Math: preserve signs in the Matrix[34]::scaling() getter.

This is a breaking change, but I think it is worth doing. Because now

    Matrix4::scaling(vec).scaling() == vec

which was true also for translation and other.

doc/changelog.dox

@@ -127,6 +127,12 @@ See also:
     in order to allow passing runtime-sized lists (see
     [mosra/magnum#403](https://github.com/mosra/magnum/pull/403))
 
+@subsubsection changelog-latest-changes-math Math library
+
+-   @ref Math::Matrix3::scaling() const and @ref Math::Matrix4::scaling() const
+    now return a signed scaling vector, taking into account negative scaling as
+    well
+
 @subsubsection changelog-latest-changes-meshtools MeshTools library
 
 -   Added @ref MeshTools::subdivideInPlace() that operates on a partially
@@ -223,6 +229,10 @@ See also:
     @ref Animation library was changed to allow mutable access to the keys &
     values it references. Existing code needs to be changed to say
     @cpp TrackView<const K, const V> @ce instead of @cpp TrackView<K, V> @ce.
+-   @ref Math::Matrix3::scaling() const and @ref Math::Matrix4::scaling() const
+    now return a signed scaling vector instead of unsigned in order to prevent
+    information loss. To restore the old behavior, apply @ref Math::abs() on
+    the result.
 -   @ref Platform::GlfwApplication::setMinWindowSize() /
     @ref Platform::GlfwApplication::setMaxWindowSize() and equivalent APIs in
     @ref Platform::Sdl2Application now premultiply the value with

src/Magnum/Math/Matrix3.h

@@ -430,11 +430,11 @@ template<class T> class Matrix3: public Matrix3x3<T> {
         /**
          * @brief Non-uniform scaling part of the matrix
          *
-         * Length of vectors in upper-left 2x2 part of the matrix. Use the
+         * *Signed* length of vectors in upper-left 2x2 part of the matrix. Use
-         * faster alternative @ref scalingSquared() where possible. Assuming
+         * the faster alternative @ref scalingSquared() where possible.
-         * the following matrix, with the upper-left 2x2 part represented by
+         * Assuming the following matrix, with the upper-left 2x2 part
-         * column vectors @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$:
+         * represented by column vectors @f$ \boldsymbol{a} @f$ and
-         * @f[
+         * @f$ \boldsymbol{b} @f$: @f[
          *      \begin{pmatrix}
          *          \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
          *          \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
@@ -446,8 +446,8 @@ template<class T> class Matrix3: public Matrix3x3<T> {
          *
          * the resulting scaling vector is: @f[
          *      \boldsymbol{s} = \begin{pmatrix}
-         *          | \boldsymbol{a} | \\
+         *          \operatorname{sgn}(a_x) | \boldsymbol{a} | \\
-         *          | \boldsymbol{b} |
+         *          \operatorname{sgn}(a_y) | \boldsymbol{b} |
          *      \end{pmatrix}
          * @f]
          *
@@ -455,8 +455,8 @@ template<class T> class Matrix3: public Matrix3x3<T> {
          *      @ref rotation() const, @ref Matrix4::scaling() const
          */
         Vector2<T> scaling() const {
-            return {(*this)[0].xy().length(),
+            return {(*this)[0].xy().length()*T((*this)[0][0] < T(0) ? -1 : 1),
-                    (*this)[1].xy().length()};
+                    (*this)[1].xy().length()*T((*this)[1][1] < T(0) ? -1 : 1)};
         }
 
