12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package sun.java2d.marlin; 27 28 final class Curve { 29 30 float ax, ay, bx, by, cx, cy, dx, dy; 31 float dax, day, dbx, dby; 32 // shared iterator instance 33 private final BreakPtrIterator iterator = new BreakPtrIterator(); 34 35 Curve() { 36 } 37 38 void set(float[] points, int type) { 39 switch(type) { 40 case 8: 41 set(points[0], points[1], 42 points[2], points[3], 43 points[4], points[5], 44 points[6], points[7]); 45 return; 46 case 6: 47 set(points[0], points[1], 48 points[2], points[3], 49 points[4], points[5]); 50 return; 51 default: 52 throw new InternalError("Curves can only be cubic or quadratic"); 53 } 54 } 55 56 void set(float x1, float y1, 57 float x2, float y2, 58 float x3, float y3, 59 float x4, float y4) 60 { 61 ax = 3f * (x2 - x3) + x4 - x1; 62 ay = 3f * (y2 - y3) + y4 - y1; 63 bx = 3f * (x1 - 2f * x2 + x3); 64 by = 3f * (y1 - 2f * y2 + y3); 65 cx = 3f * (x2 - x1); 66 cy = 3f * (y2 - y1); 67 dx = x1; 68 dy = y1; 69 dax = 3f * ax; day = 3f * ay; 70 dbx = 2f * bx; dby = 2f * by; 71 } 72 73 void set(float x1, float y1, 74 float x2, float y2, 75 float x3, float y3) 76 { 77 ax = 0f; ay = 0f; 78 bx = x1 - 2f * x2 + x3; 79 by = y1 - 2f * y2 + y3; 80 cx = 2f * (x2 - x1); 81 cy = 2f * (y2 - y1); 82 dx = x1; 83 dy = y1; 84 dax = 0f; day = 0f; 85 dbx = 2f * bx; dby = 2f * by; 86 } 87 88 float xat(float t) { 89 return t * (t * (t * ax + bx) + cx) + dx; 90 } 91 float yat(float t) { 92 return t * (t * (t * ay + by) + cy) + dy; 93 } 94 95 float dxat(float t) { 96 return t * (t * dax + dbx) + cx; 97 } 98 99 float dyat(float t) { 100 return t * (t * day + dby) + cy; 101 } 102 103 int dxRoots(float[] roots, int off) { 104 return Helpers.quadraticRoots(dax, dbx, cx, roots, off); 105 } 106 107 int dyRoots(float[] roots, int off) { 108 return Helpers.quadraticRoots(day, dby, cy, roots, off); 109 } 110 111 int infPoints(float[] pts, int off) { 112 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0 113 // Fortunately, this turns out to be quadratic, so there are at 114 // most 2 inflection points. 115 final float a = dax * dby - dbx * day; 116 final float b = 2f * (cy * dax - day * cx); 117 final float c = cy * dbx - cx * dby; 118 119 return Helpers.quadraticRoots(a, b, c, pts, off); 120 } 121 122 // finds points where the first and second derivative are 123 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where 124 // * is a dot product). Unfortunately, we have to solve a cubic. 125 private int perpendiculardfddf(float[] pts, int off) { 126 assert pts.length >= off + 4; 127 128 // these are the coefficients of some multiple of g(t) (not g(t), 129 // because the roots of a polynomial are not changed after multiplication 130 // by a constant, and this way we save a few multiplications). 131 final float a = 2f * (dax*dax + day*day); 132 final float b = 3f * (dax*dbx + day*dby); 133 final float c = 2f * (dax*cx + day*cy) + dbx*dbx + dby*dby; 134 final float d = dbx*cx + dby*cy; 135 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0f, 1f); 136 } 137 138 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses 139 // a variant of the false position algorithm to find the roots. False 140 // position requires that 2 initial values x0,x1 be given, and that the 141 // function must have opposite signs at those values. To find such 142 // values, we need the local extrema of the ROC function, for which we 143 // need the roots of its derivative; however, it's harder to find the 144 // roots of the derivative in this case than it is to find the roots 145 // of the original function. So, we find all points where this curve's 146 // first and second derivative are perpendicular, and we pretend these 147 // are our local extrema. There are at most 3 of these, so we will check 148 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection 149 // points, so roc-w can have at least 6 roots. This shouldn't be a 150 // problem for what we're trying to do (draw a nice looking curve). 