Use right handed coordinate system for rotation matrices and quaternions. Also fixes Euler angles (XYZ convention, which is used as default by Blender).

Furthermore, functions which expect a rotation matrix will now give an error simply, rather than trying to orthonormalize such matrices. The documentation for such functions has be updated accordingly.

This commit breaks code using 3D rotations, and is a part of the breaking changes in 2.1 -> 3.0 transition. The code affected within Godot code base is fixed in this commit.

16 files changed, 211 insertions, 135 deletions

diff --git a/core/math/math_funcs.h b/core/math/math_funcs.h
index ec0ed39471..24081528f0 100644
--- a/core/math/math_funcs.h
+++ b/core/math/math_funcs.h
@@ -96,6 +96,15 @@ public:
 
 	static double random(double from, double to);
 
+	static _FORCE_INLINE_ bool isequal_approx(real_t a, real_t b) {
+		// TODO: Comparing floats for approximate-equality is non-trivial.
+		// Using epsilon should cover the typical cases in Godot (where a == b is used to compare two reals), such as matrix and vector comparison operators.
+		// A proper implementation in terms of ULPs should eventually replace the contents of this function.
+		// See https://randomascii.wordpress.com/2012/02/25/comparing-floating-point-numbers-2012-edition/ for details.
+
+		return abs(a-b) < CMP_EPSILON;
+	}
+
 
 	static _FORCE_INLINE_ real_t abs(real_t g) {
 
diff --git a/core/math/matrix3.cpp b/core/math/matrix3.cpp
index c30401cc24..a985e29abb 100644
--- a/core/math/matrix3.cpp
+++ b/core/math/matrix3.cpp
@@ -73,6 +73,7 @@ void Matrix3::invert() {
 }
 
 void Matrix3::orthonormalize() {
+	ERR_FAIL_COND(determinant() == 0);
 
 	// Gram-Schmidt Process
 
@@ -99,6 +100,17 @@ Matrix3 Matrix3::orthonormalized() const {
 	return c;
 }
 
+bool Matrix3::is_orthogonal() const {
+	Matrix3 id;
+	Matrix3 m = (*this)*transposed();
+
+	return isequal_approx(id,m);
+}
+
+bool Matrix3::is_rotation() const {
+	return Math::isequal_approx(determinant(), 1) && is_orthogonal();
+}
+
 
 Matrix3 Matrix3::inverse() const {
 
@@ -150,42 +162,58 @@ Vector3 Matrix3::get_scale() const {
 	);
 
 }
-void Matrix3::rotate(const Vector3& p_axis, real_t p_phi) {
 
+// Matrix3::rotate and Matrix3::rotated return M * R(axis,phi), and is a convenience function. They do *not* perform proper matrix rotation.
+void Matrix3::rotate(const Vector3& p_axis, real_t p_phi) {
+	// TODO: This function should also be renamed as the current name is misleading: rotate does *not* perform matrix rotation.
+	// Same problem affects Matrix3::rotated.
+	// A similar problem exists in 2D math, which will be handled separately.
+	// After Matrix3 is renamed to Basis, this comments needs to be revised.
 	*this = *this * Matrix3(p_axis, p_phi);
 }
 
 Matrix3 Matrix3::rotated(const Vector3& p_axis, real_t p_phi) const {
-
 	return *this * Matrix3(p_axis, p_phi);
 
 }
 
+// get_euler returns a vector containing the Euler angles in the format
+// (a1,a2,a3), where a3 is the angle of the first rotation, and a1 is the last
+// (following the convention they are commonly defined in the literature).
+//
+// The current implementation uses XYZ convention (Z is the first rotation),
+// so euler.z is the angle of the (first) rotation around Z axis and so on,
+//
+// And thus, assuming the matrix is a rotation matrix, this function returns
+// the angles in the decomposition R = X(a1).Y(a2).Z(a3) where Z(a) rotates
+// around the z-axis by a and so on.
 Vector3 Matrix3::get_euler() const {
 
+	// Euler angles in XYZ convention.
+	// See https://en.wikipedia.org/wiki/Euler_angles#Rotation_matrix
+	//
 	// rot =  cy*cz          -cy*sz           sy
-	    //        cz*sx*sy+cx*sz  cx*cz-sx*sy*sz -cy*sx
-	    //       -cx*cz*sy+sx*sz  cz*sx+cx*sy*sz  cx*cy
-
-	Matrix3 m = *this;
-	m.orthonormalize();
+	//        cz*sx*sy+cx*sz  cx*cz-sx*sy*sz -cy*sx
+	//       -cx*cz*sy+sx*sz  cz*sx+cx*sy*sz  cx*cy
 
 	Vector3 euler;
 
-	euler.y = Math::asin(m[0][2]);
+	ERR_FAIL_COND_V(is_rotation() == false, euler);
+
+	euler.y = Math::asin(elements[0][2]);
 	if ( euler.y < Math_PI*0.5) {
 		if ( euler.y > -Math_PI*0.5) {
-			euler.x = Math::atan2(-m[1][2],m[2][2]);
-			euler.z = Math::atan2(-m[0][1],m[0][0]);
+			euler.x = Math::atan2(-elements[1][2],elements[2][2]);
+			euler.z = Math::atan2(-elements[0][1],elements[0][0]);
 
 		} else {
-			real_t r = Math::atan2(m[1][0],m[1][1]);
+			real_t r = Math::atan2(elements[1][0],elements[1][1]);
 			euler.z = 0.0;
 			euler.x = euler.z - r;
 
 		}
 	} else {
-		real_t r = Math::atan2(m[0][1],m[1][1]);
+		real_t r = Math::atan2(elements[0][1],elements[1][1]);
 		euler.z = 0;
 		euler.x = r - euler.z;
 	}
@@ -195,6 +223,9 @@ Vector3 Matrix3::get_euler() const {
 
