summaryrefslogtreecommitdiffstats
path: root/src/gui/painting/qtransform.cpp
blob: 2383272ad8c20d6703289ab46b37abde6cea5b17 (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
973
974
975
976
977
978
979
980
981
982
983
984
985
986
987
988
989
990
991
992
993
994
995
996
997
998
999
1000
1001
1002
1003
1004
1005
1006
1007
1008
1009
1010
1011
1012
1013
1014
1015
1016
1017
1018
1019
1020
1021
1022
1023
1024
1025
1026
1027
1028
1029
1030
1031
1032
1033
1034
1035
1036
1037
1038
1039
1040
1041
1042
1043
1044
1045
1046
1047
1048
1049
1050
1051
1052
1053
1054
1055
1056
1057
1058
1059
1060
1061
1062
1063
1064
1065
1066
1067
1068
1069
1070
1071
1072
1073
1074
1075
1076
1077
1078
1079
1080
1081
1082
1083
1084
1085
1086
1087
1088
1089
1090
1091
1092
1093
1094
1095
1096
1097
1098
1099
1100
1101
1102
1103
1104
1105
1106
1107
1108
1109
1110
1111
1112
1113
1114
1115
1116
1117
1118
1119
1120
1121
1122
1123
1124
1125
1126
1127
1128
1129
1130
1131
1132
1133
1134
1135
1136
1137
1138
1139
1140
1141
1142
1143
1144
1145
1146
1147
1148
1149
1150
1151
1152
1153
1154
1155
1156
1157
1158
1159
1160
1161
1162
1163
1164
1165
1166
1167
1168
1169
1170
1171
1172
1173
1174
1175
1176
1177
1178
1179
1180
1181
1182
1183
1184
1185
1186
1187
1188
1189
1190
1191
1192
1193
1194
1195
1196
1197
1198
1199
1200
1201
1202
1203
1204
1205
1206
1207
1208
1209
1210
1211
1212
1213
1214
1215
1216
1217
1218
1219
1220
1221
1222
1223
1224
1225
1226
1227
1228
1229
1230
1231
1232
1233
1234
1235
1236
1237
1238
1239
1240
1241
1242
1243
1244
1245
1246
1247
1248
1249
1250
1251
1252
1253
1254
1255
1256
1257
1258
1259
1260
1261
1262
1263
1264
1265
1266
1267
1268
1269
1270
1271
1272
1273
1274
1275
1276
1277
1278
1279
1280
1281
1282
1283
1284
1285
1286
1287
1288
1289
1290
1291
1292
1293
1294
1295
1296
1297
1298
1299
1300
1301
1302
1303
1304
1305
1306
1307
1308
1309
1310
1311
1312
1313
1314
1315
1316
1317
1318
1319
1320
1321
1322
1323
1324
1325
1326
1327
1328
1329
1330
1331
1332
1333
1334
1335
1336
1337
1338
1339
1340
1341
1342
1343
1344
1345
1346
1347
1348
1349
1350
1351
1352
1353
1354
1355
1356
1357
1358
1359
1360
1361
1362
1363
1364
1365
1366
1367
1368
1369
1370
1371
1372
1373
1374
1375
1376
1377
1378
1379
1380
1381
1382
1383
1384
1385
1386
1387
1388
1389
1390
1391
1392
1393
1394
1395
1396
1397
1398
1399
1400
1401
1402
1403
1404
1405
1406
1407
1408
1409
1410
1411
1412
1413
1414
1415
1416
1417
1418
1419
1420
1421
1422
1423
1424
1425
1426
1427
1428
1429
1430
1431
1432
1433
1434
1435
1436
1437
1438
1439
1440
1441
1442
1443
1444
1445
1446
1447
1448
1449
1450
1451
1452
1453
1454
1455
1456
1457
1458
1459
1460
1461
1462
1463
1464
1465
1466
1467
1468
1469
1470
1471
1472
1473
1474
1475
1476
1477
1478
1479
1480
1481
1482
1483
1484
1485
1486
1487
1488
1489
1490
1491
1492
1493
1494
1495
1496
1497
1498
1499
1500
1501
1502
1503
1504
1505
1506
1507
1508
1509
1510
1511
1512
1513
1514
1515
1516
1517
1518
1519
1520
1521
1522
1523
1524
1525
1526
1527
1528
1529
1530
1531
1532
1533
1534
1535
1536
1537
1538
1539
1540
1541
1542
1543
1544
1545
1546
1547
1548
1549
1550
1551
1552
1553
1554
1555
1556
1557
1558
1559
1560
1561
1562
1563
1564
1565
1566
1567
1568
1569
1570
1571
1572
1573
1574
1575
1576
1577
1578
1579
1580
1581
1582
1583
1584
1585
1586
1587
1588
1589
1590
1591
1592
1593
1594
1595
1596
1597
1598
1599
1600
1601
1602
1603
1604
1605
1606
1607
1608
1609
1610
1611
1612
1613
1614
1615
1616
1617
1618
1619
1620
1621
1622
1623
1624
1625
1626
1627
1628
1629
1630
1631
1632
1633
1634
1635
1636
1637
1638
1639
1640
1641
1642
1643
1644
1645
1646
1647
1648
1649
1650
1651
1652
1653
1654
1655
1656
1657
1658
1659
1660
1661
1662
1663
1664
1665
1666
1667
1668
1669
1670
1671
1672
1673
1674
1675
1676
1677
1678
1679
1680
1681
1682
1683
1684
1685
1686
1687
1688
1689
1690
1691
1692
1693
1694
1695
1696
1697
1698
1699
1700
1701
1702
1703
1704
1705
1706
1707
1708
1709
1710
1711
1712
1713
1714
1715
1716
1717
1718
1719
1720
1721
1722
1723
1724
1725
1726
1727
1728
1729
1730
1731
1732
1733
1734
1735
1736
1737
1738
1739
1740
1741
1742
1743
1744
1745
1746
1747
1748
1749
1750
1751
1752
1753
1754
1755
1756
1757
1758
1759
1760
1761
1762
1763
1764
1765
1766
1767
1768
1769
1770
1771
1772
1773
1774
1775
1776
1777
1778
1779
1780
1781
1782
1783
1784
1785
1786
1787
1788
1789
1790
1791
1792
1793
1794
1795
1796
1797
1798
1799
1800
1801
1802
1803
1804
1805
1806
1807
1808
1809
1810
1811
1812
1813
1814
1815
1816
1817
1818
1819
1820
1821
1822
1823
1824
1825
1826
1827
1828
1829
1830
1831
1832
1833
1834
1835
1836
1837
1838
1839
1840
1841
1842
1843
1844
1845
1846
1847
1848
1849
1850
1851
1852
1853
1854
1855
1856
1857
1858
1859
1860
1861
1862
1863
1864
1865
1866
1867
1868
1869
1870
1871
1872
1873
1874
1875
1876
1877
1878
1879
1880
1881
1882
1883
1884
1885
1886
1887
1888
1889
1890
1891
1892
1893
1894
1895
1896
1897
1898
1899
1900
1901
1902
1903
1904
1905
1906
1907
1908
1909
1910
1911
1912
1913
1914
1915
1916
1917
1918
1919
1920
1921
1922
1923
1924
1925
1926
1927
1928
1929
1930
1931
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
2019
2020
2021
2022
2023
2024
2025
2026
2027
2028
2029
2030
2031
2032
2033
2034
2035
2036
2037
2038
2039
2040
2041
2042
2043
2044
2045
2046
2047
2048
2049
2050
2051
2052
2053
2054
2055
2056
2057
2058
2059
2060
2061
2062
2063
2064
2065
2066
2067
2068
2069
2070
2071
2072
2073
2074
2075
2076
2077
2078
2079
2080
2081
2082
2083
2084
2085
2086
2087
2088
2089
2090
2091
2092
2093
2094
2095
2096
2097
2098
2099
2100
2101
2102
2103
2104
2105
2106
2107
2108
/****************************************************************************
**
** Copyright (C) 2009 Nokia Corporation and/or its subsidiary(-ies).
** Contact: Qt Software Information (qt-info@nokia.com)
**
** This file is part of the QtGui module of the Qt Toolkit.
**
** $QT_BEGIN_LICENSE:LGPL$
** No Commercial Usage
** This file contains pre-release code and may not be distributed.
** You may use this file in accordance with the terms and conditions
** contained in the either Technology Preview License Agreement or the
** Beta Release License Agreement.
**
** GNU Lesser General Public License Usage
** Alternatively, this file may be used under the terms of the GNU Lesser
** General Public License version 2.1 as published by the Free Software
** Foundation and appearing in the file LICENSE.LGPL included in the
** packaging of this file.  Please review the following information to
** ensure the GNU Lesser General Public License version 2.1 requirements
** will be met: http://www.gnu.org/licenses/old-licenses/lgpl-2.1.html.
**
** In addition, as a special exception, Nokia gives you certain
** additional rights. These rights are described in the Nokia Qt LGPL
** Exception version 1.0, included in the file LGPL_EXCEPTION.txt in this
** package.
**
** GNU General Public License Usage
** Alternatively, this file may be used under the terms of the GNU
** General Public License version 3.0 as published by the Free Software
** Foundation and appearing in the file LICENSE.GPL included in the
** packaging of this file.  Please review the following information to
** ensure the GNU General Public License version 3.0 requirements will be
** met: http://www.gnu.org/copyleft/gpl.html.
**
** If you are unsure which license is appropriate for your use, please
** contact the sales department at qt-sales@nokia.com.
** $QT_END_LICENSE$
**
****************************************************************************/
#include "qtransform.h"

