#include <Magnum/Math/Matrix4.h>
template<class T>
Matrix4 class
3D transformation matrix
Template parameters | |
---|---|
T | Underlying data type |
Expands upon a generic Matrix4x4 with functionality for 3D transformations. A 3D transformation matrix consists of a upper-left 3x3 part describing a combined scaling, rotation and shear, and the three top-right components specifying a translation:
The , and vectors can be also thought of as the three basis vectors describing the coordinate system the matrix converts to. In case of an affine transformation, the bottom row is always . A (pure) 3D perspective projection matrix, however, can look for example like this:
The bottom row having the non-zero value in the third column instead of the fourth is, simply put, what makes perspective shortening happening along the Z axis. While perspective shortening along X or Y is technically also possible, it doesn't really have a common use, neither it is a thing in case of a 2D transformation with Matrix3.
Usage
See Math type system, Operations with matrices and vectors and 2D and 3D transformations first for an introduction into using transformation matrices.
While it's possible to create the matrix directly from the components, the recommended usage is by creating elementary transformation matrices with translation(), rotation() and variants, scaling(), reflection(), shearingXY() and variants, lookAt() and orthographicProjection() / perspectiveProjection() and multiplying them together to form the final transformation — the rightmost transformation is applied first, leftmost last:
using namespace Math::Literals; Matrix4 transformation = Matrix4::rotationZ(15.0_degf)* Matrix4::translation({10.0f, 3.0f, -1.5f})* Matrix4::scaling(Vector3::yScale(2.0f));
Conversely, the transformation parts can be extracted back using the member rotation(), scaling() and their variants, and translation(). The basis vectors can be accessed using right(), up() and backward(). Matrices that combine non-uniform scaling and/or shear with rotation can't be trivially decomposed back, for these you might want to consider using Algorithms::
When a lot of transformations gets composed together over time (for example with a camera movement), a floating-point drift accumulates, causing the rotation part to no longer be orthogonal. This can be accounted for using Algorithms::
Base classes
-
template<std::class Matrix<T>
size_t size, class T> - 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 the X axis
- static auto rotationY(Rad<T> angle) -> Matrix4<T>
- 3D rotation matrix around the Y axis
- static auto rotationZ(Rad<T> angle) -> Matrix4<T>
- 3D rotation matrix around the 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 the XY plane
- static auto shearingXZ(T amountX, T amountZ) -> Matrix4<T> constexpr
- 3D shearing matrix along the XZ plane
- static auto shearingYZ(T amountY, T amountZ) -> Matrix4<T> constexpr
- 3D shearing matrix along the YZ plane
- static auto orthographicProjection(const Vector2<T>& size, T near, T far) -> Matrix4<T>
- 3D orthographic projection matrix
- static auto orthographicProjection(const Vector2<T>& bottomLeft, const Vector2<T>& topRight, T near, T far) -> Matrix4<T> new in Git master
- 3D off-center 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 perspectiveProjection(const Vector2<T>& bottomLeft, const Vector2<T>& topRight, T near, T far) -> Matrix4<T> new in 2019.10
- 3D off-center 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 a matrix from a rotation/scaling part and a translation part.
Constructors, destructors, conversion operators
- Matrix4() constexpr noexcept
- Default constructor.
- Matrix4(IdentityInitT, T value = T{1}) explicit constexpr noexcept
- Construct an identity matrix.
- Matrix4(ZeroInitT) explicit constexpr noexcept
- Construct a zero-filled matrix.
-
Matrix4(Magnum::
NoInitT) explicit constexpr noexcept - Construct without initializing the contents.
- Matrix4(const Vector4<T>& first, const Vector4<T>& second, const Vector4<T>& third, const Vector4<T>& fourth) constexpr noexcept
- Construct from column vectors.
- Matrix4(T value) explicit constexpr noexcept
- Construct with one value for all elements.
-
template<std::Matrix4(const T(&data)[cols_][rows_]) explicit constexpr noexcept new in Git master
size_t cols_, std:: size_t rows_> - Construct from a fixed-size array.
-
template<class U>Matrix4(const RectangularMatrix<4, 4, U>& other) explicit constexpr noexcept
- Construct from a matrix of a different type.