151 int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) { 152 // no OOB exception, because by now off<=6, and roots.length >= 10 153 assert off <= 6 && roots.length >= 10; 154 int ret = off; 155 int numPerpdfddf = perpendiculardfddf(roots, off); 156 float t0 = 0, ft0 = ROCsq(t0) - w*w; 157 roots[off + numPerpdfddf] = 1f; // always check interval end points 158 numPerpdfddf++; 159 for (int i = off; i < off + numPerpdfddf; i++) { 160 float t1 = roots[i], ft1 = ROCsq(t1) - w*w; 161 if (ft0 == 0f) { 162 roots[ret++] = t0; 163 } else if (ft1 * ft0 < 0f) { // have opposite signs 164 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because 165 // ROC(t) >= 0 for all t. 166 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err); 167 } 168 t0 = t1; 169 ft0 = ft1; 170 } 171 172 return ret - off; 173 } 174 175 private static float eliminateInf(float x) { 176 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE : 177 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x)); 178 } 179 180 // A slight modification of the false position algorithm on wikipedia. 181 // This only works for the ROCsq-x functions. It might be nice to have 182 // the function as an argument, but that would be awkward in java6. 183 // TODO: It is something to consider for java8 (or whenever lambda 203 } else { 204 side = -1; 205 } 206 } else if (fr * fs > 0) { 207 fs = fr; s = r; 208 if (side > 0) { 209 ft /= (1 << side); 210 side++; 211 } else { 212 side = 1; 213 } 214 } else { 215 break; 216 } 217 } 218 return r; 219 } 220 221 private static boolean sameSign(float x, float y) { 222 // another way is to test if x*y > 0. This is bad for small x, y. 223 return (x < 0f && y < 0f) || (x > 0f && y > 0f); 224 } 225 226 // returns the radius of curvature squared at t of this curve 227 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications) 228 private float ROCsq(final float t) { 229 // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency 230 final float dx = t * (t * dax + dbx) + cx; 231 final float dy = t * (t * day + dby) + cy; 232 final float ddx = 2f * dax * t + dbx; 233 final float ddy = 2f * day * t + dby; 234 final float dx2dy2 = dx*dx + dy*dy; 235 final float ddx2ddy2 = ddx*ddx + ddy*ddy; 236 final float ddxdxddydy = ddx*dx + ddy*dy; 237 return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy)); 238 } 239 240 // curve to be broken should be in pts 241 // this will change the contents of pts but not Ts 242 // TODO: There's no reason for Ts to be an array. All we need is a sequence 243 // of t values at which to subdivide. An array statisfies this condition, 244 // but is unnecessarily restrictive. Ts should be an Iterator<Float> instead. 245 // Doing this will also make dashing easier, since we could easily make 246 // LengthIterator an Iterator<Float> and feed it to this function to simplify 247 // the loop in Dasher.somethingTo. 248 BreakPtrIterator breakPtsAtTs(final float[] pts, final int type, 249 final float[] Ts, final int numTs) 250 { 251 assert pts.length >= 2*type && numTs <= Ts.length; 252 253 // initialize shared iterator: 254 iterator.init(pts, type, Ts, numTs); 255 256 return iterator; 257 } 258 259 static final class BreakPtrIterator { 260 private int nextCurveIdx; 261 private int curCurveOff; 262 private float prevT; 263 private float[] pts; 264 private int type; 265 private float[] ts; 266 private int numTs; 267 268 void init(final float[] pts, final int type, 269 final float[] ts, final int numTs) { 270 this.pts = pts; 271 this.type = type; 272 this.ts = ts; 273 