by, cx, cy, dx, dy; float dax, day, dbx, dby; Curve() { } void set(float[] points, int type) { switch(type) { case 8: set(points[0], points[1], points[2], points[3], points[4], points[5], points[6], points[7]); break; case 6: set(points[0], points[1], points[2], points[3], points[4], points[5]); break; default: throw new InternalError("Curves can only be cubic or quadratic"); } } void set(float x1, float y1, float x2, float y2, float x3, float y3, float x4, float y4) { ax = 3 * (x2 - x3) + x4 - x1; ay = 3 * (y2 - y3) + y4 - y1; bx = 3 * (x1 - 2 * x2 + x3); by = 3 * (y1 - 2 * y2 + y3); cx = 3 * (x2 - x1); cy = 3 * (y2 - y1); dx = x1; dy = y1; dax = 3 * ax; day = 3 * ay; dbx = 2 * bx; dby = 2 * by; } void set(float x1, float y1, float x2, float y2, float x3, float y3) { ax = ay = 0f; bx = x1 - 2 * x2 + x3; by = y1 - 2 * y2 + y3; cx = 2 * (x2 - x1); cy = 2 * (y2 - y1); dx = x1; dy = y1; dax = 0; day = 0; dbx = 2 * bx; dby = 2 * by; } float xat(float t) { return t * (t * (t * ax + bx) + cx) + dx; } float yat(float t) { return t * (t * (t * ay + by) + cy) + dy; } float dxat(float t) { return t * (t * dax + dbx) + cx; } float dyat(float t) { return t * (t * day + dby) + cy; } int dxRoots(float[] roots, int off) { return Helpers.quadraticRoots(dax, dbx, cx, roots, off); } int dyRoots(float[] roots, int off) { return Helpers.quadraticRoots(day, dby, cy, roots, off); } int infPoints(float[] pts, int off) { // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0 // Fortunately, this turns out to be quadratic, so there are at // most 2 inflection points. final float a = dax * dby - dbx * day; final float b = 2 * (cy * dax - day * cx); final float c = cy * dbx - cx * dby; return Helpers.quadraticRoots(a, b, c, pts, off); } // finds points where the first and second derivative are // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where // * is a dot product). Unfortunately, we have to solve a cubic. private int perpendiculardfddf(float[] pts, int off) { assert pts.length >= off + 4; // these are the coefficients of some multiple of g(t) (not g(t), // because the roots of a polynomial are not changed after multiplication // by a constant, and this way we save a few multiplications). final float a = 2*(dax*dax + day*day); final float b = 3*(dax*dbx + day*dby); final float c = 2*(dax*cx + day*cy) + dbx*dbx + dby*dby; final float d = dbx*cx + dby*cy; return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f); } // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses // a variant of the false position algorithm to find the roots. False // position requires that 2 initial values x0,x1 be given, and that the // function must have opposite signs at those values. To find such // values, we need the local extrema of the ROC function, for which we // need the roots of its derivative; however, it's harder to find the // roots of the derivative in this case than it is to find the roots // of the original function. So, we find all points where this curve's // first and second derivative are perpendicular, and we pretend these // are our local extrema. There are at most 3 of these, so we will check // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection // points, so roc-w can have at least 6 roots. This shouldn't be a // problem for what we're trying to do (draw a nice looking curve). int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) { // no OOB exception, because by now off<=6, and roots.length >= 10 assert off <= 6 && roots.length >= 10; int ret = off; int numPerpdfddf = perpendiculardfddf(roots, off); float t0 = 0, ft0 = ROCsq(t0) - w*w; roots[off + numPerpdfddf] = 1f; // always check interval end points numPerpdfddf++; for (int i = off; i < off + numPerpdfddf; i++) { float t1 = roots[i], ft1 = ROCsq(t1) - w*w; if (ft0 == 0f) { roots[ret++] = t0; } else if (ft1 * ft0 < 0f) { // have opposite signs // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because // ROC(t) >= 0 for all t. roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err); } t0 = t1; ft0 = ft1; } return ret - off; } private static float eliminateInf(float x) { return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE : (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x)); } // A