and has the form 
and has the form 
, extend the point with an extra 1 (not zero) and then left multiply by the matrix: 
includes all rotation, reflection, scaling, and shearing, and the vector 
adds any translation- standard matrix multiplication and inversion are correct, but we can take advantage of the special form of homogeneous transformation matrices (that the bottom row is always
) to come up with more efficient formulas - if
and 
then 
- if
then 
- the rigid transformations are still translation, rotation, and reflection
- the non-rigid transformations are still scaling and shearing
when the upper-left submatrix
is the identity matrix, the homogeneous transform reduces to a pure translation by :
- a homogeneous transform
is a pure rotation when the following conditions hold 
- is orthonormal
- The first condition is obvious (no translation). The second two conditions mean that the columns of can be visualized as a basis
for a rotated coordinate system. The columns are literally unit vectors in the direction of each coordinate axis of the rotated system, expressed as coordinate vectors in the parent frame. 
- It may not be intuitive, but the third condition ensures that the rotated coordinate frame is right-handed; the first condition alone would allow either a right or left-handed child frame. If the child frame is left-handed (assuming the parent frame is right-handed) then the transformation would not be a pure rotation, it would also include a reflection.
- for rotation in 2D, is
, and we know that it has the form 
, where 
is the CCW rotation from the parent frame 
to the rotated frame 
- we can begin to extend this to 3D by just using this as the upper left submatrix of a matrix which is otherwise the identity (ones along the major diagonal and zero everywhere else):
this has the effect of rotating the child frame by
as a shorthand for rotation matrices of this form
CCW radians about the 
axis of the parent frame, and we can define a function 
- any 3D rotation can be formed as the composition of three 2D rotations about orthogonal axes
- so far we have a rotation about , so we can complete the picture by deriving functions that produce rotation matrices about and
- using the right hand rule, notice that CCW rotation about rotates
towards 
so
(notice that the
is in the opposite location for 
)- similarly, CCW rotation about rotates
towards 
so
- finally, we can compose these together to produce a general 3D rotation matrix:
- note that the order of the composition matters
- any order is sufficient, but if someone tells you
, then you both also need to agree on the order of rotation about the three different axes - the
parameters are called Euler angles when using the specific sequence -- (also called heading (or yaw), pitch, and roll) - sometimes this term is used to refer to any three-angle convention
- any order is sufficient, but if someone tells you
- though it may not be obvious, there are a few issues with representing a 3D rotation with three angles, especially when it comes to interpolating 3D rotations in an animation; we will discuss this more later and introduce some new representations that can be helpful in that case
- in 2D we reflected about either the or coordinate axis by starting with
and then flipping one of the diagonal entries from 
to 
- in 3D we do not reflect about an axis, but rather a coordinate plane
- to reflect about the
plane, use 
- to reflect about the
plane, use 
- to reflect about the
plane, use 
- as in 2D, the composition of two reflections is no longer a reflection: it is a rotation
- the composition of three reflections is still a reflection
- in 2D we scaled along the and/or coordinate axis by starting with and then setting each diagonal entry to the desired non-negative scale factor for the corresponding axis
- we can do the same in 3D:
scales each of the three coordinate axes by the indicated amount - again, if the scale amounts are different for different axes, then the scale is non-uniform; if they are all the same then the scale is uniform
- in 2D we formed a shear matrix by starting with , selecting one of the basis vectors
to remain fixed, and then shifting the other basis vector in the direction of - the same approach works in 3D
shears the and axes in the direction of 
shears the and axes in the direction of 
shears the and axes in the direction of
and 
, then it does work to linearly interpolate each angle, e.g. 
(though see below for more discussion of this)
as above
on the pose of the child frame is exactly the same as the effect of 
, where the translation transform 
just has 
is called an axis-angle representation
for the differential rotation from one given orientation to another, then linear interpolation from 
to 
to 
) works well
and 
where 
, or 
or 
, i.e. the opposite rotation about the same axis, or the same rotation about the opposing axis
is the same rotation as 
of an arbitrary point 
(
), in part because of the need to compute 
and 
is composed of a 3D vector 
and a scalar 
: 
(sometimes quaternions are written in the opposite order, i.e. 
)
has unit magnitude iff 
, where 
, we can easily show that 
by just appending the scalar part 
be a unit quaternion corresponding to the axis-angle 
is the inverse rotation in unit quaternion form
and 
corresponds to the inverse of the rotation corresponding to 
is the same rotation as 
and 
(using trigonometric 
is also the same rotation as 
, but now we have the corresponding quaternion 
is the same rotation as 
, 
is calculated as 
are unit quaternions corresponding to 3D rotations, then 
by a rotation represented by the unit quaternion 
, and then the rotated point 
is computed as 
does apply 
to any other orientation represented as unit quaternion 
, use a procedure called spherical linear interpolation (SLERP)
corresponding to any value of a scalar parameter 
or 
return 
or 
return 