this.numTs = numTs; 274 275 nextCurveIdx = 0; 276 curCurveOff = 0; 277 prevT = 0f; 278 } 279 280 public boolean hasNext() { 281 return nextCurveIdx <= numTs; 282 } 283 284 public int next() { 285 int ret; 286 if (nextCurveIdx < numTs) { 287 float curT = ts[nextCurveIdx]; 288 float splitT = (curT - prevT) / (1f - prevT); 289 Helpers.subdivideAt(splitT, 290 pts, curCurveOff, 291 pts, 0, 292 pts, type, type); 293 prevT = curT; 294 ret = 0; 295 curCurveOff = type; 296 } else { 297 ret = curCurveOff; 298 } 299 nextCurveIdx++; 300 return ret; 301 } 302 } 303 } 304 | 12 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or 13 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License 14 * version 2 for more details (a copy is included in the LICENSE file that 15 * accompanied this code). 16 * 17 * You should have received a copy of the GNU General Public License version 18 * 2 along with this work; if not, write to the Free Software Foundation, 19 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. 20 * 21 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA 22 * or visit www.oracle.com if you need additional information or have any 23 * questions. 24 */ 25 26 package sun.java2d.marlin; 27 28 final class Curve { 29 30 float ax, ay, bx, by, cx, cy, dx, dy; 31 float dax, day, dbx, dby; 32 33 Curve() { 34 } 35 36 void set(float[] points, int type) { 37 switch(type) { 38 case 8: 39 set(points[0], points[1], 40 points[2], points[3], 41 points[4], points[5], 42 points[6], points[7]); 43 return; 44 case 6: 45 set(points[0], points[1], 46 points[2], points[3], 47 points[4], points[5]); 48 return; 49 default: 50 throw new InternalError("Curves can only be cubic or quadratic"); 51 } 52 } 53 54 void set(float x1, float y1, 55 float x2, float y2, 56 float x3, float y3, 57 float x4, float y4) 58 { 59 ax = 3.0f * (x2 - x3) + x4 - x1; 60 ay = 3.0f * (y2 - y3) + y4 - y1; 61 bx = 3.0f * (x1 - 2.0f * x2 + x3); 62 by = 3.0f * (y1 - 2.0f * y2 + y3); 63 cx = 3.0f * (x2 - x1); 64 cy = 3.0f * (y2 - y1); 65 dx = x1; 66 dy = y1; 67 dax = 3.0f * ax; day = 3.0f * ay; 68 dbx = 2.0f * bx; dby = 2.0f * by; 69 } 70 71 void set(float x1, float y1, 72 float x2, float y2, 73 float x3, float y3) 74 { 75 ax = 0.0f; ay = 0.0f; 76 bx = x1 - 2.0f * x2 + x3; 77 by = y1 - 2.0f * y2 + y3; 78 cx = 2.0f * (x2 - x1); 79 cy = 2.0f * (y2 - y1); 80 dx = x1; 81 dy = y1; 82 dax = 0.0f; day = 0.0f; 83 dbx = 2.0f * bx; dby = 2.0f * by; 84 } 85 86 float xat(float t) { 87 return t * (t * (t * ax + bx) + cx) + dx; 88 } 89 float yat(float t) { 90 return t * (t * (t * ay + by) + cy) + dy; 91 } 92 93 float dxat(float t) { 94 return t * (t * dax + dbx) + cx; 95 } 96 97 float dyat(float t) { 98 return t * (t * day + dby) + cy; 99 } 100 101 int dxRoots(float[] roots, int off) { 102 return Helpers.quadraticRoots(dax, dbx, cx, roots, off); 103 } 104 105 int dyRoots(float[] roots, int off) { 106 return Helpers.quadraticRoots(day, dby, cy, roots, off); 107 } 108 109 int infPoints(float[] pts, int off) { 110 // inflection point at t if -f'(t)x*f''(t)y + f'(t)y*f''(t)x == 0 111 // Fortunately, this turns out to be quadratic, so there are at 112 // most 2 inflection points. 113 final float a = dax * dby - dbx * day; 114 final float b = 2.0f * (cy * dax - day * cx); 115 final float c = cy * dbx - cx * dby; 116 117 return Helpers.quadraticRoots(a, b, c, pts, off); 118 } 119 120 // finds points where the first and second derivative are 121 // perpendicular. This happens when g(t) = f'(t)*f''(t) == 0 (where 122 // * is a dot product). Unfortunately, we have to solve a cubic. 123 private int perpendiculardfddf(float[] pts, int off) { 124 assert pts.length >= off + 4; 125 126 // these are the coefficients of some multiple of g(t) (not g(t), 127 // because the roots of a polynomial are not changed after multiplication 128 // by a constant, and this way we save a few multiplications). 