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* DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER.
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* published by the Free Software Foundation. Oracle designates this
* particular file as subject to the "Classpath" exception as provided
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*
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* 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).
*
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* 2 along with this work; if not, write to the Free Software Foundation,
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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() {}
};
}
}