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magnum/src/Magnum/Math/Matrix3.h

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#ifndef Magnum_Math_Matrix3_h
#define Magnum_Math_Matrix3_h
/*
This file is part of Magnum.
Copyright © 2010, 2011, 2012, 2013, 2014, 2015, 2016, 2017, 2018, 2019,
2020 Vladimír Vondruš <mosra@centrum.cz>
Permission is hereby granted, free of charge, to any person obtaining a
copy of this software and associated documentation files (the "Software"),
to deal in the Software without restriction, including without limitation
the rights to use, copy, modify, merge, publish, distribute, sublicense,
and/or sell copies of the Software, and to permit persons to whom the
Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER
DEALINGS IN THE SOFTWARE.
*/
/** @file
* @brief Class @ref Magnum::Math::Matrix3
*/
#include "Magnum/Math/Matrix.h"
#include "Magnum/Math/Vector3.h"
namespace Magnum { namespace Math {
/**
@brief 2D transformation matrix
@tparam T Underlying data type
See @ref matrix-vector and @ref transformations for brief introduction.
@see @ref Magnum::Matrix3, @ref Magnum::Matrix3d, @ref Matrix3x3,
@ref DualComplex, @ref SceneGraph::MatrixTransformation2D
@configurationvalueref{Magnum::Math::Matrix3}
*/
template<class T> class Matrix3: public Matrix3x3<T> {
public:
/**
* @brief 2D translation matrix
* @param vector Translation vector
*
* @f[
* \boldsymbol{A} = \begin{pmatrix}
* 1 & 0 & v_x \\
* 0 & 1 & v_y \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref translation() const, @ref DualComplex::translation(),
* @ref Matrix4::translation(const Vector3<T>&),
* @ref Vector2::xAxis(), @ref Vector2::yAxis()
*/
constexpr static Matrix3<T> translation(const Vector2<T>& vector) {
return {{ T(1), T(0), T(0)},
{ T(0), T(1), T(0)},
{vector.x(), vector.y(), T(1)}};
}
/**
* @brief 2D scaling matrix
* @param vector Scaling vector
*
* @f[
* \boldsymbol{A} = \begin{pmatrix}
* v_x & 0 & 0 \\
* 0 & v_y & 0 \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref scaling() const, @ref Matrix4::scaling(const Vector3<T>&),
* @ref Vector2::xScale(), @ref Vector2::yScale()
*/
constexpr static Matrix3<T> scaling(const Vector2<T>& vector) {
return {{vector.x(), T(0), T(0)},
{ T(0), vector.y(), T(0)},
{ T(0), T(0), T(1)}};
}
/**
* @brief 2D rotation matrix
* @param angle Rotation angle (counterclockwise)
*
* @f[
* \boldsymbol{A} = \begin{pmatrix}
* \cos\theta & -\sin\theta & 0 \\
* \sin\theta & \cos\theta & 0 \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref rotation() const, @ref Complex::rotation(),
* @ref DualComplex::rotation(),
* @ref Matrix4::rotation(Rad, const Vector3<T>&)
*/
static Matrix3<T> rotation(Rad<T> angle);
/**
* @brief 2D reflection matrix
* @param normal Normal of the line through which to reflect
*
* Expects that the normal is normalized. Reflection along axes can be
* done in a slightly simpler way also using @ref scaling(), e.g.
