Feature guide » 2D and 3D transformations

Representing transformations

The first and most straightforward way to represent transformations is to use homogeneous transformation matrix, i.e. Matrix3 for 2D and 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 Complex in 2D and Quaternion in 3D, or DualComplex and 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.

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.

auto a = Matrix3::rotation(23.0_degf);
auto b = Complex::rotation(Rad(Constants::piHalf()));
auto c = DualComplex::rotation(-1.57_radf);

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);

auto a = Matrix3::translation(Vector2::xAxis(-5.0f));
auto b = DualComplex::translation({-1.0f, 0.5f});

auto a = Matrix4::translation(vector);
auto b = DualQuaternion::translation(Vector3::zAxis(1.3f));

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));

auto a = Matrix3::reflection(Vector2::yAxis());
auto b = Matrix4::reflection(axis.normalized());

auto a = Matrix3::projection({4.0f, 3.0f});
auto b = Matrix4::orthographicProjection({4.0f, 3.0f}, 0.001f, 100.0f);
auto c = Matrix4::perspectiveProjection(35.0_degf, 1.333f, 0.001f, 100.0f);

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:

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);

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

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 Matrix3::transformVector() and Complex::transformVector(), in 3D using Matrix4::transformVector() and Quaternion::transformVector(). For transformation with normalized quaternion you can use faster alternative Quaternion::transformVectorNormalized(). Example:

auto transformation = Matrix3::rotation(-30.0_degf)*Matrix3::scaling(Vector2(3.0f));
Vector2 transformed = transformation.transformVector({1.5f, -7.9f});

Point transformation involves also translation, in 2D is done with Matrix3::transformPoint() and DualComplex::transformPoint(), in 3D with Matrix4::transformPoint() and DualQuaternion::transformPoint(). Also here you can use faster alternative DualQuaternion::transformPointNormalized():

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});

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:

Matrix4 transformation;
Matrix3x3 rotationScaling = transformation.rotationScaling();
Vector3 up = transformation.up();
Vector3 right = transformation.right();

Matrix3 b;
Matrix2x2 rotation = b.rotation();
Float xTranslation = b.translation().x();

Extracting scaling and rotation from arbitrary transformation matrices is harder and can be done using 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:

Matrix3 c = Matrix3::from(rotation, {1.0f, 3.0f});

Complex numbers and quaternions are far better in this regard and they allow you to extract rotation angle using Complex::angle() or Quaternion::angle() or rotation axis in 3D using Quaternion::axis(). Their dual versions allow to extract both rotation and translation part using DualComplex::rotation() const, DualQuaternion::rotation() const, DualComplex::translation() const and DualQuaternion::translation() const.

DualComplex a;
Rad rotationAngle = a.rotation().angle();
Vector2 translation = a.translation();

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

You can convert Complex and Quaternion to rotation matrix using Complex::toMatrix() and Quaternion::toMatrix() or their dual version to rotation and translation matrix using DualComplex::toMatrix() and DualQuaternion::toMatrix():

Quaternion a;
auto rotation = Matrix4::from(a.toMatrix(), {});

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

Conversion the other way around is possible only from rotation matrices using Complex::fromMatrix() or Quaternion::fromMatrix() and from rotation and translation matrices using DualComplex::fromMatrix() and DualQuaternion::fromMatrix():

Matrix3 rotation;
auto a = Complex::fromMatrix(rotation.rotationScaling());

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

Transformation interpolation

Normalizing transformations

When doing multiplicative transformations, e.g. adding rotating to a 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 Math::Algorithms::gramSchmidtOrthogonalize() (or Math::Algorithms::gramSchmidtOrthonormalize(), if you don't have any scaling). You can also use Math::Algorithms::svd() to more precisely (but way more slowly) account for the drift. Example:

Matrix4 transformation;
Math::Algorithms::gramSchmidtOrthonormalizeInPlace(transformation);

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

DualQuaternion transformation;
transformation = transformation.normalized();