slight modification of the false position algorithm on wikipedia. // This only works for the ROCsq-x functions. It might be nice to have // the function as an argument, but that would be awkward in java6. // TODO: It is something to consider for java8 (or whenever lambda // expressions make it into the language), depending on how closures // and turn out. Same goes for the newton's method // algorithm in Helpers.java private float falsePositionROCsqMinusX(float x0, float x1, final float x, final float err) { final int iterLimit = 100; int side = 0; float t = x1, ft = eliminateInf(ROCsq(t) - x); float s = x0, fs = eliminateInf(ROCsq(s) - x); float r = s, fr; for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) { r = (fs * t - ft * s) / (fs - ft); fr = ROCsq(r) - x; if (sameSign(fr, ft)) { ft = fr; t = r; if (side < 0) { fs /= (1 << (-side)); side--; } else { side = -1; } } else if (fr * fs > 0) { fs = fr; s = r; if (side > 0) { ft /= (1 << side); side++; } else { side = 1; } } else { break; } } return r; } private static boolean sameSign(double x, double y) { // another way is to test if x*y > 0. This is bad for small x, y. return (x < 0 && y < 0) || (x > 0 && y > 0); } // returns the radius of curvature squared at t of this curve // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications) private float ROCsq(final float t) { // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency final float dx = t * (t * dax + dbx) + cx; final float dy = t * (t * day + dby) + cy; final float ddx = 2 * dax * t + dbx; final float ddy = 2 * day * t + dby; final float dx2dy2 = dx*dx + dy*dy; final float ddx2ddy2 = ddx*ddx + ddy*ddy; final float ddxdxddydy = ddx*dx + ddy*dy; return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy)); } // curve to be broken should be in pts // this will change the contents of pts but not Ts // TODO: There's no reason for Ts to be an array. All we need is a sequence // of t values at which to subdivide. An array statisfies this condition, // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead. // Doing this will also make dashing easier, since we could easily make // LengthIterator an Iterator<Float> and feed it to this function to simplify // the loop in Dasher.somethingTo. static Iterator<Integer> breakPtsAtTs(final float[] pts, final int type, final float[] Ts, final int numTs) { assert pts.length >= 2*type && numTs <= Ts.length; return new Iterator<Integer>() { // these prevent object creation and destruction during autoboxing. // Because of this, the compiler should be able to completely // eliminate the boxing costs. final Integer i0 = 0; final Integer itype = type; int nextCurveIdx = 0; Integer curCurveOff = i0; float prevT = 0; @Override public boolean hasNext() { return nextCurveIdx < numTs + 1; } @Override public Integer next() { Integer ret; if (nextCurveIdx < numTs) { float curT = Ts[nextCurveIdx]; float splitT = (curT - prevT) / (1 - prevT); Helpers.subdivideAt(splitT, pts, curCurveOff, pts, 0, pts, type, type); prevT = curT; ret = i0; curCurveOff = itype; } else { ret = curCurveOff; } nextCurveIdx++; return ret; } @Override public void remove() {} }; }}
/*
* Copyright (c) 2007, 2011, Oracle and/or its affiliates. All rights reserved.
* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
*
* This code is free software; you can redistribute it and/or modify it
* under the terms of the GNU General Public License version 2 only, as
* published by the Free Software Foundation. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
* by Oracle in the LICENSE file that accompanied this code.
*
* This code is distributed in the hope that it will be useful, but WITHOUT
* ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
* FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
* version 2 for more details (a copy is included in the LICENSE file that
* accompanied this code).
*
* You should have received a copy of the GNU General Public License version
* 2 along with this work; if not, write to the Free Software Foundation,
* Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA.
*
* Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA
* or visit www.oracle.com if you need additional information or have any
* questions.