* @cpp Matrix3::reflection(Vector2::yAxis()) @ce is equivalent to
* @cpp Matrix3::scaling(Vector2::yScale(-1.0f)) @ce. @f[
* \boldsymbol{A} = \boldsymbol{I} - 2 \boldsymbol{NN}^T ~~~~~ \boldsymbol{N} = \begin{pmatrix} n_x \\ n_y \end{pmatrix}
* @f]
* @see @ref Matrix4::reflection(), @ref Vector::isNormalized(),
* @ref reflect()
*/
static Matrix3<T> reflection(const Vector2<T>& normal) {
CORRADE_ASSERT(normal.isNormalized(),
"Math::Matrix3::reflection(): normal" << normal << "is not normalized", {});
return from(Matrix2x2<T>() - T(2)*normal*RectangularMatrix<1, 2, T>(normal).transposed(), {});
}
/**
* @brief 2D shearing matrix along X axis
* @param amount Shearing amount
*
* Y axis remains unchanged. @f[
* \boldsymbol{A} = \begin{pmatrix}
* 1 & v_x & 0 \\
* 0 & 1 & 0 \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref shearingY(), @ref Matrix4::shearingXY(),
* @ref Matrix4::shearingXZ(), @ref Matrix4::shearingYZ()
*/
constexpr static Matrix3<T> shearingX(T amount) {
return {{ T(1), T(0), T(0)},
{amount, T(1), T(0)},
{ T(0), T(0), T(1)}};
}
/**
* @brief 2D shearing matrix along Y axis
* @param amount Shearing amount
*
* X axis remains unchanged. @f[
* \boldsymbol{A} = \begin{pmatrix}
* 1 & 0 & 0 \\
* v_y & 1 & 0 \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref shearingX(), @ref Matrix4::shearingXY(),
* @ref Matrix4::shearingXZ(), @ref Matrix4::shearingYZ()
*/
constexpr static Matrix3<T> shearingY(T amount) {
return {{T(1), amount, T(0)},
{T(0), T(1), T(0)},
{T(0), T(0), T(1)}};
}
/**
* @brief 2D projection matrix
* @param size Size of the view
*
* @f[
* \boldsymbol{A} = \begin{pmatrix}
* \frac{2}{s_x} & 0 & 0 \\
* 0 & \frac{2}{s_y} & 0 \\
* 0 & 0 & 1
* \end{pmatrix}
* @f]
* @see @ref Matrix4::orthographicProjection(),
* @ref Matrix4::perspectiveProjection()
* @m_keywords{gluOrtho2D()}
*/
static Matrix3<T> projection(const Vector2<T>& size) {
return scaling(2.0f/size);
}
/**
* @brief Create matrix from rotation/scaling part and translation part
* @param rotationScaling Rotation/scaling part (upper-left 2x2
* matrix)
* @param translation Translation part (first two elements of
* third column)
*
* @see @ref rotationScaling(), @ref translation() const
*/
constexpr static Matrix3<T> from(const Matrix2x2<T>& rotationScaling, const Vector2<T>& translation) {
return {{rotationScaling[0], T(0)},
{rotationScaling[1], T(0)},
{ translation, T(1)}};
}
/**
* @brief Default constructor
*
* Equivalent to @ref Matrix3(IdentityInitT, T).
*/
constexpr /*implicit*/ Matrix3() noexcept: Matrix3x3<T>{IdentityInit, T(1)} {}
/**
* @brief Construct an identity matrix
*
* The @p value allows you to specify value on diagonal.
*/
constexpr explicit Matrix3(IdentityInitT, T value = T{1}) noexcept: Matrix3x3<T>{IdentityInit, value} {}
/** @copydoc Matrix::Matrix(ZeroInitT) */
constexpr explicit Matrix3(ZeroInitT) noexcept: Matrix3x3<T>{ZeroInit} {}
/** @copydoc Matrix::Matrix(Magnum::NoInitT) */
constexpr explicit Matrix3(Magnum::NoInitT) noexcept: Matrix3x3<T>{Magnum::NoInit} {}
/** @brief Construct from column vectors */
constexpr /*implicit*/ Matrix3(const Vector3<T>& first, const Vector3<T>& second, const Vector3<T>& third) noexcept: Matrix3x3<T>(first, second, third) {}
/** @brief Construct with one value for all elements */
constexpr explicit Matrix3(T value) noexcept: Matrix3x3<T>{value} {}
/** @copydoc Matrix::Matrix(const RectangularMatrix<size, size, U>&) */
template<class U> constexpr explicit Matrix3(const RectangularMatrix<3, 3, U>& other) noexcept: Matrix3x3<T>(other) {}
/** @brief Construct from external representation */
template<class U, class V = decltype(Implementation::RectangularMatrixConverter<3, 3, T, U>::from(std::declval<U>()))> constexpr explicit Matrix3(const U& other) noexcept: Matrix3x3<T>(Implementation::RectangularMatrixConverter<3, 3, T, U>::from(other)) {}
/**
* @brief Construct by slicing or expanding a matrix of a different size
*
* If the other matrix is larger, takes only the first @cpp size @ce
* columns and rows from it; if the other matrix is smaller, it's
* expanded to an identity (ones on diagonal, zeros elsewhere).
