Added various functions basic math classes. Also enabled math checks only for debug builds.

Added set_scale, set_rotation_euler, set_rotation_axis_angle. Addresses #2565 directly.
Added an euler angle constructor for Basis in GDScript and also exposed is_normalized for vectors and quaternions.
Various other changes mostly cosmetic in nature.

13 files changed, 216 insertions, 49 deletions

diff --git a/core/math/math_2d.cpp b/core/math/math_2d.cpp
index 77cff6a052..c6f8e41ae5 100644
--- a/core/math/math_2d.cpp
+++ b/core/math/math_2d.cpp
@@ -62,7 +62,8 @@ Vector2 Vector2::normalized() const {
 }
 
 bool Vector2::is_normalized() const {
-	return Math::isequal_approx(length(), (real_t)1.0);
+	// use length_squared() instead of length() to avoid sqrt(), makes it more stringent.
+	return Math::is_equal_approx(length_squared(), 1.0);
 }
 
 real_t Vector2::distance_to(const Vector2 &p_vector2) const {
@@ -280,7 +281,7 @@ Vector2 Vector2::cubic_interpolate(const Vector2 &p_b, const Vector2 &p_pre_a, c
 
 // slide returns the component of the vector along the given plane, specified by its normal vector.
 Vector2 Vector2::slide(const Vector2 &p_n) const {
-#ifdef DEBUG_ENABLED
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(p_n.is_normalized() == false, Vector2());
 #endif
 	return *this - p_n * this->dot(p_n);
@@ -291,7 +292,7 @@ Vector2 Vector2::bounce(const Vector2 &p_n) const {
 }
 
 Vector2 Vector2::reflect(const Vector2 &p_n) const {
-#ifdef DEBUG_ENABLED
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(p_n.is_normalized() == false, Vector2());
 #endif
 	return 2.0 * p_n * this->dot(p_n) - *this;
@@ -438,7 +439,9 @@ Transform2D Transform2D::inverse() const {
 void Transform2D::affine_invert() {
 
 	real_t det = basis_determinant();
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND(det == 0);
+#endif
 	real_t idet = 1.0 / det;
 
 	SWAP(elements[0][0], elements[1][1]);
diff --git a/core/math/math_defs.h b/core/math/math_defs.h
index 08f4e27e64..74839309d4 100644
--- a/core/math/math_defs.h
+++ b/core/math/math_defs.h
@@ -34,6 +34,10 @@
 #define CMP_NORMALIZE_TOLERANCE 0.000001
 #define CMP_POINT_IN_PLANE_EPSILON 0.00001
 
+#ifdef DEBUG_ENABLED
+#define MATH_CHECKS
+#endif
+
 #define USEC_TO_SEC(m_usec) ((m_usec) / 1000000.0)
 /**
   * "Real" is a type that will be translated to either floats or fixed depending
diff --git a/core/math/math_funcs.h b/core/math/math_funcs.h
index 3e02ac0bb8..46eeb6eca2 100644
--- a/core/math/math_funcs.h
+++ b/core/math/math_funcs.h
@@ -167,7 +167,7 @@ public:
 	static float random(float from, float to);
 	static real_t random(int from, int to) { return (real_t)random((real_t)from, (real_t)to); }
 
-	static _ALWAYS_INLINE_ bool isequal_approx(real_t a, real_t b) {
+	static _ALWAYS_INLINE_ bool is_equal_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.
diff --git a/core/math/matrix3.cpp b/core/math/matrix3.cpp
index b31df2fadb..898a569cbf 100644
--- a/core/math/matrix3.cpp
+++ b/core/math/matrix3.cpp
@@ -61,8 +61,9 @@ void Basis::invert() {
 	real_t det = elements[0][0] * co[0] +
 				 elements[0][1] * co[1] +
 				 elements[0][2] * co[2];
-
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND(det == 0);
+#endif
 	real_t s = 1.0 / det;
 
 	set(co[0] * s, cofac(0, 2, 2, 1) * s, cofac(0, 1, 1, 2) * s,
@@ -71,8 +72,9 @@ void Basis::invert() {
 }
 
 void Basis::orthonormalize() {
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND(determinant() == 0);
-
+#endif
 	// Gram-Schmidt Process
 
