magnum/doc/transformations.dox

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              Vladimír Vondruš <mosra@centrum.cz>

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namespace Magnum {
/** @page transformations 2D and 3D transformations
@brief Introduction to essential operations on vectors and points.

-   Previous page: @ref matrix-vector
-   Next page: @ref plugins

Transformations are essential operations involved in scene management -- object
relations, hierarchies, animations etc. They extend basic vectors and matrices
in @ref Math namespace, see its documentation for more information about usage
with CMake.

@tableofcontents

Magnum provides classes for transformations in both 2D and 3D. Each class is
suited for different purposes, but their usage is nearly the same to make your
life simpler. This page will explain the basic operation and differences
between various representations.

@section transformations-representation Representing transformations

The first and most straightforward way to represent transformations is to use
homogeneous transformation matrix, i.e. @ref Matrix3 for 2D and @ref Matrix4
for 3D. The matrices are able to represent all possible types of
transformations -- rotation, translation, scaling, reflection etc. and also
projective transformation, thus they are used at the very core of graphics
pipeline and are supported natively in OpenGL.

On the other hand, matrices need 9 or 16 floats to represent the
transformation, which has implications on both memory usage and performance
(relatively slow matrix multiplication). It is also relatively hard to extract
transformation properties (such as rotation angle/axis) from them, interpolate
between them or compute inverse transformation. They suffer badly from
so-called floating-point drift -- e.g. after a few combined rotations the
transformation won't be pure rotation anymore, but will involve also a bit of
scaling, shearing and whatnot.

However, you can trade some transformation features for improved performance
and better behavior -- for just a rotation you can use @ref Complex in 2D and
@ref Quaternion in 3D, or @ref DualComplex and @ref DualQuaternion if you want
also translation. It is not possible to represent scaling, reflection or other
transformations with them, but they occupy only 2 or 4 floats (4 or 8 floats in
dual versions), can be easily inverted and interpolated and have many other
awesome properties. However, they are not magic so they also suffer slightly
from floating-point drift, but not too much and the drift can be accounted for
more easily than with matrices.

@section transformations-types Transformation types

Transformation matrices and (dual) complex numbers or quaternions have
completely different internals, but they share the same API to achieve the same
things, greatly simplifying their usage. In many cases it is even possible to
hot-swap the transformation class type without changing any function calls.

@subsection transformations-default Default (identity) transformation

Default-constructed @ref Matrix3, @ref Matrix4, @ref Complex, @ref Quaternion,
@ref DualComplex and @ref DualQuaternion represent identity transformation, so
you don't need to worry about them in initialization.

@subsection transformations-rotation Rotation

2D rotation is represented solely by its angle in counterclockwise direction
and rotation transformation can be created by calling @ref Matrix3::rotation(),
@ref Complex::rotation() or @ref DualComplex::rotation(), for example:
@code
auto a = Matrix3::rotation(23.0_degf);
auto b = Complex::rotation(Rad(Constants::pi()/2));
auto c = DualComplex::rotation(-1.57_radf);
@endcode

3D rotation is represented by angle and (three-dimensional) axis. The rotation
can be created by calling @ref Matrix4::rotation(), @ref Quaternion::rotation()
or @ref DualQuaternion::rotation(). The axis must be always of unit length to
avoid redundant normalization. Shortcuts @ref Vector3::xAxis(),
@ref Vector3::yAxis() and @ref Vector3::zAxis() are provided for convenience.
Matrix representation has also @ref Matrix4::rotationX(),
@ref Matrix4::rotationY() and @ref Matrix4::rotationZ() which are faster than
using the generic function for rotation around primary axes. Examples:
@code
auto a = Quaternion::rotation(60.0_degf, Vector3::xAxis());
auto b = DualQuaternion::rotation(-1.0_degf, Vector3(1.0f, 0.5f, 3.0f).normalized());
auto c = Matrix4::rotationZ(angle);
@endcode

Rotations are always around origin. Rotation about arbitrary point can be done
by applying translation to have the point at origin, performing the rotation and
then translating back. Read below for more information.
@todo DualQuaternion and rotation around arbitrary axis

@subsection transformations-translation Translation

2D translation is defined by two-dimensional vector and can be created with
@ref Matrix3::translation() or @ref DualComplex::translation(). You can use
@ref Vector2::xAxis() or @ref Vector2::yAxis() to translate only along given
axis. Examples:
@code
auto a = Matrix3::translation(Vector2::xAxis(-5.0f));
auto b = DualComplex::translation({-1.0f, 0.5f});
@endcode