 }
 
+// set_euler expects a vector containing the Euler angles in the format
+// (c,b,a), where a is the angle of the first rotation, and c is the last.
+// The current implementation uses XYZ convention (Z is the first rotation).
 void Matrix3::set_euler(const Vector3& p_euler) {
 
 	real_t c, s;
@@ -215,17 +246,30 @@ void Matrix3::set_euler(const Vector3& p_euler) {
 	*this = xmat*(ymat*zmat);
 }
 
+bool Matrix3::isequal_approx(const Matrix3& a, const Matrix3& b) const {
+
+        for (int i=0;i<3;i++) {
+                for (int j=0;j<3;j++) {
+                        if (Math::isequal_approx(a.elements[i][j],b.elements[i][j]) == false)
+                                return false;
+                }
+        }
+
+        return true;
+}
+
 bool Matrix3::operator==(const Matrix3& p_matrix) const {
 
 	for (int i=0;i<3;i++) {
 		for (int j=0;j<3;j++) {
-			if (elements[i][j]!=p_matrix.elements[i][j])
+			if (elements[i][j] != p_matrix.elements[i][j])
 				return false;
 		}
 	}
 
 	return true;
 }
+
 bool Matrix3::operator!=(const Matrix3& p_matrix) const {
 
 	return (!(*this==p_matrix));
@@ -249,11 +293,9 @@ Matrix3::operator String() const {
 }
 
 Matrix3::operator Quat() const {
+	ERR_FAIL_COND_V(is_rotation() == false, Quat());
 
-	Matrix3 m=*this;
-	m.orthonormalize();
-
-	real_t trace = m.elements[0][0] + m.elements[1][1] + m.elements[2][2];
+	real_t trace = elements[0][0] + elements[1][1] + elements[2][2];
 	real_t temp[4];
 
 	if (trace > 0.0)
@@ -262,25 +304,25 @@ Matrix3::operator Quat() const {
 		temp[3]=(s * 0.5);
 		s = 0.5 / s;
 
-		temp[0]=((m.elements[2][1] - m.elements[1][2]) * s);
-		temp[1]=((m.elements[0][2] - m.elements[2][0]) * s);
-		temp[2]=((m.elements[1][0] - m.elements[0][1]) * s);
+		temp[0]=((elements[2][1] - elements[1][2]) * s);
+		temp[1]=((elements[0][2] - elements[2][0]) * s);
+		temp[2]=((elements[1][0] - elements[0][1]) * s);
 	}
 	else
 	{
-		int i = m.elements[0][0] < m.elements[1][1] ?
-			(m.elements[1][1] < m.elements[2][2] ? 2 : 1) :
-			(m.elements[0][0] < m.elements[2][2] ? 2 : 0);
+		int i = elements[0][0] < elements[1][1] ?
+			(elements[1][1] < elements[2][2] ? 2 : 1) :
+			(elements[0][0] < elements[2][2] ? 2 : 0);
 		int j = (i + 1) % 3;
 		int k = (i + 2) % 3;
 
-		real_t s = Math::sqrt(m.elements[i][i] - m.elements[j][j] - m.elements[k][k] + 1.0);
+		real_t s = Math::sqrt(elements[i][i] - elements[j][j] - elements[k][k] + 1.0);
 		temp[i] = s * 0.5;
 		s = 0.5 / s;
 
-		temp[3] = (m.elements[k][j] - m.elements[j][k]) * s;
-		temp[j] = (m.elements[j][i] + m.elements[i][j]) * s;
-		temp[k] = (m.elements[k][i] + m.elements[i][k]) * s;
+		temp[3] = (elements[k][j] - elements[j][k]) * s;
+		temp[j] = (elements[j][i] + elements[i][j]) * s;
+		temp[k] = (elements[k][i] + elements[i][k]) * s;
 	}
 
 	return Quat(temp[0],temp[1],temp[2],temp[3]);
@@ -356,6 +398,10 @@ void Matrix3::set_orthogonal_index(int p_index){
 
 
 void Matrix3::get_axis_and_angle(Vector3 &r_axis,real_t& r_angle) const {
+	// TODO: We can handle improper matrices here too, in which case axis will also correspond to the axis of reflection.
+	// See Eq. (52) in http://scipp.ucsc.edu/~haber/ph251/rotreflect_13.pdf for example
+	// After that change, we should fail on is_orthogonal() == false.
+	ERR_FAIL_COND(is_rotation() == false);
 
 
 	double angle,x,y,z; // variables for result
@@ -423,14 +469,13 @@ void Matrix3::get_axis_and_angle(Vector3 &r_axis,real_t& r_angle) const {
 	// as we have reached here there are no singularities so we can handle normally
 	double s = Math::sqrt((elements[1][2] - elements[2][1])*(elements[1][2] - elements[2][1])
 		+(elements[2][0] - elements[0][2])*(elements[2][0] - elements[0][2])
-		+(elements[0][1] - elements[1][0])*(elements[0][1] - elements[1][0])); // used to normalise
-	if (Math::abs(s) < 0.001) s=1;
-		// prevent divide by zero, should not happen if matrix is orthogonal and should be
-		// caught by singularity test above, but I've left it in just in case
+		+(elements[0][1] - elements[1][0])*(elements[0][1] - elements[1][0])); // s=|axis||sin(angle)|, used to normalise
+
 	angle = Math::acos(( elements[0][0] + elements[1][1] + elements[2][2] - 1)/2);
-	x = (elements[1][2] - elements[2][1])/s;
-	y = (elements[2][0] - elements[0][2])/s;
-	z = (elements[0][1] - elements[1][0])/s;
+	if (angle < 0) s = -s;
+	x = (elements[2][1] - elements[1][2])/s;
+	y = (elements[0][2] - elements[2][0])/s;
+	z = (elements[1][0] - elements[0][1])/s;
 
 	r_axis=Vector3(x,y,z);
 	r_angle=angle;
@@ -457,6 +502,7 @@ Matrix3::Matrix3(const Quat& p_quat) {
 }
 