*/
template<std::size_t otherSize> constexpr explicit Matrix3(const RectangularMatrix<otherSize, otherSize, T>& other) noexcept: Matrix3x3<T>{other} {}
/** @brief Copy constructor */
constexpr /*implicit*/ Matrix3(const RectangularMatrix<3, 3, T>& other) noexcept: Matrix3x3<T>(other) {}
/**
* @brief Check whether the matrix represents a rigid transformation
*
* A [rigid transformation](https://en.wikipedia.org/wiki/Rigid_transformation)
* consists only of rotation, reflection and translation (i.e., no
* scaling, skew or projection).
* @see @ref isOrthogonal()
*/
bool isRigidTransformation() const {
return rotationScaling().isOrthogonal() && row(2) == Vector3<T>(T(0), T(0), T(1));
}
/**
* @brief 2D rotation and scaling part of the matrix
*
* Unchanged upper-left 2x2 part of the matrix. @f[
* \begin{pmatrix}
* \color{m-danger} a_x & \color{m-success} b_x & t_x \\
* \color{m-danger} a_y & \color{m-success} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* 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
* @ref Algorithms::svd() and @ref Algorithms::qr() for further info.
* See also @ref rotationShear(), @ref rotation() const and
* @ref scaling() const for extracting further properties.
*
* @see @ref from(const Matrix2x2<T>&, const Vector2<T>&),
* @ref rotation(Rad<T>), @ref Matrix4::rotationScaling()
*/
constexpr Matrix2x2<T> rotationScaling() const {
return {(*this)[0].xy(),
(*this)[1].xy()};
}
/**
* @brief 2D rotation and shear part of the matrix
*
* Normalized upper-left 2x2 part of the matrix. Assuming the following
* matrix, with the upper-left 2x2 part represented by column vectors
* @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-danger} a_x & \color{m-success} b_x & t_x \\
* \color{m-danger} a_y & \color{m-success} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting rotation is extracted as: @f[
* \boldsymbol{R} = \begin{pmatrix}
* \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} &
* \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|}
* \end{pmatrix}
* @f]
*
* This function is a counterpart to @ref rotation() const that does
* not require orthogonal input. See also @ref rotationScaling() and
* @ref scaling() const for extracting other properties. The
* @ref Algorithms::svd() and @ref Algorithms::qr() can be used to
* separate the rotation / shear properties.
*
* @see @ref from(const Matrix2x2<T>&, const Vector2<T>&),
* @ref rotation(Rad), @ref Matrix4::rotationShear() const
*/
Matrix2x2<T> rotationShear() const {
return {(*this)[0].xy().normalized(),
(*this)[1].xy().normalized()};
}
/**
* @brief 2D rotation part of the matrix
*
* Normalized upper-left 2x2 part of the matrix. Expects that the
* normalized part is orthogonal. Assuming the following matrix, with
* the upper-left 2x2 part represented by column vectors
* @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
* \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting rotation is extracted as: @f[
* \boldsymbol{R} = \begin{pmatrix}
* \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} &
* \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|}
* \end{pmatrix}
* @f]
*
* This function is equivalent to @ref rotationShear() but with the
* added orthogonality requirement. See also @ref rotationScaling() and
* @ref scaling() const for extracting other properties.
*
* @note Extracting rotation part of a matrix this way may cause
* assertions in case you have unsanitized input (for example a
* model transformation loaded from an external source) or when
* you accumulate many transformations together (for example when
* controlling a FPS camera). To mitigate this, either first
* reorthogonalize the matrix using
* @ref Algorithms::gramSchmidtOrthogonalize(), decompose it to
* basic linear transformations using @ref Algorithms::svd() or
* @ref Algorithms::qr() or use a different transformation
* representation that suffers less floating point error and can
* be easier renormalized such as @ref DualComplex. Another
* possibility is to ignore the error and extract combined
* rotation and scaling / shear with @ref rotationScaling() /
* @ref rotationShear().