129 final float a = 2.0f * (dax*dax + day*day); 130 final float b = 3.0f * (dax*dbx + day*dby); 131 final float c = 2.0f * (dax*cx + day*cy) + dbx*dbx + dby*dby; 132 final float d = dbx*cx + dby*cy; 133 return Helpers.cubicRootsInAB(a, b, c, d, pts, off, 0.0f, 1.0f); 134 } 135 136 // Tries to find the roots of the function ROC(t)-w in [0, 1). It uses 137 // a variant of the false position algorithm to find the roots. False 138 // position requires that 2 initial values x0,x1 be given, and that the 139 // function must have opposite signs at those values. To find such 140 // values, we need the local extrema of the ROC function, for which we 141 // need the roots of its derivative; however, it's harder to find the 142 // roots of the derivative in this case than it is to find the roots 143 // of the original function. So, we find all points where this curve's 144 // first and second derivative are perpendicular, and we pretend these 145 // are our local extrema. There are at most 3 of these, so we will check 146 // at most 4 sub-intervals of (0,1). ROC has asymptotes at inflection 147 // points, so roc-w can have at least 6 roots. This shouldn't be a 148 // problem for what we're trying to do (draw a nice looking curve). 149 int rootsOfROCMinusW(float[] roots, int off, final float w, final float err) { 150 // no OOB exception, because by now off<=6, and roots.length >= 10 151 assert off <= 6 && roots.length >= 10; 152 int ret = off; 153 int numPerpdfddf = perpendiculardfddf(roots, off); 154 float t0 = 0.0f, ft0 = ROCsq(t0) - w*w; 155 roots[off + numPerpdfddf] = 1.0f; // always check interval end points 156 numPerpdfddf++; 157 for (int i = off; i < off + numPerpdfddf; i++) { 158 float t1 = roots[i], ft1 = ROCsq(t1) - w*w; 159 if (ft0 == 0.0f) { 160 roots[ret++] = t0; 161 } else if (ft1 * ft0 < 0.0f) { // have opposite signs 162 // (ROC(t)^2 == w^2) == (ROC(t) == w) is true because 163 // ROC(t) >= 0 for all t. 164 roots[ret++] = falsePositionROCsqMinusX(t0, t1, w*w, err); 165 } 166 t0 = t1; 167 ft0 = ft1; 168 } 169 170 return ret - off; 171 } 172 173 private static float eliminateInf(float x) { 174 return (x == Float.POSITIVE_INFINITY ? Float.MAX_VALUE : 175 (x == Float.NEGATIVE_INFINITY ? Float.MIN_VALUE : x)); 176 } 177 178 // A slight modification of the false position algorithm on wikipedia. 179 // This only works for the ROCsq-x functions. It might be nice to have 180 // the function as an argument, but that would be awkward in java6. 181 // TODO: It is something to consider for java8 (or whenever lambda 201 } else { 202 side = -1; 203 } 204 } else if (fr * fs > 0) { 205 fs = fr; s = r; 206 if (side > 0) { 207 ft /= (1 << side); 208 side++; 209 } else { 210 side = 1; 211 } 212 } else { 213 break; 214 } 215 } 216 return r; 217 } 218 219 private static boolean sameSign(float x, float y) { 220 // another way is to test if x*y > 0. This is bad for small x, y. 221 return (x < 0.0f && y < 0.0f) || (x > 0.0f && y > 0.0f); 222 } 223 224 // returns the radius of curvature squared at t of this curve 225 // see http://en.wikipedia.org/wiki/Radius_of_curvature_(applications) 226 private float ROCsq(final float t) { 227 // dx=xat(t) and dy=yat(t). These calls have been inlined for efficiency 228 final float dx = t * (t * dax + dbx) + cx; 229 final float dy = t * (t * day + dby) + cy; 230 final float ddx = 2.0f * dax * t + dbx; 231 final float ddy = 2.0f * day * t + dby; 232 final float dx2dy2 = dx*dx + dy*dy; 233 final float ddx2ddy2 = ddx*ddx + ddy*ddy; 234 final float ddxdxddydy = ddx*dx + ddy*dy; 235 return dx2dy2*((dx2dy2*dx2dy2) / (dx2dy2 * ddx2ddy2 - ddxdxddydy*ddxdxddydy)); 236 } 237 } 238 |