--- a/jdk/src/share/classes/sun/java2d/pisces/Helpers.java Thu Feb 03 19:15:30 2011 -0800
+++ b/jdk/src/share/classes/sun/java2d/pisces/Helpers.java Tue Feb 08 09:22:49 2011 -0500
@@ -26,6 +26,12 @@
package sun.java2d.pisces;
import java.util.Arrays;
+import static java.lang.Math.PI;
+import static java.lang.Math.cos;
+import static java.lang.Math.sqrt;
+import static java.lang.Math.cbrt;
+import static java.lang.Math.acos;
+
final class Helpers {
private Helpers() {
@@ -75,100 +81,74 @@
return ret - off;
}
- // find the roots of g(t) = a*t^3 + b*t^2 + c*t + d in [A,B)
- // We will not use Cardano's method, since it is complicated and
- // involves too many square and cubic roots. We will use Newton's method.
- // TODO: this should probably return ALL roots. Then the user can do
- // his own filtering of roots outside [A,B).
- static int cubicRootsInAB(final float a, final float b,
- final float c, final float d,
- float[] pts, final int off, final float E,
+ // find the roots of g(t) = d*t^3 + a*t^2 + b*t + c in [A,B)
+ static int cubicRootsInAB(float d, float a, float b, float c,
+ float[] pts, final int off,
final float A, final float B)
{
- if (a == 0) {
- return quadraticRoots(b, c, d, pts, off);
+ if (d == 0) {
+ int num = quadraticRoots(a, b, c, pts, off);
+ return filterOutNotInAB(pts, off, num, A, B) - off;
}
- // the coefficients of g'(t). no dc variable because dc=c
- // we use these to get the critical points of g(t), which
- // we then use to chose starting points for Newton's method. These
- // should be very close to the actual roots.
- final float da = 3 * a;
- final float db = 2 * b;
- int numCritPts = quadraticRoots(da, db, c, pts, off+1);
- numCritPts = filterOutNotInAB(pts, off+1, numCritPts, A, B) - off - 1;
- // need them sorted.
- if (numCritPts == 2 && pts[off+1] > pts[off+2]) {
- float tmp = pts[off+1];
- pts[off+1] = pts[off+2];
- pts[off+2] = tmp;
+ // From Graphics Gems:
+ // http://tog.acm.org/resources/GraphicsGems/gems/Roots3And4.c
+ // (also from awt.geom.CubicCurve2D. But here we don't need as
+ // much accuracy and we don't want to create arrays so we use
+ // our own customized version).
+
+ /* normal form: x^3 + ax^2 + bx + c = 0 */
+ a /= d;
+ b /= d;
+ c /= d;
+
+ // substitute x = y - A/3 to eliminate quadratic term:
+ // x^3 +Px + Q = 0
+ //
+ // Since we actually need P/3 and Q/2 for all of the
+ // calculations that follow, we will calculate
+ // p = P/3
+ // q = Q/2
+ // instead and use those values for simplicity of the code.
+ double sq_A = a * a;
+ double p = 1.0/3 * (-1.0/3 * sq_A + b);
+ double q = 1.0/2 * (2.0/27 * a * sq_A - 1.0/3 * a * b + c);
+
+ /* use Cardano's formula */
+
+ double cb_p = p * p * p;
+ double D = q * q + cb_p;
+
+ int num;
+ if (D < 0) {
+ // see: http://en.wikipedia.org/wiki/Cubic_function#Trigonometric_.28and_hyperbolic.29_method
+ final double phi = 1.0/3 * acos(-q / sqrt(-cb_p));
+ final double t = 2 * sqrt(-p);
+
+ pts[ off+0 ] = (float)( t * cos(phi));
+ pts[ off+1 ] = (float)(-t * cos(phi + PI / 3));
+ pts[ off+2 ] = (float)(-t * cos(phi - PI / 3));
+ num = 3;
+ } else {
+ final double sqrt_D = sqrt(D);
+ final double u = cbrt(sqrt_D - q);
+ final double v = - cbrt(sqrt_D + q);
+
+ pts[ off ] = (float)(u + v);
+ num = 1;
+
+ if (within(D, 0, 1e-8)) {
+ pts[off+1] = -(pts[off] / 2);
+ num = 2;
+ }
}
- int ret = off;
-
- // we don't actually care much about the extrema themselves. We
- // only use them to ensure that g(t) is monotonic in each
- // interval [pts[i],pts[i+1] (for i in off...off+numCritPts+1).
