template<class T>
Magnum::Math::Matrix4 class

3D transformation matrix

Template parameters
T Underlying data type

See Operations with matrices and vectors and 2D and 3D transformations for brief introduction.

Base classes

template<std::size_t size, class T>
class Matrix
Square matrix.

Public static functions

static auto translation(const Vector3<T>& vector) -> Matrix4<T> constexpr
3D translation matrix
static auto scaling(const Vector3<T>& vector) -> Matrix4<T> constexpr
3D scaling matrix
static auto rotation(Rad<T> angle, const Vector3<T>& normalizedAxis) -> Matrix4<T>
3D rotation matrix around arbitrary axis
static auto rotationX(Rad<T> angle) -> Matrix4<T>
3D rotation matrix around X axis
static auto rotationY(Rad<T> angle) -> Matrix4<T>
3D rotation matrix around Y axis
static auto rotationZ(Rad<T> angle) -> Matrix4<T>
3D rotation matrix around Z axis
static auto reflection(const Vector3<T>& normal) -> Matrix4<T>
3D reflection matrix
static auto shearingXY(T amountX, T amountY) -> Matrix4<T> constexpr
3D shearing matrix along XY plane
static auto shearingXZ(T amountX, T amountZ) -> Matrix4<T> constexpr
3D shearing matrix along XZ plane
static auto shearingYZ(T amountY, T amountZ) -> Matrix4<T> constexpr
3D shearing matrix along YZ plane
static auto orthographicProjection(const Vector2<T>& size, T near, T far) -> Matrix4<T>
3D orthographic projection matrix
static auto perspectiveProjection(const Vector2<T>& size, T near, T far) -> Matrix4<T>
3D perspective projection matrix
static auto perspectiveProjection(Rad<T> fov, T aspectRatio, T near, T far) -> Matrix4<T>
3D perspective projection matrix
static auto lookAt(const Vector3<T>& eye, const Vector3<T>& target, const Vector3<T>& up) -> Matrix4<T>
Matrix oriented towards a specific point.
static auto from(const Matrix3x3<T>& rotationScaling, const Vector3<T>& translation) -> Matrix4<T> constexpr
Create matrix from rotation/scaling part and translation part.

Constructors, destructors, conversion operators

Matrix4(IdentityInitT = IdentityInit, T value = T{1}) constexpr noexcept
Default constructor.
Matrix4(ZeroInitT) explicit constexpr noexcept
Construct zero-filled matrix.



Matrix4(NoInitT) explicit constexpr noexcept
Construct matrix without initializing the contents.



Matrix4(const Vector4<T>& first, const Vector4<T>& second, const Vector4<T>& third, const Vector4<T>& fourth) constexpr noexcept
Construct matrix from column vectors.
Matrix4(T value) explicit constexpr noexcept
Construct matrix with one value for all elements.
template<class U>
Matrix4(const RectangularMatrix<4, 4, U>& other) explicit constexpr noexcept
Construct matrix from another of different type.
template<class U, class V = decltype(Implementation::RectangularMatrixConverter<4, 4, T, U>::from(std::declval<U>()))>
Matrix4(const U& other) explicit constexpr
Construct matrix from external representation.
template<std::size_t otherSize>
Matrix4(const RectangularMatrix<otherSize, otherSize, T>& other) explicit constexpr noexcept
Construct matrix by slicing or expanding another of a different size.
Matrix4(const RectangularMatrix<4, 4, T>& other) constexpr noexcept
Copy constructor.

Public functions

auto isRigidTransformation() const -> bool
Check whether the matrix represents rigid transformation.
auto rotationScaling() const -> Matrix3x3<T> constexpr
3D rotation and scaling part of the matrix
auto rotationShear() const -> Matrix3x3<T>
3D rotation and scaling part of the matrix
auto rotation() const -> Matrix3x3<T>
3D rotation part of the matrix
auto rotationNormalized() const -> Matrix3x3<T>
3D rotation part of the matrix assuming there is no scaling
auto scalingSquared() const -> Vector3<T>
Non-uniform scaling part of the matrix, squared.
auto scaling() const -> Vector3<T>
Non-uniform scaling part of the matrix, squared.
auto uniformScalingSquared() const -> T
Uniform scaling part of the matrix, squared.
auto uniformScaling() const -> T
Uniform scaling part of the matrix.
auto right() -> Vector3<T>&
Right-pointing 3D vector.
auto right() const -> Vector3<T> constexpr
auto up() -> Vector3<T>&
Up-pointing 3D vector.
auto up() const -> Vector3<T> constexpr
auto backward() -> Vector3<T>&
Backward-pointing 3D vector.
auto backward() const -> Vector3<T> constexpr
auto translation() -> Vector3<T>&
3D translation part of the matrix
auto translation() const -> Vector3<T> constexpr
auto invertedRigid() const -> Matrix4<T>
Inverted rigid transformation matrix.
auto transformVector(const Vector3<T>& vector) const -> Vector3<T>
Transform 3D vector with the matrix.
auto transformPoint(const Vector3<T>& vector) const -> Vector3<T>
Transform 3D point with the matrix.

