Quaternion Rotation UMBC Graphics for Games Representing Rotation

Quaternion Rotation UMBC Graphics for Games

Representing Rotation

Rotation Matrices • Rotation can be expressed as an orthogonal matrix • Every orthogonal matrix is a rotation and/or mirroring New XNew axis Y axis New Z axis

Rotation Matrices • Rotation can be expressed as an orthogonal matrix • Every orthogonal matrix is a rotation and/or mirroring • Fits with other transformation matrices • Translation, Scaling, Reflection, Perspective • Interpolations between rotation matrices are not rotations • Interpolates endpoint positions

Euler Angles • Rotate around series of axes • Roll, Pitch, Yaw • Intuitive, pitch/yaw form common for first person controllers • Yaw = turn left/right • Pitch = look up/down • Roll = rotate around view • Subject to gimbal lock • First axis lines up with last axis, lose ability to rotate freely • Interpolates along lines of longitude/latitude • Can be weird near poles

Quaternions • Extension of complex numbers • Interpolations on great circle • Related Geometric Algebra also works • Though not as popular in games

Complex Numbers

Imaginary number i • Define imaginary number i where • Complex number has real and imaginary components • • Interpret as points on the 2 D plane • • • Conjugate: negate the imaginary part • •

Complex Arithmetic • Addition: add components • • Multiplication • Component algebra: • Multiply magnitudes, add angles:

Complex Transformations • Point • Translation = addition • • Scale = multiply by real scalar • • Rotate = multiply by unit-length complex (magnitude = 1) • • • Component form is identical to matrix rotation

Basic Rotations • Pure imaginary = rotation by 90 • • •

Quaternions

Quaternion basics • First created by Hamilton in 1843 for physics rotations • Define orthogonal imaginary axes • • Multiplication order matters, positive in ijk order • • •

Quaternion Math • Like complex has real + imaginary, quaternion is real + vector • • Conjugate negates vector part • • Addition by component • •

Quaternion Multiplication • By component • Vector form

Quaternion Transformations • Point is pure quaternion • Translation = addition of pure quaternion • Scale = multiply by real scalar • Rotation = sandwiched multiply by unit quaternion and conjugate • • Where • For one point, algebraically optimized version with cross products • For more points, cheaper to convert to matrix

Basic Rotations • Rotation • Rotate by 0º • • • Rotate by 180º around vector • •

Interpolation • Interpolate along great circle between two vectors / quaternions • Normalized Linear Interpolation (NLERP) • • Cheap, but speeds up in the middle • Spherical Linear Interpolation (SLERP) • Interpolate in equal-angular steps • •