 Matrix3::Matrix3(const Vector3& p_axis, real_t p_phi) {
+	// Rotation matrix from axis and angle, see https://en.wikipedia.org/wiki/Rotation_matrix#Rotation_matrix_from_axis_and_angle
 
 	Vector3 axis_sq(p_axis.x*p_axis.x,p_axis.y*p_axis.y,p_axis.z*p_axis.z);
 
@@ -464,15 +510,15 @@ Matrix3::Matrix3(const Vector3& p_axis, real_t p_phi) {
 	real_t sine= Math::sin(p_phi);
 
 	elements[0][0] = axis_sq.x + cosine * ( 1.0 - axis_sq.x );
-	elements[0][1] = p_axis.x * p_axis.y *  ( 1.0 - cosine ) + p_axis.z * sine;
-	elements[0][2] = p_axis.z * p_axis.x * ( 1.0 - cosine ) - p_axis.y * sine;
+	elements[0][1] = p_axis.x * p_axis.y *  ( 1.0 - cosine ) - p_axis.z * sine;
+	elements[0][2] = p_axis.z * p_axis.x * ( 1.0 - cosine ) + p_axis.y * sine;
 
-	elements[1][0] = p_axis.x * p_axis.y * ( 1.0 - cosine ) - p_axis.z * sine;
+	elements[1][0] = p_axis.x * p_axis.y * ( 1.0 - cosine ) + p_axis.z * sine;
 	elements[1][1] = axis_sq.y + cosine  * ( 1.0 - axis_sq.y );
-	elements[1][2] = p_axis.y * p_axis.z * ( 1.0 - cosine ) + p_axis.x * sine;
+	elements[1][2] = p_axis.y * p_axis.z * ( 1.0 - cosine ) - p_axis.x * sine;
 
-	elements[2][0] = p_axis.z * p_axis.x * ( 1.0 - cosine ) + p_axis.y * sine;
-	elements[2][1] = p_axis.y * p_axis.z * ( 1.0 - cosine ) - p_axis.x * sine;
+	elements[2][0] = p_axis.z * p_axis.x * ( 1.0 - cosine ) - p_axis.y * sine;
+	elements[2][1] = p_axis.y * p_axis.z * ( 1.0 - cosine ) + p_axis.x * sine;
 	elements[2][2] = axis_sq.z + cosine  * ( 1.0 - axis_sq.z );
 
 }
diff --git a/core/math/matrix3.h b/core/math/matrix3.h
index 2792200b7d..1d967c03b8 100644
--- a/core/math/matrix3.h
+++ b/core/math/matrix3.h
@@ -91,6 +91,8 @@ public:
 		return elements[0][2] * v[0] + elements[1][2] * v[1] + elements[2][2] * v[2];
 	}
 
+	bool isequal_approx(const Matrix3& a, const Matrix3& b) const;
+
 	bool operator==(const Matrix3& p_matrix) const;
 	bool operator!=(const Matrix3& p_matrix) const;
 
@@ -102,6 +104,9 @@ public:
 	int get_orthogonal_index() const;
 	void set_orthogonal_index(int p_index);
 
+	bool is_orthogonal() const;
+	bool is_rotation() const;
+
 	operator String() const;
 
 	void get_axis_and_angle(Vector3 &r_axis,real_t& r_angle) const;
diff --git a/core/math/quat.cpp b/core/math/quat.cpp
index 8aa06a2046..afe71100e1 100644
--- a/core/math/quat.cpp
+++ b/core/math/quat.cpp
@@ -27,22 +27,40 @@
 /* SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.                */
 /*************************************************************************/
 #include "quat.h"
+#include "matrix3.h"
 #include "print_string.h"
 
+// set_euler expects a vector containing the Euler angles in the format
+// (c,b,a), where a is the angle of the first rotation, and c is the last.
+// The current implementation uses XYZ convention (Z is the first rotation).
 void Quat::set_euler(const Vector3& p_euler) {
-	real_t half_yaw = p_euler.x * 0.5;
-	real_t half_pitch = p_euler.y * 0.5;
-	real_t half_roll = p_euler.z * 0.5;
-	real_t cos_yaw = Math::cos(half_yaw);
-	real_t sin_yaw = Math::sin(half_yaw);
-	real_t cos_pitch = Math::cos(half_pitch);
-	real_t sin_pitch = Math::sin(half_pitch);
-	real_t cos_roll = Math::cos(half_roll);
-	real_t sin_roll = Math::sin(half_roll);
-	set(cos_roll * sin_pitch * cos_yaw+sin_roll * cos_pitch * sin_yaw,
-		cos_roll * cos_pitch * sin_yaw - sin_roll * sin_pitch * cos_yaw,
-		sin_roll * cos_pitch * cos_yaw - cos_roll * sin_pitch * sin_yaw,
-		cos_roll * cos_pitch * cos_yaw+sin_roll * sin_pitch * sin_yaw);
+	real_t half_a1 = p_euler.x * 0.5;
+	real_t half_a2 = p_euler.y * 0.5;
+	real_t half_a3 = p_euler.z * 0.5;
+
+	// R = X(a1).Y(a2).Z(a3) convention for Euler angles.
+	// Conversion to quaternion as listed in https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19770024290.pdf (page A-2)
+	// a3 is the angle of the first rotation, following the notation in this reference.
+
+	real_t cos_a1 = Math::cos(half_a1);
+	real_t sin_a1 = Math::sin(half_a1);
+	real_t cos_a2 = Math::cos(half_a2);
+	real_t sin_a2 = Math::sin(half_a2);
+	real_t cos_a3 = Math::cos(half_a3);
+	real_t sin_a3 = Math::sin(half_a3);
+
+	set(sin_a1*cos_a2*cos_a3 + sin_a2*sin_a3*cos_a1,
+		-sin_a1*sin_a3*cos_a2 + sin_a2*cos_a1*cos_a3,
+		sin_a1*sin_a2*cos_a3 + sin_a3*cos_a1*cos_a2,
+		-sin_a1*sin_a2*sin_a3 + cos_a1*cos_a2*cos_a3);
+}
+
+// get_euler returns a vector containing the Euler angles in the format
+// (a1,a2,a3), where a3 is the angle of the first rotation, and a1 is the last.
+// The current implementation uses XYZ convention (Z is the first rotation).
+Vector3 Quat::get_euler() const {
+	Matrix3 m(*this);
+	return m.get_euler();
 }
 