Function documentation

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::translation(const Vector3<T>& vector) constexpr

3D translation matrix

Parameters
vector Translation vector
\[ \boldsymbol{A} = \begin{pmatrix} 1 & 0 & 0 & v_x \\ 0 & 1 & 0 & v_y \\ 0 & 0 & 1 & v_z \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::scaling(const Vector3<T>& vector) constexpr

3D scaling matrix

Parameters
vector Scaling vector
\[ \boldsymbol{A} = \begin{pmatrix} v_x & 0 & 0 & 0 \\ 0 & v_y & 0 & 0 \\ 0 & 0 & v_z & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::rotation(Rad<T> angle, const Vector3<T>& normalizedAxis)

3D rotation matrix around arbitrary axis

Parameters
angle Rotation angle (counterclockwise)
normalizedAxis Normalized rotation axis

Expects that the rotation axis is normalized. If possible, use faster alternatives like rotationX(), rotationY() and rotationZ().

\[ \boldsymbol{A} = \begin{pmatrix} v_{x}v_{x}(1 - \cos\theta) + \cos\theta & v_{y}v_{x}(1 - \cos\theta) - v_{z}\sin \theta & v_{z}v_{x}(1 - \cos\theta) + v_{y}\sin\theta & 0 \\ v_{x}v_{y}(1 - \cos\theta) + v_{z}\sin\theta & v_{y}v_{y}(1 - \cos\theta) + \cos\theta & v_{z}v_{y}(1 - \cos\theta) - v_{x}\sin\theta & 0 \\ v_{x}v_{z}(1 - \cos\theta) - v_{y}\sin\theta & v_{y}v_{z}(1 - \cos\theta)+v_{x}\sin\theta & v_{z}v_{z}(1 - \cos\theta) + \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::rotationX(Rad<T> angle)

3D rotation matrix around X axis

Parameters
angle Rotation angle (counterclockwise)

Faster than calling Matrix4::rotation(angle, Vector3::xAxis()).

\[ \boldsymbol{A} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & \cos\theta & -\sin\theta & 0 \\ 0 & \sin\theta & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::rotationY(Rad<T> angle)

3D rotation matrix around Y axis

Parameters
angle Rotation angle (counterclockwise)

Faster than calling Matrix4::rotation(angle, Vector3::yAxis()).

\[ \boldsymbol{A} = \begin{pmatrix} \cos\theta & 0 & \sin\theta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\theta & 0 & \cos\theta & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::rotationZ(Rad<T> angle)

3D rotation matrix around Z axis

Parameters
angle Rotation angle (counterclockwise)

Faster than calling Matrix4::rotation(angle, Vector3::zAxis()).

\[ \boldsymbol{A} = \begin{pmatrix} \cos\theta & -\sin\theta & 0 & 0 \\ \sin\theta & \cos\theta & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::reflection(const Vector3<T>& normal)

3D reflection matrix

Parameters
normal Normal of the plane through which to reflect

Expects that the normal is normalized. Reflection along axes can be done in a slightly simpler way also using scaling(), e.g. Matrix4::reflection(Vector3::yAxis()) is equivalent to Matrix4::scaling(Vector3::yScale(-1.0f)).

\[ \boldsymbol{A} = \boldsymbol{I} - 2 \boldsymbol{NN}^T ~~~~~ \boldsymbol{N} = \begin{pmatrix} n_x \\ n_y \\ n_z \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::shearingXY(T amountX, T amountY) constexpr

3D shearing matrix along XY plane

Parameters
amountX Amount of shearing along X axis
amountY Amount of shearing along Y axis

Z axis remains unchanged.

\[ \boldsymbol{A} = \begin{pmatrix} 1 & 0 & v_x & 0 \\ 0 & 1 & v_y & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::shearingXZ(T amountX, T amountZ) constexpr

3D shearing matrix along XZ plane

Parameters
amountX Amount of shearing along X axis
amountZ Amount of shearing along Z axis

Y axis remains unchanged.

\[ \boldsymbol{A} = \begin{pmatrix} 1 & v_x & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & v_z & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::shearingYZ(T amountY, T amountZ) constexpr