#include "qdatastream.h"
#include "qdebug.h"
#include "qmatrix.h"
#include "qregion.h"
#include "qpainterpath.h"
#include "qvariant.h"
#include <qmath.h>

QT_BEGIN_NAMESPACE

#define Q_NEAR_CLIP 0.000001


#define MAP(x, y, nx, ny) \
    do { \
        qreal FX_ = x; \
        qreal FY_ = y; \
        switch(t) {   \
        case TxNone:  \
            nx = FX_;   \
            ny = FY_;   \
            break;    \
        case TxTranslate:    \
            nx = FX_ + affine._dx;                \
            ny = FY_ + affine._dy;                \
            break;                              \
        case TxScale:                           \
            nx = affine._m11 * FX_ + affine._dx;  \
            ny = affine._m22 * FY_ + affine._dy;  \
            break;                              \
        case TxRotate:                          \
        case TxShear:                           \
        case TxProject:                                      \
            nx = affine._m11 * FX_ + affine._m21 * FY_ + affine._dx;        \
            ny = affine._m12 * FX_ + affine._m22 * FY_ + affine._dy;        \
            if (t == TxProject) {                                       \
                qreal w = 1./(m_13 * FX_ + m_23 * FY_ + m_33);              \
                nx *= w;                                                \
                ny *= w;                                                \
            }                                                           \
        }                                                               \
    } while (0)

/*!
    \class QTransform
    \brief The QTransform class specifies 2D transformations of a coordinate system.
    \since 4.3
    \ingroup multimedia

    A transformation specifies how to translate, scale, shear, rotate
    or project the coordinate system, and is typically used when
    rendering graphics.

    QTransform differs from QMatrix in that it is a true 3x3 matrix,
    allowing perspective transformations. QTransform's toAffine()
    method allows casting QTransform to QMatrix. If a perspective
    transformation has been specified on the matrix, then the
    conversion to an affine QMatrix will cause loss of data.

    QTransform is the recommended transformation class in Qt.

    A QTransform object can be built using the setMatrix(), scale(),
    rotate(), translate() and shear() functions.  Alternatively, it
    can be built by applying \l {QTransform#Basic Matrix
    Operations}{basic matrix operations}. The matrix can also be
    defined when constructed, and it can be reset to the identity
    matrix (the default) using the reset() function.

    The QTransform class supports mapping of graphic primitives: A given
    point, line, polygon, region, or painter path can be mapped to the
    coordinate system defined by \e this matrix using the map()
    function. In case of a rectangle, its coordinates can be
    transformed using the mapRect() function. A rectangle can also be
    transformed into a \e polygon (mapped to the coordinate system
    defined by \e this matrix), using the mapToPolygon() function.

    QTransform provides the isIdentity() function which returns true if
    the matrix is the identity matrix, and the isInvertible() function
    which returns true if the matrix is non-singular (i.e. AB = BA =
    I). The inverted() function returns an inverted copy of \e this
    matrix if it is invertible (otherwise it returns the identity
    matrix). In addition, QTransform provides the det() function
    returning the matrix's determinant.

    Finally, the QTransform class supports matrix multiplication, and
    objects of the class can be streamed as well as compared.

    \tableofcontents

    \section1 Rendering Graphics

    When rendering graphics, the matrix defines the transformations
    but the actual transformation is performed by the drawing routines
    in QPainter.

    By default, QPainter operates on the associated device's own
    coordinate system.  The standard coordinate system of a
    QPaintDevice has its origin located at the top-left position. The
    \e x values increase to the right; \e y values increase
    downward. For a complete description, see the \l {The Coordinate
    System}{coordinate system} documentation.

    QPainter has functions to translate, scale, shear and rotate the
    coordinate system without using a QTransform. For example:

    \table 100%
    \row
    \o \inlineimage qtransform-simpletransformation.png
    \o
    \snippet doc/src/snippets/transform/main.cpp 0
    \endtable

    Although these functions are very convenient, it can be more
    efficient to build a QTransform and call QPainter::setTransform() if you
    want to perform more than a single transform operation. For
    example:

    \table 100%
    \row
    \o \inlineimage qtransform-combinedtransformation.png
    \o
    \snippet doc/src/snippets/transform/main.cpp 1
    \endtable

    \section1 Basic Matrix Operations

    \image qtransform-representation.png

    A QTransform object contains a 3 x 3 matrix.  The \c m31 (\c dx) and
    \c m32 (\c dy) elements specify horizontal and vertical translation.
    The \c m11 and \c m22 elements specify horizontal and vertical scaling.
    The \c m21 and \c m12 elements specify horizontal and vertical \e shearing.
    And finally, the \c m13 and \c m23 elements specify horizontal and vertical
    projection, with \c m33 as an additional projection factor.

    QTransform transforms a point in the plane to another point using the
    following formulas:

    \snippet doc/src/snippets/code/src_gui_painting_qtransform.cpp 0

    The point \e (x, y) is the original point, and \e (x', y') is the
    transformed point. \e (x', y') can be transformed back to \e (x,
    y) by performing the same operation on the inverted() matrix.

    The various matrix elements can be set when constructing the
    matrix, or by using the setMatrix() function later on. They can also
    be manipulated using the translate(), rotate(), scale() and
    shear() convenience functions, The currently set values can be
    retrieved using the m11(), m12(), m13(), m21(), m22(), m23(),
    m31(), m32(), m33(), dx() and dy() functions.

    Translation is the simplest transformation. Setting \c dx and \c
    dy will move the coordinate system \c dx units along the X axis
    and \c dy units along the Y axis.  Scaling can be done by setting
    \c m11 and \c m22. For example, setting \c m11 to 2 and \c m22 to
    1.5 will double the height and increase the width by 50%.  The
    identity matrix has \c m11, \c m22, and \c m33 set to 1 (all others are set
    to 0) mapping a point to itself. Shearing is controlled by \c m12
    and \c m21. Setting these elements to values different from zero
    will twist the coordinate system. Rotation is achieved by
    carefully setting both the shearing factors and the scaling
    factors. Perspective transformation is achieved by carefully setting
    both the projection factors and the scaling factors.

    Here's the combined transformations example using basic matrix
    operations:

    \table 100%
    \row
    \o \inlineimage qtransform-combinedtransformation2.png
    \o
    \snippet doc/src/snippets/transform/main.cpp 2
    \endtable

    \sa QPainter, {The Coordinate System}, {demos/affine}{Affine
    Transformations Demo}, {Transformations Example}
*/

/*!
    \enum QTransform::TransformationType

    \value TxNone
    \value TxTranslate
    \value TxScale
    \value TxRotate
    \value TxShear
    \value TxProject
*/

/*!
    Constructs an identity matrix.

    All elements are set to zero except \c m11 and \c m22 (specifying
    the scale) and \c m13 which are set to 1.

    \sa reset()
*/
QTransform::QTransform()
    : m_13(0), m_23(0), m_33(1)
    , m_type(TxNone)
    , m_dirty(TxNone)
{

}

/*!
    \fn QTransform::QTransform(qreal m11, qreal m12, qreal m13, qreal m21, qreal m22, qreal m23, qreal m31, qreal m32, qreal m33)

    Constructs a matrix with the elements, \a m11, \a m12, \a m13,
    \a m21, \a m22, \a m23, \a m31, \a m32, \a m33.

    \sa setMatrix()
*/
QTransform::QTransform(qreal h11, qreal h12, qreal h13,
                       qreal h21, qreal h22, qreal h23,
                       qreal h31, qreal h32, qreal h33)
    : affine(h11, h12, h21, h22, h31, h32),
      m_13(h13), m_23(h23), m_33(h33)
    , m_type(TxNone)
    , m_dirty(TxProject)
{

}

/*!
    \fn QTransform::QTransform(qreal m11, qreal m12, qreal m21, qreal m22, qreal dx, qreal dy)

    Constructs a matrix with the elements, \a m11, \a m12, \a m21, \a m22, \a dx and \a dy.