*
* @see @ref rotationNormalized(), @ref scaling() const,
* @ref rotation(Rad<T>), @ref Matrix4::rotation() const
*/
Matrix2x2<T> rotation() const;
/**
* @brief 2D rotation part of the matrix assuming there is no scaling
*
* Similar to @ref rotation(), but expects that the rotation part is
* orthogonal, saving the extra renormalization. Assuming the
* following matrix, with the upper-left 2x2 part represented by column
* vectors @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-danger} a_x & \color{m-success} b_x & t_x \\
* \color{m-danger} a_y & \color{m-success} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting rotation is extracted as: @f[
* \boldsymbol{R} = \begin{pmatrix}
* \cfrac{\boldsymbol{a}}{|\boldsymbol{a}|} &
* \cfrac{\boldsymbol{b}}{|\boldsymbol{b}|}
* \end{pmatrix} = \begin{pmatrix}
* \boldsymbol{a} &
* \boldsymbol{b}
* \end{pmatrix}
* @f]
*
* In particular, for an orthogonal matrix, @ref rotationScaling(),
* @ref rotationShear(), @ref rotation() const and
* @ref rotationNormalized() all return the same value.
*
* @see @ref isOrthogonal(), @ref uniformScaling(),
* @ref Matrix4::rotationNormalized()
*/
Matrix2x2<T> rotationNormalized() const;
/**
* @brief Non-uniform scaling part of the matrix, squared
*
* Squared length of vectors in upper-left 2x2 part of the matrix.
* Faster alternative to @ref scaling() const, because it doesn't
* calculate the square root. Assuming the following matrix, with the
* upper-left 2x2 part represented by column vectors
* @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
* \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting scaling vector, squared, is: @f[
* \boldsymbol{s}^2 = \begin{pmatrix}
* \boldsymbol{a} \cdot \boldsymbol{a} \\
* \boldsymbol{b} \cdot \boldsymbol{b}
* \end{pmatrix}
* @f]
*
* @see @ref scaling() const, @ref uniformScalingSquared(),
* @ref rotation() const, @ref Matrix4::scalingSquared()
*/
Vector2<T> scalingSquared() const {
return {(*this)[0].xy().dot(),
(*this)[1].xy().dot()};
}
/**
* @brief Non-uniform scaling part of the matrix
*
* Length of vectors in upper-left 2x2 part of the matrix. Use the
* faster alternative @ref scalingSquared() where possible. Assuming
* the following matrix, with the upper-left 2x2 part represented by
* column vectors @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$:
* @f[
* \begin{pmatrix}
* \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
* \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting scaling vector is: @f[
* \boldsymbol{s} = \begin{pmatrix}
* | \boldsymbol{a} | \\
* | \boldsymbol{b} |
* \end{pmatrix}
* @f]
*
* Note that the returned vector is sign-less and the signs are instead
* contained in @ref rotation() const / @ref rotationShear() const in
* order to ensure @f$ \boldsymbol{R} \boldsymbol{S} = \boldsymbol{M} @f$
* for @f$ \boldsymbol{R} @f$ and @f$ \boldsymbol{S} @f$ extracted out
* of @f$ \boldsymbol{M} @f$. The signs can be extracted for example by
* applying @ref Math::sign() on a @ref diagonal(), but keep in mind
* that the signs can be negative even for pure rotation matrices.
*
* @see @ref scalingSquared(), @ref uniformScaling(),
* @ref rotation() const, @ref Matrix4::scaling() const
*/
Vector2<T> scaling() const {
return {(*this)[0].xy().length(),
(*this)[1].xy().length()};
}
/**
* @brief Uniform scaling part of the matrix, squared
*
* Squared length of vectors in upper-left 2x2 part of the matrix.
* Expects that the scaling is the same in all axes. Faster alternative
* to @ref uniformScaling(), because it doesn't compute the square
* root. Assuming the following matrix, with the upper-left 2x2 part
* represented by column vectors @f$ \boldsymbol{a} @f$ and
* @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
* \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting uniform scaling, squared, is: @f[
* s^2 = \boldsymbol{a} \cdot \boldsymbol{a}
* = \boldsymbol{b} \cdot \boldsymbol{b}
* @f]
*
* @note Extracting uniform scaling of a matrix this way may cause
* assertions in case you have unsanitized input (for example a
* model transformation loaded from an external source) or when
* you accumulate many transformations together (for example when
* controlling a FPS camera). To mitigate this, either first
* reorthogonalize the matrix using
* @ref Algorithms::gramSchmidtOrthogonalize(), decompose it to
* basic linear transformations using @ref Algorithms::svd() or
* @ref Algorithms::qr() or extract a non-uniform scaling using
* @ref scalingSquared().