-
template<class U, class = decltype(Implementation::RectangularMatrixConverter<4, 4, T, U>::from(std::Matrix4(const U& other) explicit constexpr
declval<U>()))> - Construct a matrix from external representation.
-
template<std::Matrix4(IdentityInitT, const RectangularMatrix<otherCols, otherRows, T>& other, T value = T(1)) explicit constexpr noexcept new in Git master
size_t otherCols, std:: size_t otherRows> - Construct by slicing or expanding a matrix of different size, leaving the rest at identity.
-
template<std::Matrix4(ZeroInitT, const RectangularMatrix<otherCols, otherRows, T>& other) explicit constexpr noexcept new in Git master
size_t otherCols, std:: size_t otherRows> - Construct by slicing or expanding a matrix of different size, leaving the rest at zero.
-
template<std::Matrix4(const RectangularMatrix<otherCols, otherRows, T>& other, T value = T(1)) explicit constexpr noexcept new in Git master
size_t otherCols, std:: size_t otherRows> - Construct by slicing or expanding a matrix of different size.
- Matrix4(const RectangularMatrix<4, 4, T>& other) constexpr noexcept
- Copy constructor.
Public functions
- auto isRigidTransformation() const -> bool
- Check whether the matrix represents a rigid transformation.
- auto rotationScaling() const -> Matrix3x3<T> constexpr
- 3D rotation and scaling part of the matrix
- auto rotationShear() const -> Matrix3x3<T>
- 3D rotation, reflection and shear part of the matrix
- auto rotation() const -> Matrix3x3<T>
- 3D rotation and reflection part of the matrix
- auto rotationNormalized() const -> Matrix3x3<T>
- 3D rotation and reflection 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.
- auto uniformScalingSquared() const -> T
- Uniform scaling part of the matrix, squared.
- auto uniformScaling() const -> T
- Uniform scaling part of the matrix.
- auto normalMatrix() const -> Matrix3x3<T> new in 2019.10
- Normal 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 orthographicProjectionNear() const -> Float new in Git master
- Distance to near plane of an orthographic projection matrix.
- auto orthographicProjectionFar() const -> Float new in Git master
- Distance to far plane of an orthographic projection matrix.
- auto perspectiveProjectionNear() const -> Float new in Git master
- Distance to near plane of a perspective projection matrix.
- auto perspectiveProjectionFar() const -> Float new in Git master
- Distance to far plane of a perspective projection matrix.
- auto invertedRigid() const -> Matrix4<T>
- Inverted rigid transformation matrix.
- auto transformVector(const Vector3<T>& vector) const -> Vector3<T>
- Transform a 3D vector with the matrix.
- auto transformPoint(const Vector3<T>& vector) const -> Vector3<T>
- Transform a 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 |
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().
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))
.
template<class T>
static Matrix4<T> Magnum:: Math:: Matrix4<T>:: shearingXY(T amountX,
T amountY) constexpr
3D shearing matrix along the XY plane
Parameters | |
---|---|
amountX | Amount of shearing along the X axis |
amountY | Amount of shearing along the Y axis |
Z axis remains unchanged.
template<class T>
static Matrix4<T> Magnum:: Math:: Matrix4<T>:: shearingXZ(T amountX,
T amountZ) constexpr
3D shearing matrix along the XZ plane
Parameters | |
---|---|
amountX | Amount of shearing along the X axis |
amountZ | Amount of shearing along the Z axis |
Y axis remains unchanged.
template<class T>
static Matrix4<T> Magnum:: Math:: Matrix4<T>:: shearingYZ(T amountY,
T amountZ) constexpr
3D shearing matrix along the YZ plane
Parameters | |
---|---|
amountY | Amount of shearing along the Y axis |
amountZ | Amount of shearing along the Z axis |
X axis remains unchanged.
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 |
If you need an off-center projection (as with the classic glOrtho() function), use orthographicProjection(const Vector2<T>&, const Vector2<T>&, T, T).
template<class T>
static Matrix4<T> Magnum:: Math:: Matrix4<T>:: orthographicProjection(const Vector2<T>& bottomLeft,
const Vector2<T>& topRight,
T near,
T far) new in Git master
3D off-center orthographic projection matrix
Parameters | |
---|---|
bottomLeft | Bottom left corner of the clipping plane |
topRight | Top right corner of the clipping plane |
near | Distance to near clipping plane, positive is ahead |
far | Distance to far clipping plane, positive is ahead |
Equivalent to the classic glOrtho() function. If bottomLeft
and topRight
are a negation of each other, this function is equivalent to orthographicProjection(const Vector2<T>&, T, T).