    \sa setMatrix()
*/
QTransform::QTransform(qreal h11, qreal h12, qreal h21,
                       qreal h22, qreal dx, qreal dy)
    : affine(h11, h12, h21, h22, dx, dy),
      m_13(0), m_23(0), m_33(1)
    , m_type(TxNone)
    , m_dirty(TxShear)
{

}

/*!
    \fn QTransform::QTransform(const QMatrix &matrix)

    Constructs a matrix that is a copy of the given \a matrix.
    Note that the \c m13, \c m23, and \c m33 elements are set to 0, 0,
    and 1 respectively.
 */
QTransform::QTransform(const QMatrix &mtx)
    : affine(mtx),
      m_13(0), m_23(0), m_33(1)
    , m_type(TxNone)
    , m_dirty(TxShear)
{

}

/*!
    Returns the adjoint of this matrix.
*/
QTransform QTransform::adjoint() const
{
    qreal h11, h12, h13,
        h21, h22, h23,
        h31, h32, h33;
    h11 = affine._m22*m_33 - m_23*affine._dy;
    h21 = m_23*affine._dx - affine._m21*m_33;
    h31 = affine._m21*affine._dy - affine._m22*affine._dx;
    h12 = m_13*affine._dy - affine._m12*m_33;
    h22 = affine._m11*m_33 - m_13*affine._dx;
    h32 = affine._m12*affine._dx - affine._m11*affine._dy;
    h13 = affine._m12*m_23 - m_13*affine._m22;
    h23 = m_13*affine._m21 - affine._m11*m_23;
    h33 = affine._m11*affine._m22 - affine._m12*affine._m21;

    return QTransform(h11, h12, h13,
                      h21, h22, h23,
                      h31, h32, h33);
}

/*!
    Returns the transpose of this matrix.
*/
QTransform QTransform::transposed() const
{
    QTransform t(affine._m11, affine._m21, affine._dx,
                 affine._m12, affine._m22, affine._dy,
                 m_13, m_23, m_33);
    t.m_type = m_type;
    t.m_dirty = m_dirty;
    return t;
}

/*!
    Returns an inverted copy of this matrix.

    If the matrix is singular (not invertible), the returned matrix is
    the identity matrix. If \a invertible is valid (i.e. not 0), its
    value is set to true if the matrix is invertible, otherwise it is
    set to false.

    \sa isInvertible()
*/
QTransform QTransform::inverted(bool *invertible) const
{
    QTransform invert;
    bool inv = true;
    qreal det;

    switch(type()) {
    case TxNone:
        break;
    case TxTranslate:
        invert.affine._dx = -affine._dx;
        invert.affine._dy = -affine._dy;
        break;
    case TxScale:
        inv = !qFuzzyCompare(affine._m11 + 1, 1);
        inv &= !qFuzzyCompare(affine._m22 + 1, 1);
        if (inv) {
            invert.affine._m11 = 1 / affine._m11;
            invert.affine._m22 = 1 / affine._m22;
            invert.affine._dx = -affine._dx * invert.affine._m11;
            invert.affine._dy = -affine._dy * invert.affine._m22;
        }
        break;
    case TxRotate:
    case TxShear:
        invert.affine = affine.inverted(&inv);
        break;
    default:
        // general case
        det = determinant();
        inv = !qFuzzyCompare(det + 1, 1);
        if (inv)
            invert = adjoint() / det;
        break;
    }

    if (invertible)
        *invertible = inv;

    if (inv) {
        // inverting doesn't change the type
        invert.m_type = m_type;
        invert.m_dirty = m_dirty;
    }

    return invert;
}

/*!
    Moves the coordinate system \a dx along the x axis and \a dy along
    the y axis, and returns a reference to the matrix.

    \sa setMatrix()
*/
QTransform & QTransform::translate(qreal dx, qreal dy)
{
    if (dx == 0 && dy == 0)
        return *this;

    switch(type()) {
    case TxNone:
        affine._dx = dx;
        affine._dy = dy;
        break;
    case TxTranslate:
        affine._dx += dx;
        affine._dy += dy;
        break;
    case TxScale:
        affine._dx += dx*affine._m11;
        affine._dy += dy*affine._m22;
        break;
    case TxProject:
        m_33 += dx*m_13 + dy*m_23;
        // Fall through
    case TxShear:
    case TxRotate:
        affine._dx += dx*affine._m11 + dy*affine._m21;
        affine._dy += dy*affine._m22 + dx*affine._m12;
        break;
    }
    m_dirty |= TxTranslate;
    return *this;
}

/*!
    Creates a matrix which corresponds to a translation of \a dx along
    the x axis and \a dy along the y axis. This is the same as
    QTransform().translate(dx, dy) but slightly faster.

    \since 4.5
*/
QTransform QTransform::fromTranslate(qreal dx, qreal dy)
{
    QTransform transform(1, 0, 0, 1, dx, dy);
    if (dx == 0 && dy == 0)
        transform.m_dirty = TxNone;
    else
        transform.m_dirty = TxTranslate;
    return transform;
}

/*!
    Scales the coordinate system by \a sx horizontally and \a sy
    vertically, and returns a reference to the matrix.

    \sa setMatrix()
*/
QTransform & QTransform::scale(qreal sx, qreal sy)
{
    if (sx == 1 && sy == 1)
        return *this;

    switch(type()) {
    case TxNone:
    case TxTranslate:
        affine._m11 = sx;
        affine._m22 = sy;
        break;
    case TxProject:
        m_13 *= sx;
        m_23 *= sy;
        // fall through
    case TxRotate:
    case TxShear:
        affine._m12 *= sx;
        affine._m21 *= sy;
        // fall through
    case TxScale:
        affine._m11 *= sx;
        affine._m22 *= sy;
        break;
    }
    m_dirty |= TxScale;
    return *this;
}

/*!
    Creates a matrix which corresponds to a scaling of
    \a sx horizontally and \a sy vertically.
    This is the same as QTransform().scale(sx, sy) but slightly faster.

    \since 4.5
*/
QTransform QTransform::fromScale(qreal sx, qreal sy)
{
    QTransform transform(sx, 0, 0, sy, 0, 0);
    if (sx == 1 && sy == 1)
        transform.m_dirty = TxNone;
    else
        transform.m_dirty = TxScale;
    return transform;
}

/*!
    Shears the coordinate system by \a sh horizontally and \a sv
    vertically, and returns a reference to the matrix.

    \sa setMatrix()
*/
QTransform & QTransform::shear(qreal sh, qreal sv)
{
    switch(type()) {
    case TxNone:
    case TxTranslate:
        affine._m12 = sv;
        affine._m21 = sh;
        break;
    case TxScale:
        affine._m12 = sv*affine._m22;
        affine._m21 = sh*affine._m11;
        break;
    case TxProject: {
        qreal tm13 = sv*m_23;
        qreal tm23 = sh*m_13;
        m_13 += tm13;
        m_23 += tm23;
    }
        // fall through
    case TxRotate:
    case TxShear: {
        qreal tm11 = sv*affine._m21;
        qreal tm22 = sh*affine._m12;
        qreal tm12 = sv*affine._m22;
        qreal tm21 = sh*affine._m11;
        affine._m11 += tm11; affine._m12 += tm12;
        affine._m21 += tm21; affine._m22 += tm22;
        break;
    }
    }
    m_dirty |= TxShear;
    return *this;
}

const qreal deg2rad = qreal(0.017453292519943295769);        // pi/180
const qreal inv_dist_to_plane = 1. / 1024.;

/*!
    \fn QTransform &QTransform::rotate(qreal angle, Qt::Axis axis)

    Rotates the coordinate system counterclockwise by the given \a angle
    about the specified \a axis and returns a reference to the matrix.

    Note that if you apply a QTransform to a point defined in widget
    coordinates, the direction of the rotation will be clockwise
    because the y-axis points downwards.

    The angle is specified in degrees.