- // This will allow us to determine intervals containing exactly
- // one root.
- // The end points of the interval are always local extrema.
- pts[off] = A;
- pts[off + numCritPts + 1] = B;
- numCritPts += 2;
-
- float x0 = pts[off], fx0 = evalCubic(a, b, c, d, x0);
- for (int i = off; i < off + numCritPts - 1; i++) {
- float x1 = pts[i+1], fx1 = evalCubic(a, b, c, d, x1);
- if (fx0 == 0f) {
- pts[ret++] = x0;
- } else if (fx1 * fx0 < 0f) { // have opposite signs
- pts[ret++] = CubicNewton(a, b, c, d,
- x0 + fx0 * (x1 - x0) / (fx0 - fx1), E);
- }
- x0 = x1;
- fx0 = fx1;
- }
- return ret - off;
- }
+ final float sub = 1.0f/3 * a;
- // precondition: the polynomial to be evaluated must not be 0 at x0.
- static float CubicNewton(final float a, final float b,
- final float c, final float d,
- float x0, final float err)
- {
- // considering how this function is used, 10 should be more than enough
- final int itlimit = 10;
- float fx0 = evalCubic(a, b, c, d, x0);
- float x1;
- int count = 0;
- while(true) {
- x1 = x0 - (fx0 / evalCubic(0, 3 * a, 2 * b, c, x0));
- if (Math.abs(x1 - x0) < err * Math.abs(x1 + x0) || count == itlimit) {
- break;
- }
- x0 = x1;
- fx0 = evalCubic(a, b, c, d, x0);
- count++;
+ for (int i = 0; i < num; ++i) {
+ pts[ off+i ] -= sub;
}
- return x1;
- }
- // fills the input array with numbers 0, INC, 2*INC, ...
- static void fillWithIdxes(final float[] data, final int[] idxes) {
- if (idxes.length > 0) {
- idxes[0] = 0;
- for (int i = 1; i < idxes.length; i++) {
- idxes[i] = idxes[i-1] + (int)data[idxes[i-1]];
- }
- }
- }
-
- static void fillWithIdxes(final int[] idxes, final int inc) {
- if (idxes.length > 0) {
- idxes[0] = 0;
- for (int i = 1; i < idxes.length; i++) {
- idxes[i] = idxes[i-1] + inc;
- }
- }
+ return filterOutNotInAB(pts, off, num, A, B) - off;
}
// These use a hardcoded factor of 2 for increasing sizes. Perhaps this
@@ -182,6 +162,7 @@
}
return Arrays.copyOf(in, 2 * (cursize + numToAdd));
}
+
static int[] widenArray(int[] in, final int cursize, final int numToAdd) {
if (in.length >= cursize + numToAdd) {
return in;
@@ -208,7 +189,7 @@
{
int ret = off;
for (int i = off; i < off + len; i++) {
- if (nums[i] > a && nums[i] < b) {
+ if (nums[i] >= a && nums[i] < b) {
nums[ret++] = nums[i];
}
}
@@ -225,7 +206,9 @@
}
static float linelen(float x1, float y1, float x2, float y2) {
- return (float)Math.hypot(x2 - x1, y2 - y1);
+ final float dx = x2 - x1;
+ final float dy = y2 - y1;
+ return (float)Math.sqrt(dx*dx + dy*dy);
}
static void subdivide(float[] src, int srcoff, float[] left, int leftoff,