*/
package sun.java2d.pisces;
import java.util.Iterator;
final class Curve {
float ax, ay, bx, by, cx, cy, dx, dy;
float dax, day, dbx, dby;
Curve() {
}
void set(float[] points, int type) {
switch(type) {
case 8:
set(points[0], points[1],
points[2], points[3],
points[4], points[5],
points[6], points[7]);
break;
case 6:
set(points[0], points[1],
points[2], points[3],
points[4], points[5]);
break;
default:
throw new InternalError("Curves can only be cubic or quadratic");
}
}
void set(float x1, float y1,
float x2, float y2,
float x3, float y3,
float x4, float y4)
{
ax = 3 * (x2 - x3) + x4 - x1;
ay = 3 * (y2 - y3) + y4 - y1;
bx = 3 * (x1 - 2 * x2 + x3);
by = 3 * (y1 - 2 * y2 + y3);
cx = 3 * (x2 - x1);
cy = 3 * (y2 - y1);
dx = x1;
dy = y1;
dax = 3 * ax; day = 3 * ay;
dbx = 2 * bx; dby = 2 * by;
}
void set(float x1, float y1,
float x2, float y2,
float x3, float y3)
{
ax = ay = 0f;
bx = x1 - 2 * x2 + x3;
by = y1 - 2 * y2 + y3;
cx = 2 * (x2 - x1);
cy = 2 * (y2 - y1);
dx = x1;
dy = y1;
dax = 0; day = 0;
dbx = 2 * bx; dby = 2 * by;
}
float xat(float t) {
return t * (t * (t * ax + bx) + cx) + dx;
}
float yat(float t) {
return t * (t * (t * ay + by) + cy) + dy;
}
float dxat(float t) {
return t * (t * dax + dbx) + cx;
}
float dyat(float t) {
return t * (t * day + dby) + cy;
}
int dxRoots(float[] roots, int off) {
return Helpers.quadraticRoots(dax, dbx, cx, roots, off);
}
int dyRoots(float[] roots, int off) {
return Helpers.quadraticRoots(day, dby, cy, roots, off);
}
int infPoints(float[] pts, int off) {
// inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0
// Fortunately, this turns out to be quadratic, so there are at
// most 2 inflection points.
final float a = dax * dby - dbx * day;
final float b = 2 * (cy * dax - day * cx);
final float c = cy * dbx - cx * dby;
return Helpers.quadraticRoots(a, b, c, pts, off);
}
// finds points where the first and second derivative are
// perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where
// * is a dot product). Unfortunately, we have to solve a cubic.
private int perpendiculardfddf(float[] pts, int off) {
assert pts.length >= off + 4;
// these are the coefficients of some multiple of g(t) (not g(t),
// because the roots of a polynomial are not changed after multiplication
// by a constant, and this way we save a few multiplications).
final float a = 2*(dax*dax + day*day);
final float b = 3*(dax*dbx + day*dby);
final float c = 2*(dax*cx + day*cy) + dbx*dbx + dby*dby;
final float d = dbx*cx + dby*cy;
return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f);
}
// Tries to find the roots of the function ROC(t)-w in [0, 1). It uses
// a variant of the false position algorithm to find the roots. False
// position requires that 2 initial values x0,x1 be given, and that the
// function must have opposite signs at those values. To find such
// values, we need the local extrema of the ROC function, for which we
// need the roots of its derivative; however, it's harder to find the
// roots of the derivative in this case than it is to find the roots
// of the original function. So, we find all points where this curve's
// first and second derivative are perpendicular, and we pretend these
// are our local extrema. There are at most 3 of these, so we will check
// at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection
// points, so roc-w can have at least 6 roots. This shouldn't be a
// problem for what we're trying to do (draw a nice looking curve).
int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) {
// no OOB exception, because by now off<=6, and roots.length >= 10
assert off <= 6 && roots.length >= 10;
int ret = off;
int numPerpdfddf = perpendiculardfddf(roots, off);
float t0 = 0, ft0 = ROCsq(t0) - w*w;
roots[off + numPerpdfddf] = 1f; // always check interval end points
numPerpdfddf++;
for (int i = off; i < off + numPerpdfddf; i++) {
float t1 = roots[i], ft1 = ROCsq(t1) - w*w;
if (ft0 == 0f) {
roots[ret++] = t0;
} else if (ft1 * ft0 < 0f) { // have opposite signs
// (ROC(t)^2 == w^2) == (ROC(t) == w) is true because
// ROC(t) >= 0 for all t.
roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err);
}
t0 = t1;
ft0 = ft1;
}
return ret - off;
}
private static float eliminateInf(float x) {
return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE :
(x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x));
}
// A slight modification of the false position algorithm on wikipedia.