 	Vector3 x = get_axis(0);
@@ -101,20 +103,20 @@ bool Basis::is_orthogonal() const {
 	Basis id;
 	Basis m = (*this) * transposed();
 
-	return isequal_approx(id, m);
+	return is_equal_approx(id, m);
 }
 
 bool Basis::is_rotation() const {
-	return Math::isequal_approx(determinant(), 1) && is_orthogonal();
+	return Math::is_equal_approx(determinant(), 1) && is_orthogonal();
 }
 
 bool Basis::is_symmetric() const {
 
-	if (Math::abs(elements[0][1] - elements[1][0]) > CMP_EPSILON)
+	if (!Math::is_equal_approx(elements[0][1], elements[1][0]))
 		return false;
-	if (Math::abs(elements[0][2] - elements[2][0]) > CMP_EPSILON)
+	if (!Math::is_equal_approx(elements[0][2], elements[2][0]))
 		return false;
-	if (Math::abs(elements[1][2] - elements[2][1]) > CMP_EPSILON)
+	if (!Math::is_equal_approx(elements[1][2], elements[2][1]))
 		return false;
 
 	return true;
@@ -122,11 +124,11 @@ bool Basis::is_symmetric() const {
 
 Basis Basis::diagonalize() {
 
-	//NOTE: only implemented for symmetric matrices
-	//with the Jacobi iterative method method
-
+//NOTE: only implemented for symmetric matrices
+//with the Jacobi iterative method method
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(!is_symmetric(), Basis());
-
+#endif
 	const int ite_max = 1024;
 
 	real_t off_matrix_norm_2 = elements[0][1] * elements[0][1] + elements[0][2] * elements[0][2] + elements[1][2] * elements[1][2];
@@ -159,7 +161,7 @@ Basis Basis::diagonalize() {
 
 		// Compute the rotation angle
 		real_t angle;
-		if (Math::abs(elements[j][j] - elements[i][i]) < CMP_EPSILON) {
+		if (Math::is_equal_approx(elements[j][j], elements[i][i])) {
 			angle = Math_PI / 4;
 		} else {
 			angle = 0.5 * Math::atan(2 * elements[i][j] / (elements[j][j] - elements[i][i]));
@@ -225,11 +227,25 @@ Basis Basis::scaled(const Vector3 &p_scale) const {
 }
 
 Vector3 Basis::get_scale() const {
-	// We are assuming M = R.S, and performing a polar decomposition to extract R and S.
-	// FIXME: We eventually need a proper polar decomposition.
-	// As a cheap workaround until then, to ensure that R is a proper rotation matrix with determinant +1
-	// (such that it can be represented by a Quat or Euler angles), we absorb the sign flip into the scaling matrix.
-	// As such, it works in conjunction with get_rotation().
+	// FIXME: We are assuming M = R.S (R is rotation and S is scaling), and use polar decomposition to extract R and S.
+	// A polar decomposition is M = O.P, where O is an orthogonal matrix (meaning rotation and reflection) and
+	// P is a positive semi-definite matrix (meaning it contains absolute values of scaling along its diagonal).
+	//
+	// Despite being different from what we want to achieve, we can nevertheless make use of polar decomposition
+	// here as follows. We can split O into a rotation and a reflection as O = R.Q, and obtain M = R.S where
+	// we defined S = Q.P. Now, R is a proper rotation matrix and S is a (signed) scaling matrix,
+	// which can involve negative scalings. However, there is a catch: unlike the polar decomposition of M = O.P,
+	// the decomposition of O into a rotation and reflection matrix as O = R.Q is not unique.
+	// Therefore, we are going to do this decomposition by sticking to a particular convention.
+	// This may lead to confusion for some users though.
+	//
+	// The convention we use here is to absorb the sign flip into the scaling matrix.
+	// The same convention is also used in other similar functions such as set_scale,
+	// get_rotation_axis_angle, get_rotation, set_rotation_axis_angle, set_rotation_euler, ...
+	//
+	// A proper way to get rid of this issue would be to store the scaling values (or at least their signs)
+	// as a part of Basis. However, if we go that path, we need to disable direct (write) access to the
+	// matrix elements.
 	real_t det_sign = determinant() > 0 ? 1 : -1;
 	return det_sign * Vector3(
 							  Vector3(elements[0][0], elements[1][0], elements[2][0]).length(),
@@ -237,6 +253,17 @@ Vector3 Basis::get_scale() const {
 							  Vector3(elements[0][2], elements[1][2], elements[2][2]).length());
 }
 