    \sa setMatrix()
*/
QTransform & QTransform::rotate(qreal a, Qt::Axis axis)
{
    if (a == 0)
        return *this;

    qreal sina = 0;
    qreal cosa = 0;
    if (a == 90. || a == -270.)
        sina = 1.;
    else if (a == 270. || a == -90.)
        sina = -1.;
    else if (a == 180.)
        cosa = -1.;
    else{
        qreal b = deg2rad*a;          // convert to radians
        sina = qSin(b);               // fast and convenient
        cosa = qCos(b);
    }

    if (axis == Qt::ZAxis) {
        switch(type()) {
        case TxNone:
        case TxTranslate:
            affine._m11 = cosa;
            affine._m12 = sina;
            affine._m21 = -sina;
            affine._m22 = cosa;
            break;
        case TxScale: {
            qreal tm11 = cosa*affine._m11;
            qreal tm12 = sina*affine._m22;
            qreal tm21 = -sina*affine._m11;
            qreal tm22 = cosa*affine._m22;
            affine._m11 = tm11; affine._m12 = tm12;
            affine._m21 = tm21; affine._m22 = tm22;
            break;
        }
        case TxProject: {
            qreal tm13 = cosa*m_13 + sina*m_23;
            qreal tm23 = -sina*m_13 + cosa*m_23;
            m_13 = tm13;
            m_23 = tm23;
            // fall through
        }
        case TxRotate:
        case TxShear: {
            qreal tm11 = cosa*affine._m11 + sina*affine._m21;
            qreal tm12 = cosa*affine._m12 + sina*affine._m22;
            qreal tm21 = -sina*affine._m11 + cosa*affine._m21;
            qreal tm22 = -sina*affine._m12 + cosa*affine._m22;
            affine._m11 = tm11; affine._m12 = tm12;
            affine._m21 = tm21; affine._m22 = tm22;
            break;
        }
        }
        m_dirty |= TxRotate;
    } else {
        QTransform result;
        if (axis == Qt::YAxis) {
            result.affine._m11 = cosa;
            result.m_13 = -sina * inv_dist_to_plane;
        } else {
            result.affine._m22 = cosa;
            result.m_23 = -sina * inv_dist_to_plane;
        }
        result.m_type = TxProject;
        *this = result * *this;
    }

    return *this;
}

/*!
    \fn QTransform & QTransform::rotateRadians(qreal angle, Qt::Axis axis)

    Rotates the coordinate system counterclockwise by the given \a angle
    about the specified \a axis and returns a reference to the matrix.

    Note that if you apply a QTransform to a point defined in widget
    coordinates, the direction of the rotation will be clockwise
    because the y-axis points downwards.

    The angle is specified in radians.

    \sa setMatrix()
*/
QTransform & QTransform::rotateRadians(qreal a, Qt::Axis axis)
{
    qreal sina = qSin(a);
    qreal cosa = qCos(a);

    if (axis == Qt::ZAxis) {
        switch(type()) {
        case TxNone:
        case TxTranslate:
            affine._m11 = cosa;
            affine._m12 = sina;
            affine._m21 = -sina;
            affine._m22 = cosa;
            break;
        case TxScale: {
            qreal tm11 = cosa*affine._m11;
            qreal tm12 = sina*affine._m22;
            qreal tm21 = -sina*affine._m11;
            qreal tm22 = cosa*affine._m22;
            affine._m11 = tm11; affine._m12 = tm12;
            affine._m21 = tm21; affine._m22 = tm22;
            break;
        }
        case TxProject: {
            qreal tm13 = cosa*m_13 + sina*m_23;
            qreal tm23 = -sina*m_13 + cosa*m_23;
            m_13 = tm13;
            m_23 = tm23;
            // fall through
        }
        case TxRotate:
        case TxShear: {
            qreal tm11 = cosa*affine._m11 + sina*affine._m21;
            qreal tm12 = cosa*affine._m12 + sina*affine._m22;
            qreal tm21 = -sina*affine._m11 + cosa*affine._m21;
            qreal tm22 = -sina*affine._m12 + cosa*affine._m22;
            affine._m11 = tm11; affine._m12 = tm12;
            affine._m21 = tm21; affine._m22 = tm22;
            break;
        }
        }
        m_dirty |= TxRotate;
    } else {
        QTransform result;
        if (axis == Qt::YAxis) {
            result.affine._m11 = cosa;
            result.m_13 = -sina * inv_dist_to_plane;
        } else {
            result.affine._m22 = cosa;
            result.m_23 = -sina * inv_dist_to_plane;
        }
        result.m_type = TxProject;
        *this = result * *this;
    }
    return *this;
}

/*!
    \fn bool QTransform::operator==(const QTransform &matrix) const
    Returns true if this matrix is equal to the given \a matrix,
    otherwise returns false.
*/
bool QTransform::operator==(const QTransform &o) const
{
#define qFZ qFuzzyCompare
    return qFZ(affine._m11, o.affine._m11) &&  qFZ(affine._m12, o.affine._m12) &&  qFZ(m_13, o.m_13)
        && qFZ(affine._m21, o.affine._m21) &&  qFZ(affine._m22, o.affine._m22) &&  qFZ(m_23, o.m_23)
        && qFZ(affine._dx, o.affine._dx) &&  qFZ(affine._dy, o.affine._dy) &&  qFZ(m_33, o.m_33);
#undef qFZ
}

/*!
    \fn bool QTransform::operator!=(const QTransform &matrix) const
    Returns true if this matrix is not equal to the given \a matrix,
    otherwise returns false.
*/
bool QTransform::operator!=(const QTransform &o) const
{
    return !operator==(o);
}

/*!
    \fn QTransform & QTransform::operator*=(const QTransform &matrix)
    \overload

    Returns the result of multiplying this matrix by the given \a
    matrix.
*/
QTransform & QTransform::operator*=(const QTransform &o)
{
    const TransformationType otherType = o.type();
    if (otherType == TxNone)
        return *this;

    const TransformationType thisType = type();
    if (thisType == TxNone)
        return operator=(o);

    TransformationType t = qMax(thisType, otherType);
    switch(t) {
    case TxNone:
        break;
    case TxTranslate:
        affine._dx += o.affine._dx;
        affine._dy += o.affine._dy;
        break;
    case TxScale:
    {
        qreal m11 = affine._m11*o.affine._m11;
        qreal m22 = affine._m22*o.affine._m22;

        qreal m31 = affine._dx*o.affine._m11 + o.affine._dx;
        qreal m32 = affine._dy*o.affine._m22 + o.affine._dy;

        affine._m11 = m11;
        affine._m22 = m22;
        affine._dx = m31; affine._dy = m32;
        break;
    }
    case TxRotate:
    case TxShear:
    {
        qreal m11 = affine._m11*o.affine._m11 + affine._m12*o.affine._m21;
        qreal m12 = affine._m11*o.affine._m12 + affine._m12*o.affine._m22;

        qreal m21 = affine._m21*o.affine._m11 + affine._m22*o.affine._m21;
        qreal m22 = affine._m21*o.affine._m12 + affine._m22*o.affine._m22;

        qreal m31 = affine._dx*o.affine._m11 + affine._dy*o.affine._m21 + o.affine._dx;
        qreal m32 = affine._dx*o.affine._m12 + affine._dy*o.affine._m22 + o.affine._dy;

        affine._m11 = m11; affine._m12 = m12;
        affine._m21 = m21; affine._m22 = m22;
        affine._dx = m31; affine._dy = m32;
        break;
    }
    case TxProject:
    {
        qreal m11 = affine._m11*o.affine._m11 + affine._m12*o.affine._m21 + m_13*o.affine._dx;
        qreal m12 = affine._m11*o.affine._m12 + affine._m12*o.affine._m22 + m_13*o.affine._dy;
        qreal m13 = affine._m11*o.m_13 + affine._m12*o.m_23 + m_13*o.m_33;

        qreal m21 = affine._m21*o.affine._m11 + affine._m22*o.affine._m21 + m_23*o.affine._dx;
        qreal m22 = affine._m21*o.affine._m12 + affine._m22*o.affine._m22 + m_23*o.affine._dy;
        qreal m23 = affine._m21*o.m_13 + affine._m22*o.m_23 + m_23*o.m_33;

        qreal m31 = affine._dx*o.affine._m11 + affine._dy*o.affine._m21 + m_33*o.affine._dx;
        qreal m32 = affine._dx*o.affine._m12 + affine._dy*o.affine._m22 + m_33*o.affine._dy;
        qreal m33 = affine._dx*o.m_13 + affine._dy*o.m_23 + m_33*o.m_33;

        affine._m11 = m11; affine._m12 = m12; m_13 = m13;
        affine._m21 = m21; affine._m22 = m22; m_23 = m23;
        affine._dx = m31; affine._dy = m32; m_33 = m33;
    }
    }

    m_dirty = t;
    m_type = t;

    return *this;
}

/*!
    \fn QTransform QTransform::operator*(const QTransform &matrix) const
    Returns the result of multiplying this matrix by the given \a
    matrix.

