The roll, pitch and yaw of an object relative to another is complex to compute. We use 3D complex number to compute them which makes the computation easier and more intuitive.

Roll, pitch and yaw are angles of orientation of an object in space and the conversion of these angles among different reference frames is not easy, as this example illustrates:

We will use 3D complex number to compute the roll,
pitch and yaw of a telephon relative to a car. We label the car as object a and the telephon as object
b. Their reference frames are labeled as
frame A and B and these frames are orientated
with respect to the ground whose frame is
labeled as frame G. The base vectors of the frame A are** a**_{x},** a**_{y} and** a**_{z},
that of the frame Bare **b**_{x},** b**_{y} and** b**_{z} and that of the
frame G are** g**_{x},** g**_{y}
and** g**_{z}. The vectors **a**_{x},** a**_{y} and** a**_{z}
and** b**_{x},** b**_{y} and** b**_{z} are expressedwith** g**_{x},** g**_{y}
and** g**_{z} in equations (1) and (2) with** M**_{a}and **M**_{b} being the matrices of transformation.

The roll, pitch and yaw of the
object b relative to the object a define the
orientation of the frame B in the frame A**.** This orientation is defined by the angles of rotation q, j and y, see Figure 1. For computing these angles we express the vectors **b**_{x},** b**_{y} and** b**_{z}
with the vectors **a**_{x},** a**_{y} and** a**_{z} in (3), (4) and (5). In (10), (11) and (12) the coefficients in the parentheses are dot products between
**a**_{x},** a**_{y} and** a**_{z}
and** b**_{x},** b**_{y} and** b**_{z}. The equations (10) , (11) and (12) are written in matrix form in
(13) from which we extract the
matrix of transformation** M**_{ba} shown in (14), see (3), (4) and (5).

From the 5 underscored coefficients in (14) we derive the angles q, j and y using (15), (16) and (17), knowing that is always positive because j is between -p/2 and p/2, see Figure 1. Roll, pitch and yaw of an object are the rotation angles of the object around the x, y and z axis respectively, see Figure 1 and the page https://en.wikipedia.org/wiki/Flight_dynamics_(fixed-wing_aircraft).

So, the conversion formula between roll, pitch and yaw and the angles q, j and y is equation (18).

The main advancement given by our method using 3D complex number is to have related the roll, pitch and yaw of an
object to the 3D complex number that represent the
x axis** **of the object. This mathematical discovery makes the
conversion of roll, pitch and yaw between
arbitrary frames easier and more intuitive.

The angles of
rotation q, j and y are computed with the 5 underscored
coefficients of the matrix **M**_{ba} in (14) saving thus the computation time for the
4 other coefficients which are not needed.
This will make the computation faster and the motion of moving
images on screen smoother. So, our method using 3D complex number would be
beneficial to applications such as video
games or street view of digital map etc.

For more detail of this study please read the complete paper here:

«Determination of the relative roll, pitch and yaw between arbitrary objects using 3D complex number»