*
* @see @ref rotation() const, @ref scaling() const,
* @ref scaling(const Vector2<T>&),
* @ref Matrix4::uniformScalingSquared()
*/
T uniformScalingSquared() const;
/**
* @brief Uniform scaling part of the matrix
*
* Length of vectors in upper-left 2x2 part of the matrix. Expects that
* the scaling is the same in all axes. Use faster alternative
* @ref uniformScalingSquared() where possible. Assuming the following
* matrix, with the upper-left 3x3 part represented by column vectors
* @f$ \boldsymbol{a} @f$ and @f$ \boldsymbol{b} @f$: @f[
* \begin{pmatrix}
* \color{m-warning} a_x & \color{m-warning} b_x & t_x \\
* \color{m-warning} a_y & \color{m-warning} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @m_class{m-noindent}
*
* the resulting uniform scaling is: @f[
* s = | \boldsymbol{a} | = | \boldsymbol{b} |
* @f]
*
* @note Extracting uniform scaling of a matrix this way may cause
* assertions in case you have unsanitized input (for example a
* model transformation loaded from an external source) or when
* you accumulate many transformations together (for example when
* controlling a FPS camera). To mitigate this, either first
* reorthogonalize the matrix using
* @ref Algorithms::gramSchmidtOrthogonalize(), decompose it to
* basic linear transformations using @ref Algorithms::svd() or
* @ref Algorithms::qr() or extract a non-uniform scaling using
* @ref scaling() const.
*
* @see @ref rotation() const, @ref scalingSquared() const,
* @ref scaling(const Vector2<T>&), @ref Matrix4::uniformScaling()
*/
T uniformScaling() const { return std::sqrt(uniformScalingSquared()); }
/**
* @brief Right-pointing 2D vector
*
* First two elements of first column. @f[
* \begin{pmatrix}
* \color{m-danger} a_x & b_x & t_x \\
* \color{m-danger} a_y & b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @see @ref up(), @ref Vector2::xAxis(), @ref Matrix4::right()
*/
Vector2<T>& right() { return (*this)[0].xy(); }
constexpr Vector2<T> right() const { return (*this)[0].xy(); } /**< @overload */
/**
* @brief Up-pointing 2D vector
*
* First two elements of second column. @f[
* \begin{pmatrix}
* a_x & \color{m-success} b_x & t_x \\
* a_y & \color{m-success} b_y & t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @see @ref right(), @ref Vector2::yAxis(), @ref Matrix4::up()
*/
Vector2<T>& up() { return (*this)[1].xy(); }
constexpr Vector2<T> up() const { return (*this)[1].xy(); } /**< @overload */
/**
* @brief 2D translation part of the matrix
*
* First two elements of third column. @f[
* \begin{pmatrix}
* a_x & b_x & \color{m-warning} t_x \\
* a_y & b_y & \color{m-warning} t_y \\
* \color{m-dim} 0 & \color{m-dim} 0 & \color{m-dim} 1
* \end{pmatrix}
* @f]
*
* @see @ref from(const Matrix2x2<T>&, const Vector2<T>&),
* @ref translation(const Vector2<T>&),
* @ref Matrix4::translation()
*/
Vector2<T>& translation() { return (*this)[2].xy(); }
constexpr Vector2<T> translation() const { return (*this)[2].xy(); } /**< @overload */
/**
* @brief Inverted rigid transformation matrix
*
* Expects that the matrix represents a [rigid transformation](https://en.wikipedia.org/wiki/Rigid_transformation)
* (i.e., no scaling, skew or projection). Significantly faster than
* the general algorithm in @ref inverted(). @f[