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:
For infinite far
, the result is:
If you need an off-center projection, use perspectiveProjection(const Vector2<T>&, const Vector2<T>&, T, T) instead.
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 |
aspectRatio | Horizontal:vertical aspect ratio |
near | Near clipping plane |
far | Far clipping plane |
If far
is finite, the result is:
For infinite far
, the result is:
This function is equivalent to calling perspectiveProjection(const Vector2<T>&, T, T) with the size
parameter calculated as
This function is similar to the classic gluPerspective(), with the difference that fov
is horizontal instead of vertical. If you need an off-center projection (as with the classic glFrustum() function), use perspectiveProjection(const Vector2<T>&, const Vector2<T>&, T, T).
template<class T>
static Matrix4<T> Magnum:: Math:: Matrix4<T>:: perspectiveProjection(const Vector2<T>& bottomLeft,
const Vector2<T>& topRight,
T near,
T far) new in 2019.10
3D off-center perspective projection matrix
Parameters | |
---|---|
bottomLeft | Bottom left corner of the near clipping plane |
topRight | Top right corner of the 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:
For infinite far
, the result is:
Equivalent to the classic glFrustum() function. If bottomLeft
and topRight
are a negation of each other, this function is equivalent to perspectiveProjection(const Vector2<T>&, T, T).
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 a matrix from a rotation/scaling part and a 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() constexpr noexcept
Default constructor.
Equivalent to Matrix4(IdentityInitT, T).
template<class T>
Magnum:: Math:: Matrix4<T>:: Matrix4(IdentityInitT,
T value = T{1}) explicit constexpr noexcept
Construct an identity matrix.
The value
allows you to specify value on diagonal.
template<class T>
template<std:: size_t cols_, std:: size_t rows_>
Magnum:: Math:: Matrix4<T>:: Matrix4(const T(&data)[cols_][rows_]) explicit constexpr noexcept new in Git master
Construct from a fixed-size array.
Use from(const T*) to reinterpret an arbitrary pointer to a matrix.
template<class T>
template<class U>
Magnum:: Math:: Matrix4<T>:: Matrix4(const RectangularMatrix<4, 4, U>& other) explicit constexpr noexcept
Construct from a matrix of a different type.
Performs only default casting on the values, no rounding or anything else. Example usage:
Matrix2x2 floatingPoint{Vector2{1.3f, 2.7f}, Vector2{-15.0f, 7.0f}}; Math::Matrix2x2<Byte> integral{floatingPoint}; // {{1, 2}, {-15, 7}}
template<class T>
template<std:: size_t otherCols, std:: size_t otherRows>
Magnum:: Math:: Matrix4<T>:: Matrix4(IdentityInitT,
const RectangularMatrix<otherCols, otherRows, T>& other,
T value = T(1)) explicit constexpr noexcept new in Git master
Construct by slicing or expanding a matrix of different size, leaving the rest at identity.
If the other matrix has less columns or rows, the corresponding vectors and components are set to either zeros or value
on the diagonal.
template<class T>
template<std:: size_t otherCols, std:: size_t otherRows>
Magnum:: Math:: Matrix4<T>:: Matrix4(ZeroInitT,
const RectangularMatrix<otherCols, otherRows, T>& other) explicit constexpr noexcept new in Git master
Construct by slicing or expanding a matrix of different size, leaving the rest at zero.
If the other matrix has less columns or rows, the corresponding vectors and components are set to zeros.
template<class T>
template<std:: size_t otherCols, std:: size_t otherRows>
Magnum:: Math:: Matrix4<T>:: Matrix4(const RectangularMatrix<otherCols, otherRows, T>& other,
T value = T(1)) explicit constexpr noexcept new in Git master
Construct by slicing or expanding a matrix of different size.
Equivalent to Matrix4(IdentityInitT, const RectangularMatrix<otherCols, otherRows, T>&, T). Note that this default is different from RectangularMatrix, where it's equivalent to the ZeroInit variant instead.
template<class T>
bool Magnum:: Math:: Matrix4<T>:: isRigidTransformation() const
Check whether the matrix represents a rigid transformation.