 void Quat::operator*=(const Quat& q) {
@@ -126,26 +144,25 @@ Quat Quat::slerp(const Quat& q, const real_t& t) const {
 	}
 #else
 
-	real_t         to1[4];
+	Quat          to1;
 	real_t        omega, cosom, sinom, scale0, scale1;
 
 
 	// calc cosine
-	cosom = x * q.x + y * q.y + z * q.z
-			+ w * q.w;
-
+	cosom = dot(q);
 
 	// adjust signs (if necessary)
 	if ( cosom <0.0 ) {
-		cosom = -cosom; to1[0] = - q.x;
-		to1[1] = - q.y;
-		to1[2] = - q.z;
-		to1[3] = - q.w;
+		cosom = -cosom;
+		to1.x = - q.x;
+		to1.y = - q.y;
+		to1.z = - q.z;
+		to1.w = - q.w;
 	} else  {
-		to1[0] = q.x;
-		to1[1] = q.y;
-		to1[2] = q.z;
-		to1[3] = q.w;
+		to1.x = q.x;
+		to1.y = q.y;
+		to1.z = q.z;
+		to1.w = q.w;
 	}
 
 
@@ -165,10 +182,10 @@ Quat Quat::slerp(const Quat& q, const real_t& t) const {
 	}
 	// calculate final values
 	return Quat(
-		scale0 * x + scale1 * to1[0],
-		scale0 * y + scale1 * to1[1],
-		scale0 * z + scale1 * to1[2],
-		scale0 * w + scale1 * to1[3]
+		scale0 * x + scale1 * to1.x,
+		scale0 * y + scale1 * to1.y,
+		scale0 * z + scale1 * to1.z,
+		scale0 * w + scale1 * to1.w
 	);
 #endif
 }
@@ -186,10 +203,10 @@ Quat Quat::slerpni(const Quat& q, const real_t& t) const {
 		newFactor	= Math::sin(t * theta) * sinT,
 		invFactor	= Math::sin((1.0f - t) * theta) * sinT;
 
-	return Quat( invFactor * from.x + newFactor * q.x,
-			   invFactor * from.y + newFactor * q.y,
-			   invFactor * from.z + newFactor * q.z,
-			   invFactor * from.w + newFactor * q.w );
+	return Quat(invFactor * from.x + newFactor * q.x,
+				invFactor * from.y + newFactor * q.y,
+				invFactor * from.z + newFactor * q.z,
+				invFactor * from.w + newFactor * q.w);
 
 #if 0
 	real_t         to1[4];
@@ -203,7 +220,7 @@ Quat Quat::slerpni(const Quat& q, const real_t& t) const {
 
 	// adjust signs (if necessary)
 	if ( cosom <0.0 && false) {
-		cosom = -cosom; to1[0] = - q.x;
+		cosom = -cosom;to1[0] = - q.x;
 		to1[1] = - q.y;
 		to1[2] = - q.z;
 		to1[3] = - q.w;
@@ -260,8 +277,10 @@ Quat::Quat(const Vector3& axis, const real_t& angle) {
 	if (d==0)
 		set(0,0,0,0);
 	else {
-		real_t s = Math::sin(-angle * 0.5) / d;
+		real_t sin_angle = Math::sin(angle * 0.5);
+		real_t cos_angle = Math::cos(angle * 0.5);
+		real_t s = sin_angle / d;
 		set(axis.x * s, axis.y * s, axis.z * s,
-			Math::cos(-angle * 0.5));
+			cos_angle);
 	}
 }
diff --git a/core/math/quat.h b/core/math/quat.h
index 9f4145cddb..40c048006f 100644
--- a/core/math/quat.h
+++ b/core/math/quat.h
@@ -49,15 +49,16 @@ public:
 	Quat inverse() const;
 	_FORCE_INLINE_ real_t dot(const Quat& q) const;
 	void set_euler(const Vector3& p_euler);
+	Vector3 get_euler() const;
 	Quat slerp(const Quat& q, const real_t& t) const;
 	Quat slerpni(const Quat& q, const real_t& t) const;
 	Quat cubic_slerp(const Quat& q, const Quat& prep, const Quat& postq,const real_t& t) const;
 