    Note that matrix multiplication is not commutative, i.e. a*b !=
    b*a.
*/
QTransform QTransform::operator*(const QTransform &m) const
{
    QTransform result = *this;
    result *= m;
    return result;
}

/*!
    \fn QTransform & QTransform::operator*=(qreal scalar)
    \overload

    Returns the result of performing an element-wise multiplication of this
    matrix with the given \a scalar.
*/

/*!
    \fn QTransform & QTransform::operator/=(qreal scalar)
    \overload

    Returns the result of performing an element-wise division of this
    matrix by the given \a scalar.
*/

/*!
    \fn QTransform & QTransform::operator+=(qreal scalar)
    \overload

    Returns the matrix obtained by adding the given \a scalar to each
    element of this matrix.
*/

/*!
    \fn QTransform & QTransform::operator-=(qreal scalar)
    \overload

    Returns the matrix obtained by subtracting the given \a scalar from each
    element of this matrix.
*/

/*!
    Assigns the given \a matrix's values to this matrix.
*/
QTransform & QTransform::operator=(const QTransform &matrix)
{
    affine._m11 = matrix.affine._m11;
    affine._m12 = matrix.affine._m12;
    affine._m21 = matrix.affine._m21;
    affine._m22 = matrix.affine._m22;
    affine._dx = matrix.affine._dx;
    affine._dy = matrix.affine._dy;
    m_13 = matrix.m_13;
    m_23 = matrix.m_23;
    m_33 = matrix.m_33;
    m_type = matrix.m_type;
    m_dirty = matrix.m_dirty;

    return *this;
}

/*!
    Resets the matrix to an identity matrix, i.e. all elements are set
    to zero, except \c m11 and \c m22 (specifying the scale) which are
    set to 1.

    \sa QTransform(), isIdentity(), {QTransform#Basic Matrix
    Operations}{Basic Matrix Operations}
*/
void QTransform::reset()
{
    affine._m11 = affine._m22 = m_33 = 1.0;
    affine._m12 = m_13 = affine._m21 = m_23 = affine._dx = affine._dy = 0;
    m_type = TxNone;
    m_dirty = TxNone;
}

#ifndef QT_NO_DATASTREAM
/*!
    \fn QDataStream &operator<<(QDataStream &stream, const QTransform &matrix)
    \since 4.3
    \relates QTransform

    Writes the given \a matrix to the given \a stream and returns a
    reference to the stream.

    \sa {Format of the QDataStream Operators}
*/
QDataStream & operator<<(QDataStream &s, const QTransform &m)
{
    s << double(m.m11())
      << double(m.m12())
      << double(m.m13())
      << double(m.m21())
      << double(m.m22())
      << double(m.m23())
      << double(m.m31())
      << double(m.m32())
      << double(m.m33());
    return s;
}

/*!
    \fn QDataStream &operator>>(QDataStream &stream, QTransform &matrix)
    \since 4.3
    \relates QTransform

    Reads the given \a matrix from the given \a stream and returns a
    reference to the stream.

    \sa {Format of the QDataStream Operators}
*/
QDataStream & operator>>(QDataStream &s, QTransform &t)
{
     double m11, m12, m13,
         m21, m22, m23,
         m31, m32, m33;

     s >> m11;
     s >> m12;
     s >> m13;
     s >> m21;
     s >> m22;
     s >> m23;
     s >> m31;
     s >> m32;
     s >> m33;
     t.setMatrix(m11, m12, m13,
                 m21, m22, m23,
                 m31, m32, m33);
     return s;
}

#endif // QT_NO_DATASTREAM

#ifndef QT_NO_DEBUG_STREAM
QDebug operator<<(QDebug dbg, const QTransform &m)
{
    dbg.nospace() << "QTransform("
                  << "11="  << m.m11()
                  << " 12=" << m.m12()
                  << " 13=" << m.m13()
                  << " 21=" << m.m21()
                  << " 22=" << m.m22()
                  << " 23=" << m.m23()
                  << " 31=" << m.m31()
                  << " 32=" << m.m32()
                  << " 33=" << m.m33()
                  << ")";
    return dbg.space();
}
#endif

/*!
    \fn QPoint operator*(const QPoint &point, const QTransform &matrix)
    \relates QTransform

    This is the same as \a{matrix}.map(\a{point}).

    \sa QTransform::map()
*/
QPoint QTransform::map(const QPoint &p) const
{
    qreal fx = p.x();
    qreal fy = p.y();

    qreal x = 0, y = 0;

    TransformationType t = type();
    switch(t) {
    case TxNone:
        x = fx;
        y = fy;
        break;
    case TxTranslate:
        x = fx + affine._dx;
        y = fy + affine._dy;
        break;
    case TxScale:
        x = affine._m11 * fx + affine._dx;
        y = affine._m22 * fy + affine._dy;
        break;
    case TxRotate:
    case TxShear:
    case TxProject:
        x = affine._m11 * fx + affine._m21 * fy + affine._dx;
        y = affine._m12 * fx + affine._m22 * fy + affine._dy;
        if (t == TxProject) {
            qreal w = 1./(m_13 * fx + m_23 * fy + m_33);
            x *= w;
            y *= w;
        }
    }
    return QPoint(qRound(x), qRound(y));
}


/*!
    \fn QPointF operator*(const QPointF &point, const QTransform &matrix)
    \relates QTransform

    Same as \a{matrix}.map(\a{point}).

    \sa QTransform::map()
*/

/*!
    \overload

    Creates and returns a QPointF object that is a copy of the given point,
    \a p, mapped into the coordinate system defined by this matrix.
*/
QPointF QTransform::map(const QPointF &p) const
{
    qreal fx = p.x();
    qreal fy = p.y();

    qreal x = 0, y = 0;

    TransformationType t = type();
    switch(t) {
    case TxNone:
        x = fx;
        y = fy;
        break;
    case TxTranslate:
        x = fx + affine._dx;
        y = fy + affine._dy;
        break;
    case TxScale:
        x = affine._m11 * fx + affine._dx;
        y = affine._m22 * fy + affine._dy;
        break;
    case TxRotate:
    case TxShear:
    case TxProject:
        x = affine._m11 * fx + affine._m21 * fy + affine._dx;
        y = affine._m12 * fx + affine._m22 * fy + affine._dy;
        if (t == TxProject) {
            qreal w = 1./(m_13 * fx + m_23 * fy + m_33);
            x *= w;
            y *= w;
        }
    }
    return QPointF(x, y);
}

/*!
    \fn QPoint QTransform::map(const QPoint &point) const
    \overload

    Creates and returns a QPoint object that is a copy of the given \a
    point, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/

/*!
    \fn QLineF operator*(const QLineF &line, const QTransform &matrix)
    \relates QTransform

    This is the same as \a{matrix}.map(\a{line}).

    \sa QTransform::map()
*/

/*!
    \fn QLine operator*(const QLine &line, const QTransform &matrix)
    \relates QTransform

    This is the same as \a{matrix}.map(\a{line}).

    \sa QTransform::map()
*/

/*!
    \overload

    Creates and returns a QLineF object that is a copy of the given line,
    \a l, mapped into the coordinate system defined by this matrix.
*/
QLine QTransform::map(const QLine &l) const
{
    qreal fx1 = l.x1();
    qreal fy1 = l.y1();
    qreal fx2 = l.x2();
    qreal fy2 = l.y2();

    qreal x1 = 0, y1 = 0, x2 = 0, y2 = 0;

    TransformationType t = type();
    switch(t) {
    case TxNone:
        x1 = fx1;
        y1 = fy1;
        x2 = fx2;
        y2 = fy2;
        break;
    case TxTranslate:
        x1 = fx1 + affine._dx;
        y1 = fy1 + affine._dy;
        x2 = fx2 + affine._dx;
        y2 = fy2 + affine._dy;
        break;
    case TxScale:
        x1 = affine._m11 * fx1 + affine._dx;
        y1 = affine._m22 * fy1 + affine._dy;
        x2 = affine._m11 * fx2 + affine._dx;
        y2 = affine._m22 * fy2 + affine._dy;
        break;
    case TxRotate:
    case TxShear:
    case TxProject:
        x1 = affine._m11 * fx1 + affine._m21 * fy1 + affine._dx;
        y1 = affine._m12 * fx1 + affine._m22 * fy1 + affine._dy;
        x2 = affine._m11 * fx2 + affine._m21 * fy2 + affine._dx;
        y2 = affine._m12 * fx2 + affine._m22 * fy2 + affine._dy;
        if (t == TxProject) {
            qreal w = 1./(m_13 * fx1 + m_23 * fy1 + m_33);
            x1 *= w;
            y1 *= w;
            w = 1./(m_13 * fx2 + m_23 * fy2 + m_33);
            x2 *= w;
            y2 *= w;
        }
    }
    return QLine(qRound(x1), qRound(y1), qRound(x2), qRound(y2));
}

/*!
    \overload

    \fn QLineF QTransform::map(const QLineF &line) const

    Creates and returns a QLine object that is a copy of the given \a
    line, mapped into the coordinate system defined by this matrix.
    Note that the transformed coordinates are rounded to the nearest
    integer.
*/

QLineF QTransform::map(const QLineF &l) const
{
    qreal fx1 = l.x1();
    qreal fy1 = l.y1();
    qreal fx2 = l.x2();
    qreal fy2 = l.y2();

    qreal x1 = 0, y1 = 0, x2 = 0, y2 = 0;