+// Sets scaling while preserving rotation.
+// This requires some care when working with matrices with negative determinant,
+// since we're using a particular convention for "polar" decomposition in get_scale and get_rotation.
+// For details, see the explanation in get_scale.
+void Basis::set_scale(const Vector3 &p_scale) {
+	Vector3 e = get_euler();
+	Basis(); // reset to identity
+	scale(p_scale);
+	rotate(e);
+}
+
 // Multiplies the matrix from left by the rotation matrix: M -> R.M
 // Note that this does *not* rotate the matrix itself.
 //
@@ -259,6 +286,7 @@ void Basis::rotate(const Vector3 &p_euler) {
 	*this = rotated(p_euler);
 }
 
+// TODO: rename this to get_rotation_euler
 Vector3 Basis::get_rotation() const {
 	// Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
 	// and returns the Euler angles corresponding to the rotation part, complementing get_scale().
@@ -273,6 +301,42 @@ Vector3 Basis::get_rotation() const {
 	return m.get_euler();
 }
 
+void Basis::get_rotation_axis_angle(Vector3 &p_axis, real_t &p_angle) const {
+	// Assumes that the matrix can be decomposed into a proper rotation and scaling matrix as M = R.S,
+	// and returns the Euler angles corresponding to the rotation part, complementing get_scale().
+	// See the comment in get_scale() for further information.
+	Basis m = orthonormalized();
+	real_t det = m.determinant();
+	if (det < 0) {
+		// Ensure that the determinant is 1, such that result is a proper rotation matrix which can be represented by Euler angles.
+		m.scale(Vector3(-1, -1, -1));
+	}
+
+	m.get_axis_angle(p_axis, p_angle);
+}
+
+// Sets rotation while preserving scaling.
+// This requires some care when working with matrices with negative determinant,
+// since we're using a particular convention for "polar" decomposition in get_scale and get_rotation.
+// For details, see the explanation in get_scale.
+void Basis::set_rotation_euler(const Vector3 &p_euler) {
+	Vector3 s = get_scale();
+	Basis(); // reset to identity
+	scale(s);
+	rotate(p_euler);
+}
+
+// Sets rotation while preserving scaling.
+// This requires some care when working with matrices with negative determinant,
+// since we're using a particular convention for "polar" decomposition in get_scale and get_rotation.
+// For details, see the explanation in get_scale.
+void Basis::set_rotation_axis_angle(const Vector3 &p_axis, real_t p_angle) {
+	Vector3 s = get_scale();
+	Basis(); // reset to identity
+	scale(s);
+	rotate(p_axis, p_angle);
+}
+
 // 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).
@@ -293,9 +357,9 @@ Vector3 Basis::get_euler() const {
 	//       -cx*cz*sy+sx*sz  cz*sx+cx*sy*sz  cx*cy
 
 	Vector3 euler;
-
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(is_rotation() == false, euler);
-
+#endif
 	euler.y = Math::asin(elements[0][2]);
 	if (euler.y < Math_PI * 0.5) {
 		if (euler.y > -Math_PI * 0.5) {
@@ -339,11 +403,11 @@ void Basis::set_euler(const Vector3 &p_euler) {
 	*this = xmat * (ymat * zmat);
 }
 