* A^{-1} = \begin{pmatrix} (A^{2,2})^T & (A^{2,2})^T \begin{pmatrix} a_{2,0} \\ a_{2,1} \end{pmatrix} \\ \begin{array}{cc} 0 & 0 \end{array} & 1 \end{pmatrix}
* @f]
* @f$ A^{i, j} @f$ is matrix without i-th row and j-th column, see
* @ref ij()
* @see @ref isRigidTransformation(), @ref invertedOrthogonal(),
* @ref rotationScaling(), @ref translation() const,
* @ref Matrix4::invertedRigid()
*/
Matrix3<T> invertedRigid() const;
/**
* @brief Transform a 2D vector with the matrix
*
* Unlike in @ref transformPoint(), translation is not involved in the
* transformation. @f[
* \boldsymbol v' = \boldsymbol M \begin{pmatrix} v_x \\ v_y \\ 0 \end{pmatrix}
* @f]
* @see @ref Complex::transformVector(),
* @ref Matrix4::transformVector()
* @todo extract 2x2 matrix and multiply directly? (benchmark that)
*/
Vector2<T> transformVector(const Vector2<T>& vector) const {
return ((*this)*Vector3<T>(vector, T(0))).xy();
}
/**
* @brief Transform a 2D point with the matrix
*
* Unlike in @ref transformVector(), translation is also involved in
* the transformation. @f[
* \boldsymbol v' = \boldsymbol M \begin{pmatrix} v_x \\ v_y \\ 1 \end{pmatrix}
* @f]
* @see @ref DualComplex::transformPoint(),
* @ref Matrix4::transformPoint()
*/
Vector2<T> transformPoint(const Vector2<T>& vector) const {
return ((*this)*Vector3<T>(vector, T(1))).xy();
}
MAGNUM_RECTANGULARMATRIX_SUBCLASS_IMPLEMENTATION(3, 3, Matrix3<T>)
MAGNUM_MATRIX_SUBCLASS_IMPLEMENTATION(3, Matrix3, Vector3)
};
#ifndef DOXYGEN_GENERATING_OUTPUT
MAGNUM_MATRIXn_OPERATOR_IMPLEMENTATION(3, Matrix3)
#endif
template<class T> Matrix3<T> Matrix3<T>::rotation(const Rad<T> angle) {
const T sine = std::sin(T(angle));
const T cosine = std::cos(T(angle));
return {{ cosine, sine, T(0)},
{ -sine, cosine, T(0)},
{ T(0), T(0), T(1)}};
}
template<class T> Matrix2x2<T> Matrix3<T>::rotation() const {
Matrix2x2<T> rotation{(*this)[0].xy().normalized(),
(*this)[1].xy().normalized()};
CORRADE_ASSERT(rotation.isOrthogonal(),
"Math::Matrix3::rotation(): the normalized rotation part is not orthogonal:" << Corrade::Utility::Debug::newline << rotation, {});
return rotation;
}
template<class T> Matrix2x2<T> Matrix3<T>::rotationNormalized() const {
Matrix2x2<T> rotation{(*this)[0].xy(),
(*this)[1].xy()};
CORRADE_ASSERT(rotation.isOrthogonal(),
"Math::Matrix3::rotationNormalized(): the rotation part is not orthogonal:" << Corrade::Utility::Debug::newline << rotation, {});
return rotation;
}
template<class T> T Matrix3<T>::uniformScalingSquared() const {
const T scalingSquared = (*this)[0].xy().dot();
CORRADE_ASSERT(TypeTraits<T>::equals((*this)[1].xy().dot(), scalingSquared),
"Math::Matrix3::uniformScaling(): the matrix doesn't have uniform scaling:" << Corrade::Utility::Debug::newline << rotationScaling(), {});
return scalingSquared;
}
template<class T> inline Matrix3<T> Matrix3<T>::invertedRigid() const {
CORRADE_ASSERT(isRigidTransformation(),
"Math::Matrix3::invertedRigid(): the matrix doesn't represent a rigid transformation:" << Corrade::Utility::Debug::newline << *this, {});
Matrix2x2<T> inverseRotation = rotationScaling().transposed();
return from(inverseRotation, inverseRotation*-translation());
}
namespace Implementation {
template<class T> struct StrictWeakOrdering<Matrix3<T>>: StrictWeakOrdering<RectangularMatrix<3, 3, T>> {};
}
}}
#endif