3D translation is defined by three-dimensional vector and can be created with
@ref Matrix4::translation() or @ref DualQuaternion::translation(). You can use
@ref Vector3::xAxis() and friends also here. Examples:
@code
auto a = Matrix4::translation(vector);
auto b = DualQuaternion::translation(Vector3::zAxis(1.3f));
@endcode

@subsection transformations-scaling Scaling and reflection

Scaling is defined by two- or three-dimensional vector and is represented by
matrices. You can create it with @ref Matrix3::scaling() or @ref Matrix4::scaling().
You can use @ref Vector3::xScale(), @ref Vector3::yScale(), @ref Vector3::zScale()
or their 2D counterparts to scale along one axis and leave the rest unchanged
or call explicit one-parameter vector constructor to scale uniformly on all
axes. Examples:
@code
auto a = Matrix3::scaling(Vector2::xScale(2.0f));
auto b = Matrix4::scaling({2.0f, -2.0f, 1.5f});
auto c = Matrix4::scaling(Vector3(10.0f));
@endcode

Reflections are defined by normal along which to reflect (i.e., two- or
three-dimensional vector of unit length) and they are also represented by
matrices. Reflection is created with @ref Matrix3::reflection() or
@ref Matrix4::reflection(). You can use @ref Vector3::xAxis() and friends also
here. Examples:
@code
auto a = Matrix3::reflection(Vector2::yAxis());
auto b = Matrix4::reflection(axis.normalized());
@endcode

Scaling and reflection is also done relative to origin, you can use method
mentioned above to scale or reflect around arbitrary point.

Scaling and reflection can be (to some extent) also represented by complex
numbers and quaternions, but it has some bad properties and would make some
operations more expensive, so it's not implemented.

@subsection transformations-projective Projective transformations

Projective transformations eploit the full potential of transformation
matrices. In 2D there is only one projection type, which can be created with
@ref Matrix3::projection() and it is defined by area which will be projected
into unit rectangle. In 3D there is orthographic projection, created with
@ref Matrix4::orthographicProjection() and defined by volume to project into
unit cube, and perspective projection. Perspective projection is created with
@ref Matrix4::perspectiveProjection() and is defined either by field-of-view,
aspect ratio and distance to near and far plane of view frustum or by size of
near plane, its distance and distance to far plane. Some examples:
@code
auto a = Matrix3::projection({4.0f, 3.0f});
auto b = Matrix4::orthographicProjection({4.0f, 3.0f, 100.0f});
auto c = Matrix4::perspectiveProjection(35.0_degf, 1.333f, 0.001f, 100.0f);
@endcode

@section transformations-composing Composing and inverting transformations

Transformations (of the same representation) can be composed simply by
multiplying them, it works the same for matrices, complex numbers, quaternions
and their dual counterparts. Order of multiplication matters -- the
transformation on the right-hand side of multiplication is applied first, the
transformation on the left-hand side is applied second. For example, rotation
followed by translation is done like this:
@code
auto a = DualComplex::translation(Vector2::yAxis(2.0f))*
         DualComplex::rotation(25.0_degf);
auto b = Matrix4::translation(Vector3::yAxis(5.0f))*
         Matrix4::rotationY(25.0_degf);
@endcode

Inverse transformation can be computed using @ref Matrix3::inverted(),
@ref Matrix4::inverted(), @ref Complex::inverted(), @ref Quaternion::inverted(),
@ref DualComplex::inverted() or @ref DualQuaternion::inverted(). Matrix
inversion is quite costly, so if your transformation involves only translation
and rotation, you can use faster alternatives @ref Matrix3::invertedRigid() and
@ref Matrix4::invertedRigid(). If you are sure that the (dual) complex number
or (dual) quaternion is of unit length, you can use @ref Complex::invertedNormalized(),
@ref Quaternion::invertedNormalized(), @ref DualComplex::invertedNormalized()
or @ref DualQuaternion::invertedNormalized() which is a little bit faster,
because it doesn't need to renormalize the result.