-bool Basis::isequal_approx(const Basis &a, const Basis &b) const {
+bool Basis::is_equal_approx(const Basis &a, const Basis &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)
+			if (Math::is_equal_approx(a.elements[i][j], b.elements[i][j]) == false)
 				return false;
 		}
 	}
@@ -386,8 +450,9 @@ Basis::operator String() const {
 }
 
 Basis::operator Quat() const {
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(is_rotation() == false, Quat());
-
+#endif
 	real_t trace = elements[0][0] + elements[1][1] + elements[2][2];
 	real_t temp[4];
 
@@ -481,9 +546,10 @@ void Basis::set_orthogonal_index(int p_index) {
 	*this = _ortho_bases[p_index];
 }
 
-void Basis::get_axis_and_angle(Vector3 &r_axis, real_t &r_angle) const {
+void Basis::get_axis_angle(Vector3 &r_axis, real_t &r_angle) const {
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND(is_rotation() == false);
-
+#endif
 	real_t angle, x, y, z; // variables for result
 	real_t epsilon = 0.01; // margin to allow for rounding errors
 	real_t epsilon2 = 0.1; // margin to distinguish between 0 and 180 degrees
@@ -572,11 +638,11 @@ Basis::Basis(const Quat &p_quat) {
 			xz - wy, yz + wx, 1.0 - (xx + yy));
 }
 
-Basis::Basis(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
-
+void Basis::set_axis_angle(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_angle
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND(p_axis.is_normalized() == false);
-
+#endif
 	Vector3 axis_sq(p_axis.x * p_axis.x, p_axis.y * p_axis.y, p_axis.z * p_axis.z);
 
 	real_t cosine = Math::cos(p_phi);
@@ -594,3 +660,7 @@ Basis::Basis(const Vector3 &p_axis, real_t p_phi) {
 	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);
 }
+
+Basis::Basis(const Vector3 &p_axis, real_t p_phi) {
+	set_axis_angle(p_axis, p_phi);
+}
diff --git a/core/math/matrix3.h b/core/math/matrix3.h
index 0240bc8610..ddf473f167 100644
--- a/core/math/matrix3.h
+++ b/core/math/matrix3.h
@@ -76,15 +76,25 @@ public:
 
 	void rotate(const Vector3 &p_euler);
 	Basis rotated(const Vector3 &p_euler) const;
+
 	Vector3 get_rotation() const;
+	void get_rotation_axis_angle(Vector3 &p_axis, real_t &p_angle) const;
 
-	void scale(const Vector3 &p_scale);
-	Basis scaled(const Vector3 &p_scale) const;
-	Vector3 get_scale() const;
+	void set_rotation_euler(const Vector3 &p_euler);
+	void set_rotation_axis_angle(const Vector3 &p_axis, real_t p_angle);
 
 	Vector3 get_euler() const;
 	void set_euler(const Vector3 &p_euler);
 
+	void get_axis_angle(Vector3 &r_axis, real_t &r_angle) const;
+	void set_axis_angle(const Vector3 &p_axis, real_t p_phi);
+
+	void scale(const Vector3 &p_scale);
+	Basis scaled(const Vector3 &p_scale) const;
+
+	Vector3 get_scale() const;
+	void set_scale(const Vector3 &p_scale);
+
 	// transposed dot products
 	_FORCE_INLINE_ real_t tdotx(const Vector3 &v) const {
 		return elements[0][0] * v[0] + elements[1][0] * v[1] + elements[2][0] * v[2];
@@ -96,7 +106,7 @@ public:
 		return elements[0][2] * v[0] + elements[1][2] * v[1] + elements[2][2] * v[2];
 	}
 
-	bool isequal_approx(const Basis &a, const Basis &b) const;
+	bool is_equal_approx(const Basis &a, const Basis &b) const;
 
 	bool operator==(const Basis &p_matrix) const;
 	bool operator!=(const Basis &p_matrix) const;
@@ -120,8 +130,6 @@ public:
 
 	operator String() const;
 
-	void get_axis_and_angle(Vector3 &r_axis, real_t &r_angle) const;
-
 	/* create / set */
 