 	_FORCE_INLINE_ void get_axis_and_angle(Vector3& r_axis, real_t &r_angle) const {
 		r_angle = 2 * Math::acos(w);
-		r_axis.x = -x / Math::sqrt(1-w*w);
-		r_axis.y = -y / Math::sqrt(1-w*w);
-		r_axis.z = -z / Math::sqrt(1-w*w);
+		r_axis.x = x / Math::sqrt(1-w*w);
+		r_axis.y = y / Math::sqrt(1-w*w);
+		r_axis.z = z / Math::sqrt(1-w*w);
 	}
 
 	void operator*=(const Quat& q);
@@ -183,12 +184,10 @@ Quat Quat::operator/(const real_t& s) const {
 
 
 bool Quat::operator==(const Quat& p_quat) const {
-
 	return x==p_quat.x && y==p_quat.y && z==p_quat.z && w==p_quat.w;
 }
 
 bool Quat::operator!=(const Quat& p_quat) const {
-
 	return x!=p_quat.x || y!=p_quat.y || z!=p_quat.z || w!=p_quat.w;
 }
 
diff --git a/core/math/vector3.h b/core/math/vector3.h
index 3f451b0ab7..14cf1bc6ca 100644
--- a/core/math/vector3.h
+++ b/core/math/vector3.h
@@ -293,7 +293,6 @@ bool Vector3::operator==(const Vector3& p_v) const {
 }
 
 bool Vector3::operator!=(const Vector3& p_v) const {
-
 	return (x!=p_v.x || y!=p_v.y || z!=p_v.z);
 }
 
diff --git a/doc/base/classes.xml b/doc/base/classes.xml
index b49c23f117..4be1666e59 100644
--- a/doc/base/classes.xml
+++ b/doc/base/classes.xml
@@ -20714,7 +20714,7 @@
 			<argument index="1" name="phi" type="float">
 			</argument>
 			<description>
-				Create a matrix from an axis vector and an angle.
+				Create a matrix which rotates around the given axis by the specified angle.
 			</description>
 		</method>
 		<method name="Matrix3">
@@ -20741,7 +20741,7 @@
 			<return type="Vector3">
 			</return>
 			<description>
-				Return euler angles from the matrix.
+				Return euler angles (in the XYZ convention: first Z, then Y, and X last) from the matrix. Returned vector contains the rotation angles in the format (third,second,first).
 			</description>
 		</method>
 		<method name="get_orthogonal_index">
@@ -20767,7 +20767,7 @@
 			<return type="Matrix3">
 			</return>
 			<description>
-				Return the orthonormalized version of the matrix (useful to call from time to time to avoid rounding error).
+				Return the orthonormalized version of the matrix (useful to call from time to time to avoid rounding error for orthogonal matrices). This performs a Gram-Schmidt orthonormalization on the basis of the matrix.
 			</description>
 		</method>
 		<method name="rotated">
@@ -20777,10 +20777,7 @@
 			</argument>
 			<argument index="1" name="phi" type="float">
 			</argument>
-			<description>
-				Return the rotated version of the matrix, by a given axis and angle.
-			</description>
-		</method>
+                </method>
 		<method name="scaled">
 			<return type="Matrix3">
 			</return>
@@ -31485,7 +31482,7 @@
 		Quaternion.
 	</brief_description>
 	<description>
-		Quaternion is a 4 dimensional vector that is used to represent a rotation. It mainly exists to perform SLERP (spherical-linear interpolation) between to rotations obtained by a Matrix3 cheaply. Adding quaternions also cheaply adds the rotations, however quaternions need to be often normalized, or else they suffer from precision issues.
+		Quaternion is a 4 dimensional vector that is used to represent a rotation. It mainly exists to perform SLERP (spherical-linear interpolation) between to rotations obtained by a Matrix3 cheaply. Multiplying quaternions also cheaply reproduces rotation sequences, however quaternions need to be often normalized, or else they suffer from precision issues.
 	</description>
 	<methods>
 		<method name="Quat">
@@ -31510,6 +31507,7 @@
 			<argument index="1" name="angle" type="float">
 			</argument>
 			<description>
+                Returns a quaternion that will rotate around the given axis by the specified angle.
 			</description>
 		</method>
 		<method name="Quat">
@@ -31518,6 +31516,7 @@
 			<argument index="0" name="from" type="Matrix3">
 			</argument>
 			<description>
+				Returns the rotation matrix corresponding to the given quaternion.
 			</description>
 		</method>
 		<method name="cubic_slerp">
@@ -31540,14 +31539,14 @@
 			<argument index="0" name="b" type="Quat">
 			</argument>
 			<description>
-				Returns the dot product between two quaternions.
+				Returns the dot product of two quaternions.
 			</description>
 		</method>
 		<method name="inverse">
 			<return type="Quat">
 			</return>
 			<description>
-				Returns the inverse of the quaternion (applies to the inverse rotation too).
+				Returns the inverse of the quaternion.
 			</description>
 		</method>
 		<method name="length">
@@ -43135,7 +43134,7 @@
 			<argument index="1" name="phi" type="float">
 			</argument>
 			<description>
-				Rotate the transform locally.
+				Rotate the transform locally. This introduces an additional pre-rotation to the transform, changing the basis to basis * Matrix3(axis, phi).
 			</description>
 		</method>
 		<method name="scaled">
diff --git a/modules/gridmap/grid_map_editor_plugin.cpp b/modules/gridmap/grid_map_editor_plugin.cpp
index eb712e49cb..dada7fcc9e 100644
--- a/modules/gridmap/grid_map_editor_plugin.cpp
+++ b/modules/gridmap/grid_map_editor_plugin.cpp
@@ -103,13 +103,13 @@ void  GridMapEditor::_menu_option(int p_option) {
 			if (input_action==INPUT_DUPLICATE) {
 
 				r.set_orthogonal_index(selection.duplicate_rot);
-				r.rotate(Vector3(0,1,0),Math_PI/2.0);
+				r.rotate(Vector3(0,1,0),-Math_PI/2.0);
 				selection.duplicate_rot=r.get_orthogonal_index();
 				_update_duplicate_indicator();
 				break;
 			}
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(0,1,0),Math_PI/2.0);
+			r.rotate(Vector3(0,1,0),-Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
@@ -118,14 +118,14 @@ void  GridMapEditor::_menu_option(int p_option) {
 			if (input_action==INPUT_DUPLICATE) {
 
 				r.set_orthogonal_index(selection.duplicate_rot);
-				r.rotate(Vector3(1,0,0),Math_PI/2.0);
+				r.rotate(Vector3(1,0,0),-Math_PI/2.0);
 				selection.duplicate_rot=r.get_orthogonal_index();
 				_update_duplicate_indicator();
 				break;
 			}
 