@section transformations-transforming Transforming vectors and points

Transformations can be used directly for transforming vectors and points.
Vector transformation does not involve translation, in 2D can be done using
@ref Matrix3::transformVector() and @ref Complex::transformVector(), in 3D
using @ref Matrix4::transformVector() and @ref Quaternion::transformVector().
For transformation with normalized quaternion you can use faster alternative
@ref Quaternion::transformVectorNormalized(). Example:
@code
auto transformation = Matrix3::rotation(-30.0_degf)*Matrix3::scaling(Vector2(3.0f));
Vector2 transformed = transformation.transformVector({1.5f, -7.9f});
@endcode

Point transformation involves also translation, in 2D is done with
@ref Matrix3::transformPoint() and @ref DualComplex::transformPoint(), in 3D
with @ref Matrix4::transformPoint() and @ref DualQuaternion::transformPoint().
Also here you can use faster alternative @ref DualQuaternion::transformPointNormalized():
@code
auto transformation = DualQuaternion::rotation(-30.0_degf, Vector3::xAxis())*
                      DualQuaternion::translation(Vector3::yAxis(3.0f));
Vector3 transformed = transformation.transformPointNormalized({1.5f, 3.0f, -7.9f});
@endcode

@section transformations-properties Transformation properties and conversion

It is possible to extract some transformation properties from transformation
matrices, particularly translation vector, rotation/scaling part of the matrix
(or pure rotation if the matrix has uniform scaling) and also base vectors:
@code
Matrix4 a;
auto rotationScaling = transformation.rotationScaling();
Vector3 up = transformation.up();
Vector3 right = transformation.right();

Matrix3 b;
auto rotation = b.rotation();
Float xTranslation = b.translation().x();
@endcode
Extracting scaling and rotation from arbitrary transformation matrices is
harder and can be done using @ref Math::Algorithms::svd(). Extracting rotation
angle (and axis in 3D) from rotation part is possible using by converting it to
complex number or quaternion, see below.

You can also recreate transformation matrix from rotation and translation
parts:
@code
Matrix3 c = Matrix3::from(rotation, {1.0f, 3.0f});
@endcode

Complex numbers and quaternions are far better in this regard and they allow
you to extract rotation angle using @ref Complex::angle() or
@ref Quaternion::angle() or rotation axis in 3D using @ref Quaternion::axis().
Their dual versions allow to extract both rotation and translation part using
@link DualComplex::rotation() const @endlink, @link DualQuaternion::rotation() const @endlink,
@link DualComplex::translation() const @endlink and
@link DualQuaternion::translation() const @endlink.
@todoc Remove workaround when Doxygen can handle const
@code
DualComplex a;
Rad rotationAngle = a.rotation().angle();
Vector2 translation = a.translation();

Quaternion b;
Vector3 rotationAxis = b.axis();
@endcode

You can convert Complex and Quaternion to rotation matrix using
@ref Complex::toMatrix() and @ref Quaternion::toMatrix() or their dual version
to rotation and  translation matrix using @ref DualComplex::toMatrix() and
@ref DualQuaternion::toMatrix():
@code
Quaternion a;
auto rotation = Matrix4::from(a.toMatrix(), {});

DualComplex b;
Matrix3 transformation = b.toMatrix();
@endcode

Conversion the other way around is possible only from rotation matrices using
@ref Complex::fromMatrix() or @ref Quaternion::fromMatrix() and from rotation
and translation matrices using @ref DualComplex::fromMatrix() and
@ref DualQuaternion::fromMatrix():
@code
Matrix3 rotation;
auto a = Complex::fromMatrix(rotation.rotationScaling());

Matrix4 transformation;
auto b = DualQuaternion::fromMatrix(transformation);
@endcode

@section transformations-interpolation Transformation interpolation

@todoc Write this when interpolation is done also for (dual) complex numbers and
    dual quaternions

@section transformations-normalization Normalizing transformations

When doing multiplicative transformations, e.g. adding rotating to an
transformation many times during an animation, the resulting transformation will
accumulate rounding errors and behave strangely. For transformation matrices
this can't always be fixed, because they can represent any transformation (and
thus no algorithm can't tell if the transformation is in expected form or not).
If you restrict yourselves (e.g. only uniform scaling and no skew), the matrix
can be reorthogonalized using @ref Math::Algorithms::gramSchmidtOrthogonalize()
(or @ref Math::Algorithms::gramSchmidtOrthonormalize(), if you don't have any
scaling). You can also use @ref Math::Algorithms::svd() to more precisely (but
way more slowly) account for the drift. Example:
@code
Matrix4 transformation;
Math::Algorithms::gramSchmidtOrthonormalizeInPlace(transformation);
@endcode

For quaternions and complex number this problem can be solved far more easily
using @ref Complex::normalized(), @ref Quaternion::normalized(),
@ref DualComplex::normalized() and @ref DualQuaternion::normalized().
Transformation quaternions and complex numbers are always of unit length, thus
normalizing them reduces the drift.
@code
DualQuaternion transformation;
transformation = transformation.normalized();
@endcode

-   Previous page: @ref matrix-vector
-   Next page: @ref plugins
*/
}