 	_FORCE_INLINE_ void set(real_t xx, real_t xy, real_t xz, real_t yx, real_t yy, real_t yz, real_t zx, real_t zy, real_t zz) {
diff --git a/core/math/quat.cpp b/core/math/quat.cpp
index b990e9184f..6c06317cf1 100644
--- a/core/math/quat.cpp
+++ b/core/math/quat.cpp
@@ -91,6 +91,10 @@ Quat Quat::normalized() const {
 	return *this / length();
 }
 
+bool Quat::is_normalized() const {
+	return Math::is_equal_approx(length(), 1.0);
+}
+
 Quat Quat::inverse() const {
 	return Quat(-x, -y, -z, w);
 }
diff --git a/core/math/quat.h b/core/math/quat.h
index 3fc843b83a..dca1c8701f 100644
--- a/core/math/quat.h
+++ b/core/math/quat.h
@@ -47,6 +47,7 @@ public:
 	real_t length() const;
 	void normalize();
 	Quat normalized() const;
+	bool is_normalized() const;
 	Quat inverse() const;
 	_FORCE_INLINE_ real_t dot(const Quat &q) const;
 	void set_euler(const Vector3 &p_euler);
@@ -55,7 +56,7 @@ public:
 	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 {
+	_FORCE_INLINE_ void get_axis_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);
diff --git a/core/math/vector3.h b/core/math/vector3.h
index 8550ae7009..22d45400b1 100644
--- a/core/math/vector3.h
+++ b/core/math/vector3.h
@@ -388,7 +388,8 @@ Vector3 Vector3::normalized() const {
 }
 
 bool Vector3::is_normalized() const {
-	return Math::isequal_approx(length(), (real_t)1.0);
+	// use length_squared() instead of length() to avoid sqrt(), makes it more stringent.
+	return Math::is_equal_approx(length_squared(), 1.0);
 }
 
 Vector3 Vector3::inverse() const {
@@ -403,7 +404,7 @@ void Vector3::zero() {
 
 // slide returns the component of the vector along the given plane, specified by its normal vector.
 Vector3 Vector3::slide(const Vector3 &p_n) const {
-#ifdef DEBUG_ENABLED
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(p_n.is_normalized() == false, Vector3());
 #endif
 	return *this - p_n * this->dot(p_n);
@@ -414,7 +415,7 @@ Vector3 Vector3::bounce(const Vector3 &p_n) const {
 }
 
 Vector3 Vector3::reflect(const Vector3 &p_n) const {
-#ifdef DEBUG_ENABLED
+#ifdef MATH_CHECKS
 	ERR_FAIL_COND_V(p_n.is_normalized() == false, Vector3());
 #endif
 	return 2.0 * p_n * this->dot(p_n) - *this;
diff --git a/core/variant_call.cpp b/core/variant_call.cpp
index 9c435ea0e5..b8b2651068 100644
--- a/core/variant_call.cpp
+++ b/core/variant_call.cpp
@@ -327,6 +327,7 @@ struct _VariantCall {
 	VCALL_LOCALMEM0R(Vector2, normalized);
 	VCALL_LOCALMEM0R(Vector2, length);
 	VCALL_LOCALMEM0R(Vector2, length_squared);
+	VCALL_LOCALMEM0R(Vector2, is_normalized);
 	VCALL_LOCALMEM1R(Vector2, distance_to);
 	VCALL_LOCALMEM1R(Vector2, distance_squared_to);
 	VCALL_LOCALMEM1R(Vector2, angle_to);
@@ -361,6 +362,7 @@ struct _VariantCall {
 	VCALL_LOCALMEM0R(Vector3, max_axis);
 	VCALL_LOCALMEM0R(Vector3, length);
 	VCALL_LOCALMEM0R(Vector3, length_squared);
+	VCALL_LOCALMEM0R(Vector3, is_normalized);
 	VCALL_LOCALMEM0R(Vector3, normalized);
 	VCALL_LOCALMEM0R(Vector3, inverse);
 	VCALL_LOCALMEM1R(Vector3, snapped);
@@ -417,6 +419,7 @@ struct _VariantCall {
 	VCALL_LOCALMEM0R(Quat, length);
 	VCALL_LOCALMEM0R(Quat, length_squared);
 	VCALL_LOCALMEM0R(Quat, normalized);
+	VCALL_LOCALMEM0R(Quat, is_normalized);
 	VCALL_LOCALMEM0R(Quat, inverse);
 	VCALL_LOCALMEM1R(Quat, dot);
 	VCALL_LOCALMEM1R(Quat, xform);
@@ -703,6 +706,9 @@ struct _VariantCall {
 	VCALL_PTR1R(Basis, scaled);
 	VCALL_PTR0R(Basis, get_scale);
 	VCALL_PTR0R(Basis, get_euler);
+	VCALL_PTR1(Basis, set_scale);
+	VCALL_PTR1(Basis, set_rotation_euler);
+	VCALL_PTR2(Basis, set_rotation_axis_angle);
 	VCALL_PTR1R(Basis, tdotx);
 	VCALL_PTR1R(Basis, tdoty);
 	VCALL_PTR1R(Basis, tdotz);
@@ -874,6 +880,11 @@ struct _VariantCall {
 		r_ret = Basis(p_args[0]->operator Vector3(), p_args[1]->operator real_t());
 	}
 