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(1,0,0),Math_PI/2.0);
+			r.rotate(Vector3(1,0,0),-Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
@@ -134,35 +134,35 @@ void  GridMapEditor::_menu_option(int p_option) {
 			if (input_action==INPUT_DUPLICATE) {
 
 				r.set_orthogonal_index(selection.duplicate_rot);
-				r.rotate(Vector3(0,0,1),Math_PI/2.0);
+				r.rotate(Vector3(0,0,1),-Math_PI/2.0);
 				selection.duplicate_rot=r.get_orthogonal_index();
 				_update_duplicate_indicator();
 				break;
 			}
 
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(0,0,1),Math_PI/2.0);
+			r.rotate(Vector3(0,0,1),-Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
 		case MENU_OPTION_CURSOR_BACK_ROTATE_Y: {
 			Matrix3 r;
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(0,1,0),-Math_PI/2.0);
+			r.rotate(Vector3(0,1,0),Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
 		case MENU_OPTION_CURSOR_BACK_ROTATE_X: {
 			Matrix3 r;
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(1,0,0),-Math_PI/2.0);
+			r.rotate(Vector3(1,0,0),Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
 		case MENU_OPTION_CURSOR_BACK_ROTATE_Z: {
 			Matrix3 r;
 			r.set_orthogonal_index(cursor_rot);
-			r.rotate(Vector3(0,0,1),-Math_PI/2.0);
+			r.rotate(Vector3(0,0,1),Math_PI/2.0);
 			cursor_rot=r.get_orthogonal_index();
 			_update_cursor_transform();
 		} break;
diff --git a/scene/3d/character_camera.cpp b/scene/3d/character_camera.cpp
index e8b7759a98..b4cd46bd35 100644
--- a/scene/3d/character_camera.cpp
+++ b/scene/3d/character_camera.cpp
@@ -255,8 +255,8 @@ void CharacterCamera::_compute_camera() {
 			orbit.x=max_orbit_x;
 
 		Matrix3 m;
-		m.rotate(Vector3(0,1,0),Math::deg2rad(orbit.y));
-		m.rotate(Vector3(1,0,0),Math::deg2rad(orbit.x));
+		m.rotate(Vector3(0,1,0),-Math::deg2rad(orbit.y));
+		m.rotate(Vector3(1,0,0),-Math::deg2rad(orbit.x));
 
 		new_pos = (m.get_axis(2) * distance) + character_pos;
 
@@ -432,8 +432,8 @@ void CharacterCamera::set_orbit(const Vector2& p_orbit) {
 		float d = char_pos.distance_to(follow_pos);
 
 		Matrix3 m;
-		m.rotate(Vector3(0,1,0),orbit.y);
-		m.rotate(Vector3(1,0,0),orbit.x);
+		m.rotate(Vector3(0,1,0),-orbit.y);
+		m.rotate(Vector3(1,0,0),-orbit.x);
 
 		follow_pos=char_pos + m.get_axis(2) * d;
 
@@ -475,8 +475,8 @@ void CharacterCamera::rotate_orbit(const Vector2& p_relative) {
 	if (type == CAMERA_FOLLOW && is_inside_scene()) {
 
 		Matrix3 m;
-		m.rotate(Vector3(0,1,0),Math::deg2rad(p_relative.y));
-		m.rotate(Vector3(1,0,0),Math::deg2rad(p_relative.x));
+		m.rotate(Vector3(0,1,0),-Math::deg2rad(p_relative.y));
+		m.rotate(Vector3(1,0,0),-Math::deg2rad(p_relative.x));
 
 		Vector3 char_pos = get_global_transform().origin;
 		char_pos.y+=height;
diff --git a/tools/collada/collada.cpp b/tools/collada/collada.cpp
index c62affe5ad..204de45082 100644
--- a/tools/collada/collada.cpp
+++ b/tools/collada/collada.cpp
@@ -144,7 +144,7 @@ Transform Collada::Node::compute_transform(Collada &state) const {
 			case XForm::OP_ROTATE: {
 				if (xf.data.size()>=4) {
 
-					xform_step.rotate(Vector3(xf.data[0],xf.data[1],xf.data[2]),-Math::deg2rad(xf.data[3]));
+					xform_step.rotate(Vector3(xf.data[0],xf.data[1],xf.data[2]),Math::deg2rad(xf.data[3]));
 				}
 			} break;
 			case XForm::OP_SCALE: {
@@ -1604,7 +1604,7 @@ Collada::Node* Collada::_parse_visual_instance_camera(XMLParser& parser) {
 	cam->camera= _uri_to_id(parser.get_attribute_value_safe("url"));
 
 	if (state.up_axis==Vector3::AXIS_Z) //collada weirdness
-		cam->post_transform.basis.rotate(Vector3(1,0,0),Math_PI*0.5);
+		cam->post_transform.basis.rotate(Vector3(1,0,0),-Math_PI*0.5);
 