3D shearing matrix along YZ plane

Parameters
amountY Amount of shearing along Y axis
amountZ Amount of shearing along Z axis

X axis remains unchanged.

\[ \boldsymbol{A} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ v_y & 1 & 0 & 0 \\ v_z & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::orthographicProjection(const Vector2<T>& size, T near, T far)

3D orthographic projection matrix

Parameters
size Size of the view
near Distance to near clipping plane, positive is ahead
far Distance to far clipping plane, positive is ahead
\[ \boldsymbol{A} = \begin{pmatrix} \frac{2}{s_x} & 0 & 0 & 0 \\ 0 & \frac{2}{s_y} & 0 & 0 \\ 0 & 0 & \frac{2}{n - f} & \frac{n + f}{n - f} \\ 0 & 0 & 0 & 1 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::perspectiveProjection(const Vector2<T>& size, T near, T far)

3D perspective projection matrix

Parameters
size Size of near clipping plane
near Distance to near clipping plane, positive is ahead
far Distance to far clipping plane, positive is ahead

If far is finite, the result is:

\[ \boldsymbol{A} = \begin{pmatrix} \frac{2n}{s_x} & 0 & 0 & 0 \\ 0 & \frac{2n}{s_y} & 0 & 0 \\ 0 & 0 & \frac{n + f}{n - f} & \frac{2nf}{n - f} \\ 0 & 0 & -1 & 0 \end{pmatrix} \]

For infinite far, the result is:

\[ \boldsymbol{A} = \begin{pmatrix} \frac{2n}{s_x} & 0 & 0 & 0 \\ 0 & \frac{2n}{s_y} & 0 & 0 \\ 0 & 0 & -1 & -2n \\ 0 & 0 & -1 & 0 \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::perspectiveProjection(Rad<T> fov, T aspectRatio, T near, T far)

3D perspective projection matrix

Parameters
fov Horizontal field of view angle $ \theta $
aspectRatio Horizontal:vertical aspect ratio $ a $
near Near clipping plane $ n $
far Far clipping plane $ f $

If far is finite, the result is:

\[ \boldsymbol{A} = \begin{pmatrix} \frac{1}{\tan \left(\frac{\theta}{2} \right)} & 0 & 0 & 0 \\ 0 & \frac{a}{\tan \left(\frac{\theta}{2} \right)} & 0 & 0 \\ 0 & 0 & \frac{n + f}{n - f} & \frac{2nf}{n - f} \\ 0 & 0 & -1 & 0 \end{pmatrix} \]

For infinite far, the result is:

\[ \boldsymbol{A} = \begin{pmatrix} \frac{1}{\tan \left( \frac{\theta}{2} \right) } & 0 & 0 & 0 \\ 0 & \frac{a}{\tan \left( \frac{\theta}{2} \right) } & 0 & 0 \\ 0 & 0 & -1 & -2n \\ 0 & 0 & -1 & 0 \end{pmatrix} \]

This function is equivalent to calling perspectiveProjection(const Vector2<T>&, T, T) with the size parameter calculated as

\[ \boldsymbol{s} = 2 n \tan \left(\tfrac{\theta}{2} \right) \begin{pmatrix} 1 \\ \frac{1}{a} \end{pmatrix} \]

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::lookAt(const Vector3<T>& eye, const Vector3<T>& target, const Vector3<T>& up)

Matrix oriented towards a specific point.

Parameters
eye Location to place the matrix
target Location towards which the matrix is oriented
up Vector as a guide of which way is up (should not be the same direction as target - eye)

template<class T>
static Matrix4<T> Magnum::Math::Matrix4<T>::from(const Matrix3x3<T>& rotationScaling, const Vector3<T>& translation) constexpr

Create matrix from rotation/scaling part and translation part.

Parameters
rotationScaling Rotation/scaling part (upper-left 3x3 matrix)
translation Translation part (first three elements of fourth column)

template<class T>
Magnum::Math::Matrix4<T>::Matrix4(IdentityInitT = IdentityInit, T value = T{1}) constexpr noexcept

Default constructor.

Creates identity matrix. value allows you to specify value on diagonal.

template<class T> template<class U>
Magnum::Math::Matrix4<T>::Matrix4(const RectangularMatrix<4, 4, U>& other) explicit constexpr noexcept

Construct matrix from another of different type.