+	static void Basis_init3(Variant &r_ret, const Variant **p_args) {
+
+		r_ret = Basis(p_args[0]->operator Vector3());
+	}
+
 	static void Transform_init1(Variant &r_ret, const Variant **p_args) {
 
 		Transform t;
@@ -1428,6 +1439,7 @@ void register_variant_methods() {
 	ADDFUNC0(VECTOR2, REAL, Vector2, length, varray());
 	ADDFUNC0(VECTOR2, REAL, Vector2, angle, varray());
 	ADDFUNC0(VECTOR2, REAL, Vector2, length_squared, varray());
+	ADDFUNC0(VECTOR2, BOOL, Vector2, is_normalized, varray());
 	ADDFUNC1(VECTOR2, REAL, Vector2, distance_to, VECTOR2, "to", varray());
 	ADDFUNC1(VECTOR2, REAL, Vector2, distance_squared_to, VECTOR2, "to", varray());
 	ADDFUNC1(VECTOR2, REAL, Vector2, angle_to, VECTOR2, "to", varray());
@@ -1461,6 +1473,7 @@ void register_variant_methods() {
 	ADDFUNC0(VECTOR3, INT, Vector3, max_axis, varray());
 	ADDFUNC0(VECTOR3, REAL, Vector3, length, varray());
 	ADDFUNC0(VECTOR3, REAL, Vector3, length_squared, varray());
+	ADDFUNC0(VECTOR3, BOOL, Vector3, is_normalized, varray());
 	ADDFUNC0(VECTOR3, VECTOR3, Vector3, normalized, varray());
 	ADDFUNC0(VECTOR3, VECTOR3, Vector3, inverse, varray());
 	ADDFUNC1(VECTOR3, VECTOR3, Vector3, snapped, REAL, "by", varray());
@@ -1496,6 +1509,7 @@ void register_variant_methods() {
 	ADDFUNC0(QUAT, REAL, Quat, length, varray());
 	ADDFUNC0(QUAT, REAL, Quat, length_squared, varray());
 	ADDFUNC0(QUAT, QUAT, Quat, normalized, varray());
+	ADDFUNC0(QUAT, BOOL, Quat, is_normalized, varray());
 	ADDFUNC0(QUAT, QUAT, Quat, inverse, varray());
 	ADDFUNC1(QUAT, REAL, Quat, dot, QUAT, "b", varray());
 	ADDFUNC1(QUAT, VECTOR3, Quat, xform, VECTOR3, "v", varray());
@@ -1691,6 +1705,9 @@ void register_variant_methods() {
 	ADDFUNC0(BASIS, REAL, Basis, determinant, varray());
 	ADDFUNC2(BASIS, BASIS, Basis, rotated, VECTOR3, "axis", REAL, "phi", varray());
 	ADDFUNC1(BASIS, BASIS, Basis, scaled, VECTOR3, "scale", varray());
+	ADDFUNC1(BASIS, NIL, Basis, set_scale, VECTOR3, "scale", varray());
+	ADDFUNC1(BASIS, NIL, Basis, set_rotation_euler, VECTOR3, "euler", varray());
+	ADDFUNC2(BASIS, NIL, Basis, set_rotation_axis_angle, VECTOR3, "axis", REAL, "angle", varray());
 	ADDFUNC0(BASIS, VECTOR3, Basis, get_scale, varray());
 	ADDFUNC0(BASIS, VECTOR3, Basis, get_euler, varray());
 	ADDFUNC1(BASIS, REAL, Basis, tdotx, VECTOR3, "with", varray());
@@ -1748,6 +1765,7 @@ void register_variant_methods() {
 