         /**

src/Magnum/Math/Matrix4.h

@@ -684,11 +684,11 @@ template<class T> class Matrix4: public Matrix4x4<T> {
         /**
          * @brief Non-uniform scaling part of the matrix
          *
-         * Length of vectors in upper-left 3x3 part of the matrix. Use the
+         * *Signed* length of vectors in upper-left 3x3 part of the matrix. Use
-         * faster alternative @ref scalingSquared() where possible. Assuming
+         * the faster alternative @ref scalingSquared() where possible.
-         * the following matrix, with the upper-left 3x3 part represented by
+         * Assuming the following matrix, with the upper-left 3x3 part
-         * column vectors @f$ \boldsymbol{a} @f$, @f$ \boldsymbol{b} @f$ and
+         * represented by column vectors @f$ \boldsymbol{a} @f$,
-         * @f$ \boldsymbol{c} @f$: @f[
+         * @f$ \boldsymbol{b} @f$ and @f$ \boldsymbol{c} @f$: @f[
          *      \begin{pmatrix}
          *          \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\
          *          \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\
@@ -701,9 +701,9 @@ template<class T> class Matrix4: public Matrix4x4<T> {
          *
          * the resulting scaling vector is: @f[
          *      \boldsymbol{s} = \begin{pmatrix}
-         *          | \boldsymbol{a} | \\
+         *          \operatorname{sgn}(a_x) | \boldsymbol{a} | \\
-         *          | \boldsymbol{b} | \\
+         *          \operatorname{sgn}(b_y) | \boldsymbol{b} | \\
-         *          | \boldsymbol{c} |
+         *          \operatorname{sgn}(c_z) | \boldsymbol{c} |
          *      \end{pmatrix}
          * @f]
          *
@@ -711,9 +711,9 @@ template<class T> class Matrix4: public Matrix4x4<T> {
          *      @ref rotation() const, @ref Matrix3::scaling() const
          */
         Vector3<T> scaling() const {
-            return {(*this)[0].xyz().length(),
+            return {(*this)[0].xyz().length()*T((*this)[0][0] < T(0) ? -1 : 1),
-                    (*this)[1].xyz().length(),
+                    (*this)[1].xyz().length()*T((*this)[1][1] < T(0) ? -1 : 1),
-                    (*this)[2].xyz().length()};
+                    (*this)[2].xyz().length()*T((*this)[2][2] < T(0) ? -1 : 1)};
         }
 
         /**

src/Magnum/Math/Test/Matrix3Test.cpp

@@ -526,10 +526,18 @@ void Matrix3Test::scalingPart() {
     Matrix3 translationRotationScaling =
         Matrix3::translation({2.0f, -3.0f})*
         Matrix3::rotation(15.0_degf)*
-        Matrix3::scaling({0.5f, 3.5f});
+        Matrix3::scaling({0.5f, -3.5f});
 
-    CORRADE_COMPARE(translationRotationScaling.scaling(), (Vector2{0.5f, 3.5f}));
+    CORRADE_COMPARE(translationRotationScaling.scaling(), (Vector2{0.5f, -3.5f}));
     CORRADE_COMPARE(translationRotationScaling.scalingSquared(), (Vector2{0.25f, 12.25f}));
+
+    Matrix3 rotationScaling =
+        Matrix3::rotation(215.0_degf)*
+        Matrix3::scaling({-0.5f, 3.5f});
+
+    /* Sign inverted because of the rotation */
+    CORRADE_COMPARE(rotationScaling.scaling(), (Vector2{0.5f, -3.5f}));
+    CORRADE_COMPARE(rotationScaling.scalingSquared(), (Vector2{0.25f, 12.25f}));
 }
 
 void Matrix3Test::uniformScalingPart() {

src/Magnum/Math/Test/Matrix4Test.cpp

@@ -760,10 +760,18 @@ void Matrix4Test::scalingPart() {
     Matrix4 translationRotationScaling =
         Matrix4::translation({2.0f, 5.0f, -3.0f})*
         Matrix4::rotation(-74.0_degf, Vector3(-1.0f, 2.0f, 2.0f).normalized())*
-        Matrix4::scaling({0.5f, 3.5f, 1.2f});
+        Matrix4::scaling({0.5f, -3.5f, 1.2f});
 
-    CORRADE_COMPARE(translationRotationScaling.scaling(), (Vector3{0.5f, 3.5f, 1.2f}));
+    CORRADE_COMPARE(translationRotationScaling.scaling(), (Vector3{0.5f, -3.5f, 1.2f}));
     CORRADE_COMPARE(translationRotationScaling.scalingSquared(), (Vector3{0.25f, 12.25f, 1.44f}));
+
+    Matrix4 rotationScaling =
+        Matrix4::rotation(-174.0_degf, Vector3(-1.0f, 2.0f, 2.0f).normalized())*
+        Matrix4::scaling({-0.5f, 3.5f, 1.2f});
+
+    /* Sign inverted because of the rotation */
+    CORRADE_COMPARE(rotationScaling.scaling(), (Vector3{0.5f, -3.5f, -1.2f}));
+    CORRADE_COMPARE(rotationScaling.scalingSquared(), (Vector3{0.25f, 12.25f, 1.44f}));
 }
 
 void Matrix4Test::uniformScalingPart() {