Doc: Improve QTransform documentation

* Explain QTransform's model of vector/matrix operations.
* Improve snippets used to illuminate QTransform's behavior.

Fixes: QTBUG-83869
Pick-to: 6.2 6.1 5.15
Change-Id: I84c3b1a221c139ee992f82c3ee4aebadeef8ee63
Reviewed-by: Edward Welbourne <edward.welbourne@qt.io>
Reviewed-by: Topi Reiniö <topi.reinio@qt.io>
Reviewed-by: Andy Nichols <andy.nichols@qt.io>

src/gui/doc/snippets/code/src_gui_painting_qtransform.cpp

@@ -54,7 +54,7 @@ namespace src_gui_painting_qtransform {
 //! [0]
 x' = m11*x + m21*y + dx
 y' = m22*y + m12*x + dy
-if (is not affine) {
+if (!isAffine()) {
     w' = m13*x + m23*y + m33
     x' /= w'
     y' /= w'
@@ -65,7 +65,7 @@ if (is not affine) {
 //! [1]
 x' = m11*x + m21*y + dx
 y' = m22*y + m12*x + dy
-if (is not affine) {
+if (!isAffine()) {
     w' = m13*x + m23*y + m33
     x' /= w'
     y' /= w'
@@ -76,7 +76,7 @@ if (is not affine) {
 //! [2]
 x' = m11*x + m21*y + dx
 y' = m22*y + m12*x + dy
-if (is not affine) {
+if (!isAffine()) {
     w' = m13*x + m23*y + m33
     x' /= w'
     y' /= w'
@@ -87,7 +87,7 @@ if (is not affine) {
 //! [3]
 x' = m11*x + m21*y + dx
 y' = m22*y + m12*x + dy
-if (is not affine) {
+if (!isAffine()) {
     w' = m13*x + m23*y + m33
     x' /= w'
     y' /= w'

src/gui/doc/snippets/transform/main.cpp

@@ -48,6 +48,7 @@
 **
 ****************************************************************************/
 #include <QApplication>
+#include <QMath>
 #include <QPainter>
 #include <QVBoxLayout>
 #include <QWidget>
@@ -105,18 +106,15 @@ class BasicOperations : public QWidget
 //! [2]
 void BasicOperations::paintEvent(QPaintEvent *)
 {
-    double pi = 3.14;
-
-    double a    = pi/180 * 45.0;
+    const double a = qDegreesToRadians(45.0);
     double sina = sin(a);
     double cosa = cos(a);
 
-    QTransform translationTransform(1, 0, 0, 1, 50.0, 50.0);
-    QTransform rotationTransform(cosa, sina, -sina, cosa, 0, 0);
-    QTransform scalingTransform(0.5, 0, 0, 1.0, 0, 0);
+    QTransform scale(0.5, 0, 0, 1.0, 0, 0);
+    QTransform rotate(cosa, sina, -sina, cosa, 0, 0);
+    QTransform translate(1, 0, 0, 1, 50.0, 50.0);
 
-    QTransform transform;
-    transform = scalingTransform * rotationTransform * translationTransform;
+    QTransform transform = scale * rotate * translate;
 
     QPainter painter(this);
     painter.setPen(QPen(Qt::blue, 1, Qt::DashLine));

src/gui/painting/qtransform.cpp

@@ -213,6 +213,7 @@ static void nanWarning(const char *func)
     transformation is achieved by setting both the projection factors and
     the scaling factors.
 
+    \section2 Combining Transforms
     Here's the combined transformations example using basic matrix
     operations:
 
@@ -223,6 +224,26 @@ static void nanWarning(const char *func)
     \snippet transform/main.cpp 2
     \endtable
 
+    The combined transform first scales each operand, then rotates it, and
+    finally translates it, just as in the order in which the product of its
+    factors is written. This means the point to which the transforms are
+    applied is implicitly multiplied on the left with the transform
+    to its right.
+
+    \section2 Relation to Matrix Notation
+    The matrix notation in QTransform is the transpose of a commonly-taught
+    convention which represents transforms and points as matrices and vectors.
+    That convention multiplies its matrix on the left and column vector to the
+    right. In other words, when several transforms are applied to a point, the
+    right-most matrix acts directly on the vector first. Then the next matrix
+    to the left acts on the result of the first operation - and so on. As a
+    result, that convention multiplies the matrices that make up a composite
+    transform in the reverse of the order in QTransform, as you can see in
+    \l {Combining Transforms}. Transposing the matrices, and combining them to
+    the right of a row vector that represents the point, lets the matrices of
+    transforms appear, in their product, in the order in which we think of the
+    transforms being applied to the point.
+
     \sa QPainter, {Coordinate System}, {painting/affine}{Affine
     Transformations Example}, {Transformations Example}
 */