 	_VariantCall::add_constructor(_VariantCall::Basis_init1, Variant::BASIS, "x_axis", Variant::VECTOR3, "y_axis", Variant::VECTOR3, "z_axis", Variant::VECTOR3);
 	_VariantCall::add_constructor(_VariantCall::Basis_init2, Variant::BASIS, "axis", Variant::VECTOR3, "phi", Variant::REAL);
+	_VariantCall::add_constructor(_VariantCall::Basis_init3, Variant::BASIS, "euler", Variant::VECTOR3);
 
 	_VariantCall::add_constructor(_VariantCall::Transform_init1, Variant::TRANSFORM, "x_axis", Variant::VECTOR3, "y_axis", Variant::VECTOR3, "z_axis", Variant::VECTOR3, "origin", Variant::VECTOR3);
 	_VariantCall::add_constructor(_VariantCall::Transform_init2, Variant::TRANSFORM, "basis", Variant::BASIS, "origin", Variant::VECTOR3);
diff --git a/doc/base/classes.xml b/doc/base/classes.xml
index 5bb85aedc1..aa520d3869 100644
--- a/doc/base/classes.xml
+++ b/doc/base/classes.xml
@@ -6842,6 +6842,15 @@
 		<method name="Basis">
 			<return type="Basis">
 			</return>
+			<argument index="0" name="euler" type="Vector3">
+			</argument>
+			<description>
+				Create a rotation matrix (in the XYZ convention: first Z, then Y, and X last) from the specified Euler angles, given in the vector format as (third,second,first).
+			</description>
+		</method>
+		<method name="Basis">
+			<return type="Basis">
+			</return>
 			<argument index="0" name="x_axis" type="Vector3">
 			</argument>
 			<argument index="1" name="y_axis" type="Vector3">
@@ -6863,8 +6872,7 @@
 			<return type="Vector3">
 			</return>
 			<description>
-				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).
-				This function only works if the matrix represents a proper rotation.
+				Assuming that the matrix is a proper rotation matrix (orthonormal matrix with determinant +1), return Euler angles (in the XYZ convention: first Z, then Y, and X last). Returned vector contains the rotation angles in the format (third,second,first).
 			</description>
 		</method>
 		<method name="get_orthogonal_index">
@@ -6906,6 +6914,26 @@
 				Introduce an additional rotation around the given axis by phi. Only relevant when the matrix is being used as a part of [Transform]. The axis must be a normalized vector.
 			</description>
 		</method>
+		<method name="set_rotation_euler">
+			<return type="Basis">
+			</return>
+			<argument index="0" name="euler" type="Vector3">
+			</argument>
+			<description>
+				Changes only the rotation part of the [Basis] to a rotation corresponding to given Euler angles, while preserving the scaling part (as determined by get_scale).
+			</description>
+		</method>
+		<method name="set_rotation_axis_angle">
+			<return type="Basis">
+			</return>
+			<argument index="0" name="axis" type="Vector3">
+			</argument>
+			<argument index="1" name="phi" type="float">
+			</argument>
+			<description>
+				Changes only the rotation part of the [Basis] to a rotation around given axis by phi, while preserving the scaling part (as determined by get_scale).
+			</description>
+		</method>
 		<method name="scaled">
 			<return type="Basis">
 			</return>
@@ -6915,6 +6943,15 @@
 				Introduce an additional scaling specified by the given 3D scaling factor. Only relevant when the matrix is being used as a part of [Transform].
 			</description>
 		</method>
+		<method name="set_scale">
+			<return type="Basis">
+			</return>
+			<argument index="0" name="scale" type="Vector3">
+			</argument>
+			<description>
+				Changes only the scaling part of the Basis to the specified scaling, while preserving the rotation part (as determined by get_rotation).
+			</description>
+		</method>
 		<method name="tdotx">
 			<return type="float">
 			</return>
@@ -34942,6 +34979,13 @@
 				Returns a copy of the quaternion, normalized to unit length.
 			</description>
 		</method>
+		<method name="is_normalized">
+			<return type="bool">
+			</return>
+			<description>
+				Returns whether the quaternion is normalized or not.
+			</description>
+		</method>
 		<method name="slerp">
 			<return type="Quat">
 			</return>
@@ -47093,6 +47137,13 @@ do_property].
 				Returns a normalized vector to unit length.
 			</description>
 		</method>
+		<method name="is_normalized">
+			<return type="bool">
+			</return>
+			<description>
+				Returns whether the vector is normalized or not.
+			</description>
+		</method>
 		<method name="reflect">
 			<return type="Vector2">
 			</return>
@@ -47317,6 +47368,13 @@ do_property].
 				Return a copy of the normalized vector to unit length. This is the same as v / v.length().
 			</description>
 		</method>
+		<method name="is_normalized">
+			<return type="bool">
+			</return>
+			<description>
+				Returns whether the vector is normalized or not.
+			</description>
+		</method>
 		<method name="outer">
 			<return type="Basis">
 			</return>
diff --git a/main/tests/test_math.cpp b/main/tests/test_math.cpp
index c82ae1abf5..ab06349e2f 100644
--- a/main/tests/test_math.cpp
+++ b/main/tests/test_math.cpp
@@ -601,7 +601,7 @@ MainLoop *test() {
 		print_line(q3);
 