    TransformationType t = type();
    switch(t) {
    case TxNone:
        x1 = fx1;
        y1 = fy1;
        x2 = fx2;
        y2 = fy2;
        break;
    case TxTranslate:
        x1 = fx1 + affine._dx;
        y1 = fy1 + affine._dy;
        x2 = fx2 + affine._dx;
        y2 = fy2 + affine._dy;
        break;
    case TxScale:
        x1 = affine._m11 * fx1 + affine._dx;
        y1 = affine._m22 * fy1 + affine._dy;
        x2 = affine._m11 * fx2 + affine._dx;
        y2 = affine._m22 * fy2 + affine._dy;
        break;
    case TxRotate:
    case TxShear:
    case TxProject:
        x1 = affine._m11 * fx1 + affine._m21 * fy1 + affine._dx;
        y1 = affine._m12 * fx1 + affine._m22 * fy1 + affine._dy;
        x2 = affine._m11 * fx2 + affine._m21 * fy2 + affine._dx;
        y2 = affine._m12 * fx2 + affine._m22 * fy2 + affine._dy;
        if (t == TxProject) {
            qreal w = 1./(m_13 * fx1 + m_23 * fy1 + m_33);
            x1 *= w;
            y1 *= w;
            w = 1./(m_13 * fx2 + m_23 * fy2 + m_33);
            x2 *= w;
            y2 *= w;
        }
    }
    return QLineF(x1, y1, x2, y2);
}

static QPolygonF mapProjective(const QTransform &transform, const QPolygonF &poly)
{
    if (poly.size() == 0)
        return poly;

    if (poly.size() == 1)
        return QPolygonF() << transform.map(poly.at(0));

    QPainterPath path;
    path.addPolygon(poly);

    path = transform.map(path);

    QPolygonF result;
    for (int i = 0; i < path.elementCount(); ++i)
        result << path.elementAt(i);
    return result;
}


/*!
    \fn QPolygonF operator *(const QPolygonF &polygon, const QTransform &matrix)
    \since 4.3
    \relates QTransform

    This is the same as \a{matrix}.map(\a{polygon}).

    \sa QTransform::map()
*/

/*!
    \fn QPolygon operator*(const QPolygon &polygon, const QTransform &matrix)
    \relates QTransform

    This is the same as \a{matrix}.map(\a{polygon}).

    \sa QTransform::map()
*/

/*!
    \fn QPolygonF QTransform::map(const QPolygonF &polygon) const
    \overload

    Creates and returns a QPolygonF object that is a copy of the given
    \a polygon, mapped into the coordinate system defined by this
    matrix.
*/
QPolygonF QTransform::map(const QPolygonF &a) const
{
    TransformationType t = type();
    if (t >= QTransform::TxProject)
        return mapProjective(*this, a);

    int size = a.size();
    int i;
    QPolygonF p(size);
    const QPointF *da = a.constData();
    QPointF *dp = p.data();

    for(i = 0; i < size; ++i) {
        MAP(da[i].xp, da[i].yp, dp[i].xp, dp[i].yp);
    }
    return p;
}

/*!
    \fn QPolygon QTransform::map(const QPolygon &polygon) const
    \overload

    Creates and returns a QPolygon object that is a copy of the given
    \a polygon, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/
QPolygon QTransform::map(const QPolygon &a) const
{
    TransformationType t = type();
    if (t >= QTransform::TxProject)
        return mapProjective(*this, QPolygonF(a)).toPolygon();

    int size = a.size();
    int i;
    QPolygon p(size);
    const QPoint *da = a.constData();
    QPoint *dp = p.data();

    for(i = 0; i < size; ++i) {
        qreal nx = 0, ny = 0;
        MAP(da[i].xp, da[i].yp, nx, ny);
        dp[i].xp = qRound(nx);
        dp[i].yp = qRound(ny);
    }
    return p;
}

/*!
    \fn QRegion operator*(const QRegion &region, const QTransform &matrix)
    \relates QTransform

    This is the same as \a{matrix}.map(\a{region}).

    \sa QTransform::map()
*/

extern QPainterPath qt_regionToPath(const QRegion &region);

/*!
    \fn QRegion QTransform::map(const QRegion &region) const
    \overload

    Creates and returns a QRegion object that is a copy of the given
    \a region, mapped into the coordinate system defined by this matrix.

    Calling this method can be rather expensive if rotations or
    shearing are used.
*/
QRegion QTransform::map(const QRegion &r) const
{
    TransformationType t = type();
    if (t == TxNone)
        return r;

    if (t == TxTranslate) {
        QRegion copy(r);
        copy.translate(qRound(affine._dx), qRound(affine._dy));
        return copy;
    }

    if (t == TxScale && r.numRects() == 1)
        return QRegion(mapRect(r.boundingRect()));

    QPainterPath p = map(qt_regionToPath(r));
    return p.toFillPolygon(QTransform()).toPolygon();
}

struct QHomogeneousCoordinate
{
    qreal x;
    qreal y;
    qreal w;

    QHomogeneousCoordinate() {}
    QHomogeneousCoordinate(qreal x_, qreal y_, qreal w_) : x(x_), y(y_), w(w_) {}

    const QPointF toPoint() const {
        qreal iw = 1 / w;
        return QPointF(x * iw, y * iw);
    }
};

static inline QHomogeneousCoordinate mapHomogeneous(const QTransform &transform, const QPointF &p)
{
    QHomogeneousCoordinate c;
    c.x = transform.m11() * p.x() + transform.m21() * p.y() + transform.m31();
    c.y = transform.m12() * p.x() + transform.m22() * p.y() + transform.m32();
    c.w = transform.m13() * p.x() + transform.m23() * p.y() + transform.m33();
    return c;
}

static inline bool lineTo_clipped(QPainterPath &path, const QTransform &transform, const QPointF &a, const QPointF &b, bool needsMoveTo)
{
    QHomogeneousCoordinate ha = mapHomogeneous(transform, a);
    QHomogeneousCoordinate hb = mapHomogeneous(transform, b);

    if (ha.w < Q_NEAR_CLIP && hb.w < Q_NEAR_CLIP)
        return false;

    if (hb.w < Q_NEAR_CLIP) {
        const qreal t = (Q_NEAR_CLIP - hb.w) / (ha.w - hb.w);

        hb.x += (ha.x - hb.x) * t;
        hb.y += (ha.y - hb.y) * t;
        hb.w = qreal(Q_NEAR_CLIP);
    } else if (ha.w < Q_NEAR_CLIP) {
        const qreal t = (Q_NEAR_CLIP - ha.w) / (hb.w - ha.w);

        ha.x += (hb.x - ha.x) * t;
        ha.y += (hb.y - ha.y) * t;
        ha.w = qreal(Q_NEAR_CLIP);

        const QPointF p = ha.toPoint();
        if (needsMoveTo) {
            path.moveTo(p);
            needsMoveTo = false;
        } else {
            path.lineTo(p);
        }
    }

    if (needsMoveTo)
        path.moveTo(ha.toPoint());

    path.lineTo(hb.toPoint());

    return true;
}

static inline bool cubicTo_clipped(QPainterPath &path, const QTransform &transform, const QPointF &a, const QPointF &b, const QPointF &c, const QPointF &d, bool needsMoveTo)
{
    const QHomogeneousCoordinate ha = mapHomogeneous(transform, a);
    const QHomogeneousCoordinate hb = mapHomogeneous(transform, b);
    const QHomogeneousCoordinate hc = mapHomogeneous(transform, c);
    const QHomogeneousCoordinate hd = mapHomogeneous(transform, d);

    if (ha.w < Q_NEAR_CLIP && hb.w < Q_NEAR_CLIP && hc.w < Q_NEAR_CLIP && hd.w < Q_NEAR_CLIP)
        return false;

    if (ha.w >= Q_NEAR_CLIP && hb.w >= Q_NEAR_CLIP && hc.w >= Q_NEAR_CLIP && hd.w >= Q_NEAR_CLIP) {
        if (needsMoveTo)
            path.moveTo(ha.toPoint());

        path.cubicTo(hb.toPoint(), hc.toPoint(), hd.toPoint());
        return true;
    }

    if (lineTo_clipped(path, transform, a, b, needsMoveTo))
            needsMoveTo = false;
    if (lineTo_clipped(path, transform, b, c, needsMoveTo))
            needsMoveTo = false;
    if (lineTo_clipped(path, transform, c, d, needsMoveTo))
            needsMoveTo = false;

    return !needsMoveTo;
}

static QPainterPath mapProjective(const QTransform &transform, const QPainterPath &path)
{
    QPainterPath result;

    QPointF last;
    QPointF lastMoveTo;
    bool needsMoveTo = true;
    for (int i = 0; i < path.elementCount(); ++i) {
        switch (path.elementAt(i).type) {
        case QPainterPath::MoveToElement:
            if (i > 0 && lastMoveTo != last)
                lineTo_clipped(result, transform, last, lastMoveTo, needsMoveTo);

            lastMoveTo = path.elementAt(i);
            last = path.elementAt(i);
            needsMoveTo = true;
            break;
        case QPainterPath::LineToElement:
            if (lineTo_clipped(result, transform, last, path.elementAt(i), needsMoveTo))
                needsMoveTo = false;
            last = path.elementAt(i);
            break;
        case QPainterPath::CurveToElement:
            if (cubicTo_clipped(result, transform, last, path.elementAt(i), path.elementAt(i+1), path.elementAt(i+2), needsMoveTo))
                needsMoveTo = false;
            i += 2;
            last = path.elementAt(i);
            break;
        default:
            Q_ASSERT(false);
        }
    }

    if (path.elementCount() > 0 && lastMoveTo != last)
        lineTo_clipped(result, transform, last, lastMoveTo, needsMoveTo);

    return result;
}

/*!
    \fn QPainterPath operator *(const QPainterPath &path, const QTransform &matrix)
    \since 4.3
    \relates QTransform

    This is the same as \a{matrix}.map(\a{path}).