A rigid transformation consists only of rotation, reflection and translation (i.e., no scaling, skew 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.
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 rotation() const, Algorithms::
template<class T>
Matrix3x3<T> Magnum:: Math:: Matrix4<T>:: rotationShear() const
3D rotation, reflection and shear 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 , and :
the resulting rotation is extracted as:
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::
template<class T>
Matrix3x3<T> Magnum:: Math:: Matrix4<T>:: rotation() const
3D rotation and reflection 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 , and :
the resulting rotation is extracted as:
This function is equivalent to rotationShear() but with the added orthogonality requirement. See also rotationScaling() and scaling() const for extracting other properties.
There's usually several solutions for decomposing the matrix into a rotation and a scaling that satisfy . One possibility that gives you always a pure rotation matrix without reflections (which can then be fed to Quaternion::
Matrix4 transformation = …; Matrix3x3 rotation = transformation.rotation(); Vector3 scaling = transformation.scaling(); if(rotation.determinant() < 0.0f) { rotation[0] *= -1.0f; scaling[0] *= -1.0f; }
template<class T>
Matrix3x3<T> Magnum:: Math:: Matrix4<T>:: rotationNormalized() const
3D rotation and reflection part of the matrix assuming there is no scaling
Similar to rotation() const, 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 , and :
the resulting rotation is extracted as:
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 , and :
the resulting scaling vector, squared, is:
template<class T>
Vector3<T> Magnum:: Math:: Matrix4<T>:: scaling() const
Non-uniform scaling part of the matrix.
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 , and :
the resulting scaling vector is:
Note that the returned vector is sign-less and the signs are instead contained in rotation() const / rotationShear() const, meaning these contain rotation together with a potential reflection. See rotation() const for an example of decomposing a rotation + reflection matrix into a pure rotation and signed scaling.
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 , and :
the resulting uniform scaling, squared, is:
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 , and :
the resulting uniform scaling is:
template<class T>
Matrix3x3<T> Magnum:: Math:: Matrix4<T>:: normalMatrix() const new in 2019.10
Normal matrix.
Shorthand for taking an inverse transpose of the upper-left 3x3 part of the matrix. Assuming the following matrix, with the upper-left 3x3 part represented by column vectors , and :
the normal matrix is extracted as:
The inverse transpose guarantees that the normals stay perpendicular to the surface and point in the original direction even in presence of shear, non-uniform scaling and reflection.
template<class T>
Vector3<T>& Magnum:: Math:: Matrix4<T>:: translation()
3D translation part of the matrix
First three elements of fourth column.
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>
Float Magnum:: Math:: Matrix4<T>:: orthographicProjectionNear() const new in Git master
Distance to near plane of an orthographic projection matrix.
Assuming a matrix constructed with orthographicProjection(), returns the distance to its near plane:
template<class T>
Float Magnum:: Math:: Matrix4<T>:: orthographicProjectionFar() const new in Git master
Distance to far plane of an orthographic projection matrix.
Assuming a matrix constructed with orthographicProjection(), returns the distance to its far plane:
template<class T>
Float Magnum:: Math:: Matrix4<T>:: perspectiveProjectionNear() const new in Git master
Distance to near plane of a perspective projection matrix.
Assuming a matrix constructed with perspectiveProjection(), returns the distance to its near plane:
The same equation works for a perspective projection with an infinite far plane:
template<class T>
Float Magnum:: Math:: Matrix4<T>:: perspectiveProjectionFar() const new in Git master
Distance to far plane of a perspective projection matrix.
Assuming a matrix constructed with perspectiveProjection() with a positive far value, returns the distance to its far plane:
The same equation works for a perspective projection with an infinite far plane:
template<class T>
Matrix4<T> Magnum:: Math:: Matrix4<T>:: invertedRigid() const
Inverted rigid transformation matrix.
Expects that the matrix represents a rigid transformation (i.e., no scaling, skew or projection). Significantly faster than the general algorithm in inverted().
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 a 3D vector with the matrix.
Unlike in transformPoint(), translation is not involved in the transformation.
template<class T>
Vector3<T> Magnum:: Math:: Matrix4<T>:: transformPoint(const Vector3<T>& vector) const
Transform a 3D point with the matrix.
Unlike in transformVector(), translation is also involved in the transformation.