 		print_line("before v: " + v + " a: " + rtos(a));
-		q.get_axis_and_angle(v, a);
+		q.get_axis_angle(v, a);
 		print_line("after v: " + v + " a: " + rtos(a));
 	}
 
diff --git a/scene/resources/animation.cpp b/scene/resources/animation.cpp
index 4fddd55e20..2584521171 100644
--- a/scene/resources/animation.cpp
+++ b/scene/resources/animation.cpp
@@ -1760,8 +1760,8 @@ bool Animation::_transform_track_optimize_key(const TKey<TransformKey> &t0, cons
 			Vector3 v02, v01;
 			real_t a02, a01;
 
-			r02.get_axis_and_angle(v02, a02);
-			r01.get_axis_and_angle(v01, a01);
+			r02.get_axis_angle(v02, a02);
+			r01.get_axis_angle(v01, a01);
 
 			if (Math::abs(a02) > p_max_optimizable_angle)
 				return false;
diff --git a/servers/physics/body_sw.cpp b/servers/physics/body_sw.cpp
index 6b43ec31aa..e7f152767d 100644
--- a/servers/physics/body_sw.cpp
+++ b/servers/physics/body_sw.cpp
@@ -494,7 +494,7 @@ void BodySW::integrate_forces(real_t p_step) {
 		Vector3 axis;
 		real_t angle;
 
-		rot.get_axis_and_angle(axis, angle);
+		rot.get_axis_angle(axis, angle);
 		axis.normalize();
 		angular_velocity = axis.normalized() * (angle / p_step);
 
@@ -637,7 +637,7 @@ void BodySW::simulate_motion(const Transform& p_xform,real_t p_step) {
 	Vector3 axis;
 	real_t angle;
 
-	rot.get_axis_and_angle(axis,angle);
+	rot.get_axis_angle(axis,angle);
 	axis.normalize();
 	angular_velocity=axis.normalized() * (angle/p_step);
 	linear_velocity = (p_xform.origin - get_transform().origin)/p_step;