Performs only default casting on the values, no rounding or anything else. Example usage:

Matrix2x2<Float> floatingPoint({1.3f, 2.7f},
                               {-15.0f, 7.0f});
Matrix2x2<Byte> integral(floatingPoint);
// integral == {{1, 2}, {-15, 7}}

template<class T> template<std::size_t otherSize>
Magnum::Math::Matrix4<T>::Matrix4(const RectangularMatrix<otherSize, otherSize, T>& other) explicit constexpr noexcept

Construct matrix by slicing or expanding another of a different size.

If the other matrix is larger, takes only the first size columns and rows from it; if the other matrix is smaller, it's expanded to an identity (ones on diagonal, zeros elsewhere).

template<class T>
bool Magnum::Math::Matrix4<T>::isRigidTransformation() const

Check whether the matrix represents rigid transformation.

Rigid transformation consists only of rotation and translation (i.e. no scaling or projection).

template<class T>
Matrix3x3<T> Magnum::Math::Matrix4<T>::rotationScaling() const constexpr

3D rotation and scaling part of the matrix

Unchanged upper-left 3x3 part of the matrix.

\[ \begin{pmatrix} \color{m-danger} a_x & \color{m-success} b_x & \color{m-info} c_x & t_x \\ \color{m-danger} a_y & \color{m-success} b_y & \color{m-info} c_y & t_y \\ \color{m-danger} a_z & \color{m-success} b_z & \color{m-info} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

Note that an arbitrary combination of rotation and scaling can also represent shear and reflection. Especially when non-uniform scaling is involved, decomposition of the result into primary linear transformations may have multiple equivalent solutions. See Algorithms::svd() and Algorithms::qr() for further info. See also rotationShear(), rotation() const and scaling() const for extracting further properties.

template<class T>
Matrix3x3<T> Magnum::Math::Matrix4<T>::rotationShear() const

3D rotation and scaling part of the matrix

Normalized upper-left 3x3 part of the matrix. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting rotation is extracted as:

\[ \boldsymbol{R} = \begin{pmatrix} \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} & \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|} & \cfrac{\boldsymbol{c}}{|\boldsymbol{c}|} \end{pmatrix} \]

This function is a counterpart to rotation() const that does not require orthogonal input. See also rotationScaling() and scaling() const for extracting other properties. The Algorithms::svd() and Algorithms::qr() can be used to separate the rotation / shear properties.

template<class T>
Matrix3x3<T> Magnum::Math::Matrix4<T>::rotation() const

3D rotation part of the matrix

Normalized upper-left 3x3 part of the matrix. Expects that the normalized part is orthogonal. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting rotation is extracted as:

\[ \boldsymbol{R} = \begin{pmatrix} \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} & \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|} & \cfrac{\boldsymbol{c}}{|\boldsymbol{c}|} \end{pmatrix} \]

This function is equivalent to rotationShear() but with the added orthogonality requirement. See also rotationScaling() and scaling() const for extracting other properties.

template<class T>
Matrix3x3<T> Magnum::Math::Matrix4<T>::rotationNormalized() const

3D rotation part of the matrix assuming there is no scaling

Similar to rotation(), but expects that the rotation part is orthogonal, saving the extra renormalization. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-danger} a_x & \color{m-success} b_x & \color{m-info} c_x & t_x \\ \color{m-danger} a_y & \color{m-success} b_y & \color{m-info} c_y & t_y \\ \color{m-danger} a_z & \color{m-success} b_z & \color{m-info} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting rotation is extracted as:

\[ \boldsymbol{R} = \begin{pmatrix} \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} & \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|} & \cfrac{\boldsymbol{c}}{|\boldsymbol{c}|} \end{pmatrix} = \begin{pmatrix} \boldsymbol{a} & \boldsymbol{b} & \boldsymbol{c} \end{pmatrix} \]

In particular, for an orthogonal matrix, rotationScaling(), rotationShear(), rotation() const and rotationNormalized() all return the same value.

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::scalingSquared() const

Non-uniform scaling part of the matrix, squared.

Squared length of vectors in upper-left 3x3 part of the matrix. Faster alternative to scaling() const, because it doesn't calculate the square root. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting scaling vector, squared, is:

\[ \boldsymbol{s}^2 = \begin{pmatrix} \boldsymbol{a} \cdot \boldsymbol{a} \\ \boldsymbol{b} \cdot \boldsymbol{b} \\ \boldsymbol{c} \cdot \boldsymbol{c} \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::scaling() const

Non-uniform scaling part of the matrix, squared.

Length of vectors in upper-left 3x3 part of the matrix. Use the faster alternative scalingSquared() where possible. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting scaling vector is:

\[ \boldsymbol{s} = \begin{pmatrix} | \boldsymbol{a} | \\ | \boldsymbol{b} | \\ | \boldsymbol{c} | \end{pmatrix} \]

template<class T>
T Magnum::Math::Matrix4<T>::uniformScalingSquared() const

Uniform scaling part of the matrix, squared.