// This only works for the ROCsq-x functions. It might be nice to have
// the function as an argument, but that would be awkward in java6.
// TODO: It is something to consider for java8 (or whenever lambda
// expressions make it into the language), depending on how closures
// and turn out. Same goes for the newton's method
// algorithm in Helpers.java
private float falsePositionROCsqMinusX(float x0, float x1,
final float x, final float err)
{
final int iterLimit = 100;
int side = 0;
float t = x1, ft = eliminateInf(ROCsq(t) - x);
float s = x0, fs = eliminateInf(ROCsq(s) - x);
float r = s, fr;
for (int i = 0; i < iterLimit && Math.abs(t - s) > err * Math.abs(t + s); i++) {
r = (fs * t - ft * s) / (fs - ft);
fr = ROCsq(r) - x;
if (sameSign(fr, ft)) {
ft = fr; t = r;
if (side < 0) {
fs /= (1 << (-side));
side--;
} else {
side = -1;
}
} else if (fr * fs > 0) {
fs = fr; s = r;
if (side > 0) {
ft /= (1 << side);
side++;
} else {
side = 1;
}
} else {
break;
}
}
return r;
}
private static boolean sameSign(double x, double y) {
// another way is to test if x*y > 0. This is bad for small x, y.
return (x < 0 && y < 0) || (x > 0 && y > 0);
}
// returns the radius of curvature squared at t of this curve
// see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications)
private float ROCsq(final float t) {
// dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency
final float dx = t * (t * dax + dbx) + cx;
final float dy = t * (t * day + dby) + cy;
final float ddx = 2 * dax * t + dbx;
final float ddy = 2 * day * t + dby;
final float dx2dy2 = dx*dx + dy*dy;
final float ddx2ddy2 = ddx*ddx + ddy*ddy;
final float ddxdxddydy = ddx*dx + ddy*dy;
return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy));
}
// curve to be broken should be in pts
// this will change the contents of pts but not Ts
// TODO: There's no reason for Ts to be an array. All we need is a sequence
// of t values at which to subdivide. An array statisfies this condition,
// but is unnecessarily restrictive. Ts should be an Iterator<Float> instead.
// Doing this will also make dashing easier, since we could easily make
// LengthIterator an Iterator<Float> and feed it to this function to simplify
// the loop in Dasher.somethingTo.
static Iterator<Integer> breakPtsAtTs(final float[] pts, final int type,
final float[] Ts, final int numTs)
{
assert pts.length >= 2*type && numTs <= Ts.length;
return new Iterator<Integer>() {
// these prevent object creation and destruction during autoboxing.
// Because of this, the compiler should be able to completely
// eliminate the boxing costs.
final Integer i0 = 0;
final Integer itype = type;
int nextCurveIdx = 0;
Integer curCurveOff = i0;
float prevT = 0;
@Override public boolean hasNext() {
return nextCurveIdx < numTs + 1;
}
@Override public Integer next() {
Integer ret;
if (nextCurveIdx < numTs) {
float curT = Ts[nextCurveIdx];
float splitT = (curT - prevT) / (1 - prevT);
Helpers.subdivideAt(splitT,
pts, curCurveOff,
pts, 0,
pts, type, type);
prevT = curT;
ret = i0;
curCurveOff = itype;
} else {
ret = curCurveOff;
}
nextCurveIdx++;
return ret;
}
@Override public void remove() {}
};
}
}