 	if (parser.is_empty()) //nothing else to parse...
 		return cam;
@@ -1625,7 +1625,7 @@ Collada::Node* Collada::_parse_visual_instance_light(XMLParser& parser) {
 	cam->light= _uri_to_id(parser.get_attribute_value_safe("url"));
 
 	if (state.up_axis==Vector3::AXIS_Z) //collada weirdness
-		cam->post_transform.basis.rotate(Vector3(1,0,0),Math_PI*0.5);
+		cam->post_transform.basis.rotate(Vector3(1,0,0),-Math_PI*0.5);
 
 	if (parser.is_empty()) //nothing else to parse...
 		return cam;
diff --git a/tools/editor/plugins/cube_grid_theme_editor_plugin.cpp b/tools/editor/plugins/cube_grid_theme_editor_plugin.cpp
index 0c2ec15367..a5f447cfd9 100644
--- a/tools/editor/plugins/cube_grid_theme_editor_plugin.cpp
+++ b/tools/editor/plugins/cube_grid_theme_editor_plugin.cpp
@@ -179,8 +179,8 @@ void MeshLibraryEditor::_import_scene(Node *p_scene, Ref<MeshLibrary> p_library,
 			Vector3 ofs = aabb.pos + aabb.size*0.5;
 			aabb.pos-=ofs;
 			Transform xform;
-			xform.basis=Matrix3().rotated(Vector3(0,1,0),Math_PI*0.25);
-			xform.basis = Matrix3().rotated(Vector3(1,0,0),-Math_PI*0.25)*xform.basis;
+			xform.basis=Matrix3().rotated(Vector3(0,1,0),-Math_PI*0.25);
+			xform.basis = Matrix3().rotated(Vector3(1,0,0),Math_PI*0.25)*xform.basis;
 			AABB rot_aabb = xform.xform(aabb);
 			print_line("rot_aabb: "+rot_aabb);
 			float m = MAX(rot_aabb.size.x,rot_aabb.size.y)*0.5;
diff --git a/tools/editor/plugins/editor_preview_plugins.cpp b/tools/editor/plugins/editor_preview_plugins.cpp
index fbda0776b0..c07eacf69e 100644
--- a/tools/editor/plugins/editor_preview_plugins.cpp
+++ b/tools/editor/plugins/editor_preview_plugins.cpp
@@ -807,8 +807,8 @@ Ref<Texture> EditorMeshPreviewPlugin::generate(const RES& p_from) {
 	Vector3 ofs = aabb.pos + aabb.size*0.5;
 	aabb.pos-=ofs;
 	Transform xform;
-	xform.basis=Matrix3().rotated(Vector3(0,1,0),Math_PI*0.125);
-	xform.basis = Matrix3().rotated(Vector3(1,0,0),-Math_PI*0.125)*xform.basis;
+	xform.basis=Matrix3().rotated(Vector3(0,1,0),-Math_PI*0.125);
+	xform.basis = Matrix3().rotated(Vector3(1,0,0),Math_PI*0.125)*xform.basis;
 	AABB rot_aabb = xform.xform(aabb);
 	float m = MAX(rot_aabb.size.x,rot_aabb.size.y)*0.5;
 	if (m==0)
diff --git a/tools/editor/plugins/material_editor_plugin.cpp b/tools/editor/plugins/material_editor_plugin.cpp
index 9c279fa967..3ef1818135 100644
--- a/tools/editor/plugins/material_editor_plugin.cpp
+++ b/tools/editor/plugins/material_editor_plugin.cpp
@@ -129,8 +129,8 @@ MaterialEditor::MaterialEditor() {
 	viewport->add_child(box_instance);
 
 	Transform box_xform;
-	box_xform.basis.rotate(Vector3(1,0,0),Math::deg2rad(-25));
-	box_xform.basis = box_xform.basis * Matrix3().rotated(Vector3(0,1,0),Math::deg2rad(-25));
+	box_xform.basis.rotate(Vector3(1,0,0),Math::deg2rad(25));
+	box_xform.basis = box_xform.basis * Matrix3().rotated(Vector3(0,1,0),Math::deg2rad(25));
 	box_xform.basis.scale(Vector3(0.8,0.8,0.8));
 	box_instance->set_transform(box_xform);
 
diff --git a/tools/editor/plugins/mesh_editor_plugin.cpp b/tools/editor/plugins/mesh_editor_plugin.cpp
index 785efebfc8..13f3204a05 100644
--- a/tools/editor/plugins/mesh_editor_plugin.cpp
+++ b/tools/editor/plugins/mesh_editor_plugin.cpp
@@ -82,8 +82,8 @@ void MeshEditor::_notification(int p_what) {
 void MeshEditor::_update_rotation() {
 
 	Transform t;
-	t.basis.rotate(Vector3(0, 1, 0), rot_y);
-	t.basis.rotate(Vector3(1, 0, 0), rot_x);
+	t.basis.rotate(Vector3(0, 1, 0), -rot_y);
+	t.basis.rotate(Vector3(1, 0, 0), -rot_x);
 	mesh_instance->set_transform(t);
 
 }
diff --git a/tools/editor/plugins/multimesh_editor_plugin.cpp b/tools/editor/plugins/multimesh_editor_plugin.cpp
index 22981a2877..a548f26f1e 100644
--- a/tools/editor/plugins/multimesh_editor_plugin.cpp
+++ b/tools/editor/plugins/multimesh_editor_plugin.cpp
@@ -207,10 +207,10 @@ void MultiMeshEditor::_populate() {
 
