Here we have provided a C++ class for quaternional algebra. You need these two files: quaternion.h & quaternion.c++. To be able to work with Euler angles, you also need to download Ken Shoemake's QuatTypes.h, EulerAngles.h, EulerAngles.c and define SHOEMAKE in your makefile. Here is an example how to use it all together: (test.c++) and a makefile. For more on quaternions, read Prof. George Francis's introduction lecture.
Quaternions are elements of the 4-dimensional space 
formed by the real
axis and 3 imaginary orthogonal axes 
, 
, and 
that obey Hamilton’s
rule 
. They can be written
in a standard quaternionial form as 
where 
, or as a 4D vector 
where 
is called scalar
part and 
is called vector
part. Quaternions possess the
following properties:
Addition: for 
such that 
such that 
Multiplication: for 
and 
and 
such that 
, then there exists 
such that 
where 
denotes vector
dot product and 
denotes vector cross product
, then either 
or 
, from 
follows that 
and 
is the magnitude
of 
, 
is its norm. If 
, the quaternion 
is referred to as a unit
quaternion. For 
is a unit
quaternion. Inverse of 
is defined as 
and the conjugate of 
is defined as 
. For any unit
quaternion 
we have 
. Quaternions whose
real part is zero are called pure quaternions.
Rotation of a 3D vector 
by a unit quaternion 
is defined as 
where 
is a pure quaternion
build from 
by adding a zero real
part. Sequences of rotations can be
conveniently represented as the quaternionial product. For example, if 
is rotated by 
followed by 
, the result is the same as 
rotated by 
.
Document is created by Angela Bennett and Volodymyr
Kindratenko
Last modified: Tuesday, August 29,
2000 11:00:00 AM