Matrices and Transforms
Matrix and Vector Properties
 Vectors are just 1xn or nx1 matrices
 Column or row major – depends on which kind of vector you choose
 Homogeneous coordinates – invented to allow translations (a non-linear operation) to be
represented using matrix multiplication (instead of addition)
 Representation in memory (C/C++)
Matrix Operations
 Addition and Subtraction
 Matrix multiplication
 Transpose
 Determinant (if non-zero, inverse exists; if 1, pure rotation [special orthogonal])
 Matrix inverse (same as transpose for an orthogonal matrix)
Atomic Transforms
 Translation
o As a vector operation
o Using homogenous coordinates
 Rotation
o Around one axis
o Around an arbitrary axis
 Scale
o Uniform; Non-uniform (this will hose normals)
 Shearing
o [ 1 0 shx 0 ][ 0 1 shy 0 ][ 0 0 1 0 ][ 0 0 0 1 ], gives [ (x + shx z) (y + shy z) z 1]
 Concatenation: Multiply atomic matrices to generate composite matrix – order matters!
 Types of Matrices
o Special Orthogonal (aka Isotropic): each row or col is a unit vector; the vectors
are mutually orthogonal [ = pure rotation matrices]
o Affine: Preserves parallelism of lines and relations of distances, but not lengths
and angles [ = special orthogonal + translation + (non-uniform) scale + shear]
(product of rotation, translation, scale and/or shear)
Transforming Vectors vs. Points
 Vectors ignore the translation
 In homogeneous coordinates, vectors have w=0, not w=1
Describing Position, Orientation and Scale of a Rigid Body
 Think of the rigid body as having a local coordinate system (Model Space)
o Front, Left, Up
o Roll (rotation about Front), Pitch (rot about Left), Yaw (rot about Up)
 The world has a global coordinate system (World Space)
 Pos, Orient and Scale of a rigid body is really just a matrix transforming the Model Space
coord. system into World Space (Model-to-World)
Transforming Coordinate Systems
 Matrix as (i, j, k, t) vectors
 Mc-p = [ ic jc kc tc ]
 scaling applies to upper 3x3
 REVISIT: Representation in memory (C/C++)
Alternatives for Representing Rotations
 Euler Angles
o Pros: simplest/smallest representation, intuitive (pitch, yaw, roll), can interpolate
for simple rotations about a single axis
o Cons: gimbal lock, difficult to interpolate arbitrary rotations
 Matrices
o Pros: no gimbal lock, easy to apply to rigid body representation, corresponds
directly to coordinate axes
o Cons: large (9 elements), unintuitive, difficult to interpolate
 Axis + Angle
o Pros: no gimbal lock, intuitive
o Cons: difficult to interpolate
 Quaternions
o What are they? q  qxi + qyj + qzk + qw = [ qv, qw ]
o Unit quaternions are a representation for rotations: where qw = cos(½) and
qx = qy = qz = sin(½), n = unit rotation axis; i.e. [sin(½)n, cos(½) ]
o Pros: no gimbal lock, relatively intuitive (much like axis + angle), easy to
interpolate (LERP, SLERP)
o Cons: more expensive to apply to rigid body orientations (either need mat-to-quat
and quat-to-mat, or do quat-vector and quat-quat multiplies)
o Conjugate: q = –qxi – qyj – qzk + qw
o Norm/Length: |q| = qq
o Inverse: q–1 = q / |q| = q / (qq) = q (conjugate) for unit quaternions
o Sum: (q + r) = (qx + rx)i + (qy + ry)j + (qz + rz)k + (qw + rw) = [(qv + rv), (qw + rw)]
o Product:
q r = (qw rx + qx rw + qy rz – qz ry)i
+ (qw ry + qy rw – qx rz + qz rx)j
+ (qw rz + qz rw + qx ry – qy rx)k
+ (–qx rx – qy ry – qz rz + qw rw)
= [ qv, qw ] [ rv, rw ] = [ (qwrv + rwqv + qv  rv), (qwrw – qv  rv) ]
o Rotation of a point p by quaternion q:
First, convert p into a “quaternion form” p = [ p, 0 ]
Next, find p' = qpq–1 = qpq = (qp)q
Finally, convert p' back to p' by simply extracting it from [ p', 0 ]
o Interpolation:
s1 = LERP(q, r, ) = (1 – )q + ()r (e.g.  = 0.2: 20% of the way from q to r)
s2 = SLERP(q, r, ) = ()q + ()r
where we define   cos–1(qv  rv)
and then find  = ((1 – )) / sin() and  = () / sin()
 A Note on Degrees of Freedom
o DOF = dimension – constraint count
o Euler Angles: dimension=3, constraints=0, DOF=3
o Quaternions: dimension=4, constraints=1 (unit quaternion), DOF=3
o 3x3 Matrices: dimension=9, constraints=6 (all 6 unit vectors), DOF=3
 Resources
o http://skal.planet-d.net/demo/matrixfaq.htm#Q1
o http://mathworld.wolfram.com/Quaternion.htm
o http://ggt.sourceforge.net/html/group__Interp.html#a1