    \sa QTransform::map()
*/

/*!
    \overload

    Creates and returns a QPainterPath object that is a copy of the
    given \a path, mapped into the coordinate system defined by this
    matrix.
*/
QPainterPath QTransform::map(const QPainterPath &path) const
{
    TransformationType t = type();
    if (t == TxNone || path.isEmpty())
        return path;

    if (t >= TxProject)
        return mapProjective(*this, path);

    QPainterPath copy = path;
    copy.detach();

    if (t == TxTranslate) {
        for (int i=0; i<path.elementCount(); ++i) {
            QPainterPath::Element &e = copy.d_ptr->elements[i];
            e.x += affine._dx;
            e.y += affine._dy;
        }
    } else {
        // Full xform
        for (int i=0; i<path.elementCount(); ++i) {
            QPainterPath::Element &e = copy.d_ptr->elements[i];
            MAP(e.x, e.y, e.x, e.y);
        }
    }

    return copy;
}

/*!
    \fn QPolygon QTransform::mapToPolygon(const QRect &rectangle) const

    Creates and returns a QPolygon representation of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix.

    The rectangle's coordinates are transformed using the following
    formulas:

    \snippet doc/src/snippets/code/src_gui_painting_qtransform.cpp 1

    Polygons and rectangles behave slightly differently when
    transformed (due to integer rounding), so
    \c{matrix.map(QPolygon(rectangle))} is not always the same as
    \c{matrix.mapToPolygon(rectangle)}.

    \sa mapRect(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
QPolygon QTransform::mapToPolygon(const QRect &rect) const
{
    TransformationType t = type();

    QPolygon a(4);
    qreal x[4] = { 0, 0, 0, 0 }, y[4] = { 0, 0, 0, 0 };
    if (t <= TxScale) {
        x[0] = affine._m11*rect.x() + affine._dx;
        y[0] = affine._m22*rect.y() + affine._dy;
        qreal w = affine._m11*rect.width();
        qreal h = affine._m22*rect.height();
        if (w < 0) {
            w = -w;
            x[0] -= w;
        }
        if (h < 0) {
            h = -h;
            y[0] -= h;
        }
        x[1] = x[0]+w;
        x[2] = x[1];
        x[3] = x[0];
        y[1] = y[0];
        y[2] = y[0]+h;
        y[3] = y[2];
    } else {
        qreal right = rect.x() + rect.width();
        qreal bottom = rect.y() + rect.height();
        MAP(rect.x(), rect.y(), x[0], y[0]);
        MAP(right, rect.y(), x[1], y[1]);
        MAP(right, bottom, x[2], y[2]);
        MAP(rect.x(), bottom, x[3], y[3]);
    }

    // all coordinates are correctly, tranform to a pointarray
    // (rounding to the next integer)
    a.setPoints(4, qRound(x[0]), qRound(y[0]),
                qRound(x[1]), qRound(y[1]),
                qRound(x[2]), qRound(y[2]),
                qRound(x[3]), qRound(y[3]));
    return a;
}

/*!
    Creates a transformation matrix, \a trans, that maps a unit square
    to a four-sided polygon, \a quad. Returns true if the transformation
    is constructed or false if such a transformation does not exist.

    \sa quadToSquare(), quadToQuad()
*/
bool QTransform::squareToQuad(const QPolygonF &quad, QTransform &trans)
{
    if (quad.count() != 4)
        return false;

    qreal dx0 = quad[0].x();
    qreal dx1 = quad[1].x();
    qreal dx2 = quad[2].x();
    qreal dx3 = quad[3].x();

    qreal dy0 = quad[0].y();
    qreal dy1 = quad[1].y();
    qreal dy2 = quad[2].y();
    qreal dy3 = quad[3].y();

    double ax  = dx0 - dx1 + dx2 - dx3;
    double ay  = dy0 - dy1 + dy2 - dy3;

    if (!ax && !ay) { //afine transform
        trans.setMatrix(dx1 - dx0, dy1 - dy0,  0,
                        dx2 - dx1, dy2 - dy1,  0,
                        dx0,       dy0,  1);
    } else {
        double ax1 = dx1 - dx2;
        double ax2 = dx3 - dx2;
        double ay1 = dy1 - dy2;
        double ay2 = dy3 - dy2;

        /*determinants */
        double gtop    =  ax  * ay2 - ax2 * ay;
        double htop    =  ax1 * ay  - ax  * ay1;
        double bottom  =  ax1 * ay2 - ax2 * ay1;

        double a, b, c, d, e, f, g, h;  /*i is always 1*/

        if (!bottom)
            return false;

        g = gtop/bottom;
        h = htop/bottom;

        a = dx1 - dx0 + g * dx1;
        b = dx3 - dx0 + h * dx3;
        c = dx0;
        d = dy1 - dy0 + g * dy1;
        e = dy3 - dy0 + h * dy3;
        f = dy0;

        trans.setMatrix(a, d, g,
                        b, e, h,
                        c, f, 1.0);
    }

    return true;
}

/*!
    \fn bool QTransform::quadToSquare(const QPolygonF &quad, QTransform &trans)

    Creates a transformation matrix, \a trans, that maps a four-sided polygon,
    \a quad, to a unit square. Returns true if the transformation is constructed
    or false if such a transformation does not exist.

    \sa squareToQuad(), quadToQuad()
*/
bool QTransform::quadToSquare(const QPolygonF &quad, QTransform &trans)
{
    if (!squareToQuad(quad, trans))
        return false;

    bool invertible = false;
    trans = trans.inverted(&invertible);

    return invertible;
}

/*!
    Creates a transformation matrix, \a trans, that maps a four-sided
    polygon, \a one, to another four-sided polygon, \a two.
    Returns true if the transformation is possible; otherwise returns
    false.

    This is a convenience method combining quadToSquare() and
    squareToQuad() methods. It allows the input quad to be
    transformed into any other quad.

    \sa squareToQuad(), quadToSquare()
*/
bool QTransform::quadToQuad(const QPolygonF &one,
                            const QPolygonF &two,
                            QTransform &trans)
{
    QTransform stq;
    if (!quadToSquare(one, trans))
        return false;
    if (!squareToQuad(two, stq))
        return false;
    trans *= stq;
    //qDebug()<<"Final = "<<trans;
    return true;
}

/*!
    Sets the matrix elements to the specified values, \a m11,
    \a m12, \a m13 \a m21, \a m22, \a m23 \a m31, \a m32 and
    \a m33. Note that this function replaces the previous values.
    QMatrix provides the translate(), rotate(), scale() and shear()
    convenience functions to manipulate the various matrix elements
    based on the currently defined coordinate system.

    \sa QTransform()
*/

void QTransform::setMatrix(qreal m11, qreal m12, qreal m13,
                           qreal m21, qreal m22, qreal m23,
                           qreal m31, qreal m32, qreal m33)
{
    affine._m11 = m11; affine._m12 = m12; m_13 = m13;
    affine._m21 = m21; affine._m22 = m22; m_23 = m23;
    affine._dx = m31; affine._dy = m32; m_33 = m33;
    m_type = TxNone;
    m_dirty = TxProject;
}

QRect QTransform::mapRect(const QRect &rect) const
{
    TransformationType t = type();
    if (t <= TxScale) {
        int x = qRound(affine._m11*rect.x() + affine._dx);
        int y = qRound(affine._m22*rect.y() + affine._dy);
        int w = qRound(affine._m11*rect.width());
        int h = qRound(affine._m22*rect.height());
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRect(x, y, w, h);
    } else if (t < TxProject) {
        // see mapToPolygon for explanations of the algorithm.
        qreal x = 0, y = 0;
        MAP(rect.left(), rect.top(), x, y);
        qreal xmin = x;
        qreal ymin = y;
        qreal xmax = x;
        qreal ymax = y;
        MAP(rect.right() + 1, rect.top(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        MAP(rect.right() + 1, rect.bottom() + 1, x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        MAP(rect.left(), rect.bottom() + 1, x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        return QRect(qRound(xmin), qRound(ymin), qRound(xmax)-qRound(xmin), qRound(ymax)-qRound(ymin));
    } else {
        QPainterPath path;
        path.addRect(rect);
        return map(path).boundingRect().toRect();
    }
}

/*!
    \fn QRectF QTransform::mapRect(const QRectF &rectangle) const

    Creates and returns a QRectF object that is a copy of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix.