Squared length of vectors in upper-left 3x3 part of the matrix. Expects that the scaling is the same in all axes. Faster alternative to uniformScaling(), because it doesn't calculate the square root. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting uniform scaling, squared, is:

\[ s^2 = \boldsymbol{a} \cdot \boldsymbol{a} = \boldsymbol{b} \cdot \boldsymbol{b} = \boldsymbol{c} \cdot \boldsymbol{c} \]

template<class T>
T Magnum::Math::Matrix4<T>::uniformScaling() const

Uniform scaling part of the matrix.

Length of vectors in upper-left 3x3 part of the matrix. Expects that the scaling is the same in all axes. Use the faster alternative uniformScalingSquared() where possible. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors $ \boldsymbol{a} $ , $ \boldsymbol{b} $ and $ \boldsymbol{c} $ :

\[ \begin{pmatrix} \color{m-warning} a_x & \color{m-warning} b_x & \color{m-warning} c_x & t_x \\ \color{m-warning} a_y & \color{m-warning} b_y & \color{m-warning} c_y & t_y \\ \color{m-warning} a_z & \color{m-warning} b_z & \color{m-warning} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

the resulting uniform scaling is:

\[ s = | \boldsymbol{a} | = | \boldsymbol{b} | = | \boldsymbol{c} | \]

template<class T>
Vector3<T>& Magnum::Math::Matrix4<T>::right()

Right-pointing 3D vector.

First three elements of first column.

\[ \begin{pmatrix} \color{m-danger} a_x & b_x & c_x & t_x \\ \color{m-danger} a_y & b_y & c_y & t_y \\ \color{m-danger} a_z & b_z & c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::right() const constexpr

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<class T>
Vector3<T>& Magnum::Math::Matrix4<T>::up()

Up-pointing 3D vector.

First three elements of second column.

\[ \begin{pmatrix} a_x & \color{m-success} b_x & c_x & t_x \\ a_y & \color{m-success} b_y & c_y & t_y \\ a_z & \color{m-success} b_z & c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::up() const constexpr

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<class T>
Vector3<T>& Magnum::Math::Matrix4<T>::backward()

Backward-pointing 3D vector.

First three elements of third column.

\[ \begin{pmatrix} a_x & b_x & \color{m-info} c_x & t_x \\ a_y & b_y & \color{m-info} c_y & t_y \\ a_z & b_z & \color{m-info} c_z & t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::backward() const constexpr

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<class T>
Vector3<T>& Magnum::Math::Matrix4<T>::translation()

3D translation part of the matrix

First three elements of fourth column.

\[ \begin{pmatrix} a_x & b_x & c_x & \color{m-warning} t_x \\ a_y & b_y & c_y & \color{m-warning} t_y \\ a_z & b_z & c_z & \color{m-warning} t_z \\ \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1 \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::translation() const constexpr

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

template<class T>
Matrix4<T> Magnum::Math::Matrix4<T>::invertedRigid() const

Inverted rigid transformation matrix.

Expects that the matrix represents rigid transformation. Significantly faster than the general algorithm in inverted().

\[ A^{-1} = \begin{pmatrix} (A^{3,3})^T & (A^{3,3})^T \begin{pmatrix} a_{3,0} \\ a_{3,1} \\ a_{3,2} \\ \end{pmatrix} \\ \begin{array}{ccc} 0 & 0 & 0 \end{array} & 1 \end{pmatrix} \]

$ A^{i, j} $ is matrix without i-th row and j-th column, see ij()

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::transformVector(const Vector3<T>& vector) const

Transform 3D vector with the matrix.

Unlike in transformVector(), translation is not involved in the transformation.

\[ \boldsymbol v' = \boldsymbol M \begin{pmatrix} v_x \\ v_y \\ v_z \\ 0 \end{pmatrix} \]

template<class T>
Vector3<T> Magnum::Math::Matrix4<T>::transformPoint(const Vector3<T>& vector) const

Transform 3D point with the matrix.

Unlike in transformVector(), translation is also involved in the transformation.

\[ \boldsymbol v' = \boldsymbol v''_{xyz} / v''_w ~~~~~~~~~~ \boldsymbol v'' = \begin{pmatrix} v''_x \\ v''_y \\ v''_z \\ v''_w \end{pmatrix} = \boldsymbol M \begin{pmatrix} v_x \\ v_y \\ v_z \\ 1 \end{pmatrix} \\ \]