 	Transform axis_xform;
 	if (axis==Vector3::AXIS_Z) {
-		axis_xform.rotate(Vector3(1,0,0),Math_PI*0.5);
+		axis_xform.rotate(Vector3(1,0,0),-Math_PI*0.5);
 	}
 	if (axis==Vector3::AXIS_X) {
-		axis_xform.rotate(Vector3(0,0,1),Math_PI*0.5);
+		axis_xform.rotate(Vector3(0,0,1),-Math_PI*0.5);
 	}
 
 	for(int i=0;i<instance_count;i++) {
@@ -238,9 +238,9 @@ void MultiMeshEditor::_populate() {
 
 		Matrix3 post_xform;
 
-		post_xform.rotate(xform.basis.get_axis(0),Math::random(-_tilt_random,_tilt_random)*Math_PI);
-		post_xform.rotate(xform.basis.get_axis(2),Math::random(-_tilt_random,_tilt_random)*Math_PI);
-		post_xform.rotate(xform.basis.get_axis(1),Math::random(-_rotate_random,_rotate_random)*Math_PI);
+		post_xform.rotate(xform.basis.get_axis(0),-Math::random(-_tilt_random,_tilt_random)*Math_PI);
+		post_xform.rotate(xform.basis.get_axis(2),-Math::random(-_tilt_random,_tilt_random)*Math_PI);
+		post_xform.rotate(xform.basis.get_axis(1),-Math::random(-_rotate_random,_rotate_random)*Math_PI);
 		xform.basis = post_xform * xform.basis;
 		//xform.basis.orthonormalize();
 
diff --git a/tools/editor/plugins/spatial_editor_plugin.cpp b/tools/editor/plugins/spatial_editor_plugin.cpp
index 44d3b63335..70a256fd2c 100644
--- a/tools/editor/plugins/spatial_editor_plugin.cpp
+++ b/tools/editor/plugins/spatial_editor_plugin.cpp
@@ -61,8 +61,8 @@ void SpatialEditorViewport::_update_camera() {
 
 	Transform camera_transform;
 	camera_transform.translate(cursor.pos);
-	camera_transform.basis.rotate(Vector3(0, 1, 0), cursor.y_rot);
-	camera_transform.basis.rotate(Vector3(1, 0, 0), cursor.x_rot);
+	camera_transform.basis.rotate(Vector3(0, 1, 0), -cursor.y_rot);
+	camera_transform.basis.rotate(Vector3(1, 0, 0), -cursor.x_rot);
 
 	if (orthogonal)
 		camera_transform.translate(0, 0, 4096);
@@ -474,8 +474,8 @@ Vector3 SpatialEditorViewport::_get_screen_to_space(const Vector3& p_pos) {
 
 	Transform camera_transform;
 	camera_transform.translate( cursor.pos );
-	camera_transform.basis.rotate(Vector3(0,1,0),cursor.y_rot);
-	camera_transform.basis.rotate(Vector3(1,0,0),cursor.x_rot);
+	camera_transform.basis.rotate(Vector3(0,1,0),-cursor.y_rot);
+	camera_transform.basis.rotate(Vector3(1,0,0),-cursor.x_rot);
 	camera_transform.translate(0,0,cursor.distance);
 
 	return camera_transform.xform(Vector3( ((p_pos.x/get_size().width)*2.0-1.0)*screen_w, ((1.0-(p_pos.y/get_size().height))*2.0-1.0)*screen_h,-get_znear()));
@@ -1484,7 +1484,7 @@ void SpatialEditorViewport::_sinput(const InputEvent &p_event) {
 
 
 							Transform r;
-							r.basis.rotate(plane.normal,-angle);
+							r.basis.rotate(plane.normal,angle);
 
 							List<Node*> &selection = editor_selection->get_selected_node_list();
 
@@ -1591,8 +1591,8 @@ void SpatialEditorViewport::_sinput(const InputEvent &p_event) {
 					Transform camera_transform;
 
 					camera_transform.translate(cursor.pos);
-					camera_transform.basis.rotate(Vector3(0,1,0),cursor.y_rot);
-					camera_transform.basis.rotate(Vector3(1,0,0),cursor.x_rot);
+					camera_transform.basis.rotate(Vector3(0,1,0),-cursor.y_rot);
+					camera_transform.basis.rotate(Vector3(1,0,0),-cursor.x_rot);
 					Vector3 translation(-m.relative_x*pan_speed,m.relative_y*pan_speed,0);
 					translation*=cursor.distance/DISTANCE_DEFAULT;
 					camera_transform.translate(translation);
@@ -2810,7 +2810,7 @@ void SpatialEditor::_xform_dialog_action() {
 			continue;
 		Vector3 axis;
 		axis[i]=1.0;
-		t.basis.rotate(axis,rotate[i]);
+		t.basis.rotate(axis,rotate[i]); // BUG(?): Angle not flipped; please check during the review of PR #6865.
 	}
 
 	for(int i=0;i<3;i++) {
@@ -3160,7 +3160,7 @@ void SpatialEditor::_init_indicators() {
 
 
 
-	light_transform.rotate(Vector3(1,0,0),Math_PI/5.0);
+	light_transform.rotate(Vector3(1,0,0),-Math_PI/5.0);
 	VisualServer::get_singleton()->instance_set_transform(light_instance,light_transform);
 
 
@@ -3773,8 +3773,8 @@ void SpatialEditor::_update_ambient_light_color(const Color& p_color) {
 void SpatialEditor::_update_default_light_angle() {
 
 	Transform t;
-	t.basis.rotate(Vector3(1,0,0),settings_default_light_rot_x);
-	t.basis.rotate(Vector3(0,1,0),settings_default_light_rot_y);
+	t.basis.rotate(Vector3(1,0,0),-settings_default_light_rot_x);
+	t.basis.rotate(Vector3(0,1,0),-settings_default_light_rot_y);
 	settings_dlight->set_transform(t);
 	if (light_instance.is_valid()) {
 		VS::get_singleton()->instance_set_transform(light_instance,t);