    The rectangle's coordinates are transformed using the following
    formulas:

    \snippet doc/src/snippets/code/src_gui_painting_qtransform.cpp 2

    If rotation or shearing has been specified, this function returns
    the \e bounding rectangle. To retrieve the exact region the given
    \a rectangle maps to, use the mapToPolygon() function instead.

    \sa mapToPolygon(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/
QRectF QTransform::mapRect(const QRectF &rect) const
{
    TransformationType t = type();
    if (t <= TxScale) {
        qreal x = affine._m11*rect.x() + affine._dx;
        qreal y = affine._m22*rect.y() + affine._dy;
        qreal w = affine._m11*rect.width();
        qreal h = affine._m22*rect.height();
        if (w < 0) {
            w = -w;
            x -= w;
        }
        if (h < 0) {
            h = -h;
            y -= h;
        }
        return QRectF(x, y, w, h);
    } else if (t < TxProject) {
        qreal x = 0, y = 0;
        MAP(rect.x(), rect.y(), x, y);
        qreal xmin = x;
        qreal ymin = y;
        qreal xmax = x;
        qreal ymax = y;
        MAP(rect.x() + rect.width(), rect.y(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        MAP(rect.x() + rect.width(), rect.y() + rect.height(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        MAP(rect.x(), rect.y() + rect.height(), x, y);
        xmin = qMin(xmin, x);
        ymin = qMin(ymin, y);
        xmax = qMax(xmax, x);
        ymax = qMax(ymax, y);
        return QRectF(xmin, ymin, xmax-xmin, ymax - ymin);
    } else {
        QPainterPath path;
        path.addRect(rect);
        return map(path).boundingRect();
    }
}

/*!
    \fn QRect QTransform::mapRect(const QRect &rectangle) const
    \overload

    Creates and returns a QRect object that is a copy of the given \a
    rectangle, mapped into the coordinate system defined by this
    matrix. Note that the transformed coordinates are rounded to the
    nearest integer.
*/

/*!
    Maps the given coordinates \a x and \a y into the coordinate
    system defined by this matrix. The resulting values are put in *\a
    tx and *\a ty, respectively.

    The coordinates are transformed using the following formulas:

    \snippet doc/src/snippets/code/src_gui_painting_qtransform.cpp 3

    The point (x, y) is the original point, and (x', y') is the
    transformed point.

    \sa {QTransform#Basic Matrix Operations}{Basic Matrix Operations}
*/
void QTransform::map(qreal x, qreal y, qreal *tx, qreal *ty) const
{
    TransformationType t = type();
    MAP(x, y, *tx, *ty);
}

/*!
    \overload

    Maps the given coordinates \a x and \a y into the coordinate
    system defined by this matrix. The resulting values are put in *\a
    tx and *\a ty, respectively. Note that the transformed coordinates
    are rounded to the nearest integer.
*/
void QTransform::map(int x, int y, int *tx, int *ty) const
{
    TransformationType t = type();
    qreal fx = 0, fy = 0;
    MAP(x, y, fx, fy);
    *tx = qRound(fx);
    *ty = qRound(fy);
}

/*!
  Returns the QTransform cast to a QMatrix.
 */
const QMatrix &QTransform::toAffine() const
{
    return affine;
}

/*!
  Returns the transformation type of this matrix.

  The transformation type is the highest enumeration value
  capturing all of the matrix's transformations. For example,
  if the matrix both scales and shears, the type would be \c TxShear,
  because \c TxShear has a higher enumeration value than \c TxScale.

  Knowing the transformation type of a matrix is useful for optimization:
  you can often handle specific types more optimally than handling
  the generic case.
  */
QTransform::TransformationType QTransform::type() const
{
    if (m_dirty >= m_type) {
        if (m_dirty > TxShear && (!qFuzzyCompare(m_13 + 1, 1) || !qFuzzyCompare(m_23 + 1, 1) || !qFuzzyCompare(m_33, 1)))
             m_type = TxProject;
        else if (m_dirty > TxScale && (!qFuzzyCompare(affine._m12 + 1, 1) || !qFuzzyCompare(affine._m21 + 1, 1))) {
            const qreal dot = affine._m11 * affine._m12 + affine._m21 * affine._m22;
            if (qFuzzyCompare(dot + 1, 1))
                m_type = TxRotate;
            else
                m_type = TxShear;
        } else if (m_dirty > TxTranslate && (!qFuzzyCompare(affine._m11, 1) || !qFuzzyCompare(affine._m22, 1)))
            m_type = TxScale;
        else if (m_dirty > TxNone && (!qFuzzyCompare(affine._dx + 1, 1) || !qFuzzyCompare(affine._dy + 1, 1)))
            m_type = TxTranslate;
        else
            m_type = TxNone;

        m_dirty = TxNone;
    }

    return static_cast<TransformationType>(m_type);
}

/*!

    Returns the transform as a QVariant.
*/
QTransform::operator QVariant() const
{
    return QVariant(QVariant::Transform, this);
}


/*!
    \fn bool QTransform::isInvertible() const

    Returns true if the matrix is invertible, otherwise returns false.

    \sa inverted()
*/

/*!
    \fn qreal QTransform::det() const

    Returns the matrix's determinant.
*/


/*!
    \fn qreal QTransform::m11() const

    Returns the horizontal scaling factor.

    \sa scale(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m12() const

    Returns the vertical shearing factor.

    \sa shear(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m21() const

    Returns the horizontal shearing factor.

    \sa shear(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m22() const

    Returns the vertical scaling factor.

    \sa scale(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::dx() const

    Returns the horizontal translation factor.

    \sa m31(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::dy() const

    Returns the vertical translation factor.

    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/


/*!
    \fn qreal QTransform::m13() const

    Returns the horizontal projection factor.

    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/


/*!
    \fn qreal QTransform::m23() const

    Returns the vertical projection factor.

    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m31() const

    Returns the horizontal translation factor.

    \sa dx(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m32() const

    Returns the vertical translation factor.

    \sa dy(), translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::m33() const

    Returns the division factor.

    \sa translate(), {QTransform#Basic Matrix Operations}{Basic Matrix
    Operations}
*/

/*!
    \fn qreal QTransform::determinant() const

    Returns the matrix's determinant.
*/

/*!
    \fn bool QTransform::isIdentity() const

    Returns true if the matrix is the identity matrix, otherwise
    returns false.

    \sa reset()
*/

/*!
    \fn bool QTransform::isAffine() const

    Returns true if the matrix represent an affine transformation,
    otherwise returns false.
*/

/*!
    \fn bool QTransform::isScaling() const

    Returns true if the matrix represents a scaling
    transformation, otherwise returns false.

    \sa reset()
*/

/*!
    \fn bool QTransform::isRotating() const

    Returns true if the matrix represents some kind of a
    rotating transformation, otherwise returns false.

    \sa reset()
*/

/*!
    \fn bool QTransform::isTranslating() const

    Returns true if the matrix represents a translating
    transformation, otherwise returns false.

    \sa reset()
*/

// returns true if the transform is uniformly scaling
// (same scale in x and y direction)
// scale is set to the max of x and y scaling factors
Q_GUI_EXPORT
bool qt_scaleForTransform(const QTransform &transform, qreal *scale)
{
    const QTransform::TransformationType type = transform.type();
    if (type <= QTransform::TxTranslate) {
        *scale = 1;
        return true;
    } else if (type == QTransform::TxScale) {
        const qreal xScale = qAbs(transform.m11());
        const qreal yScale = qAbs(transform.m22());
        *scale = qMax(xScale, yScale);
        return qFuzzyCompare(xScale, yScale);
    }

    const qreal xScale = transform.m11() * transform.m11()
                         + transform.m21() * transform.m21();
    const qreal yScale = transform.m12() * transform.m12()
                         + transform.m22() * transform.m22();
    *scale = qSqrt(qMax(xScale, yScale));
    return type == QTransform::TxRotate && qFuzzyCompare(xScale, yScale);
}

QT_END_NAMESPACE