Euler Angles Written by Paul Bourke June 2000

Rotations about each axis are often used to transform between different coordinate systems, for example, to direct the virtual camera in a flight simulator. These angles often go by different names, in the discussion here I will use a right hand coordinate system (y "forward", x to the right, and z upwards). As such rotation about the z axis will be referred to as direction, rotation about the y axis is roll (sometimes called bank), and rotation about the x axis is pitch. Further, a rotation will be considered positive if it is clockwise when looking down the axis towards the origin. Other conventions will be left as an exercise for the reader.

Rotation by t_x about the x axis

x' y' z'

=

1 0 0 0 cos(tx) sin(tx) 0 -sin(tx) cos(tx)

x y z

Rotation by t_y about the y axis

x' y' z'

=

cos(ty) 0 -sin(ty) 0 1 0 sin(ty) 0 cos(ty)

x y z

Rotation angle t_z about the z axis

x' y' z'

=

cos(tz) sin(tz) 0 -sin(tz) cos(tz) 0 0 0 1

x y z

A characteristic of applying these transformations is that the order is important. If the rotation matrices above are called R_x(t), R_y(t), and R_z(t) respectively then applying the rotations in the order R_z(t) R_x(t) R_y(t) will in general result in a different result to another order, say R_x(t) R_y(t) R_z(t). In what follows a particular order will be discussed and the other combinations will be left up to the reader to derive based on the same approach. The particular order of rotations applied here is to rotate about the y axis first (roll), they the x axis (pitch), then the z axis (direction). This is perhaps the most common order is usage in games and flight simulators.

x' y' z'

= Rz(t) Rx(t) Ry(t)

x y z

The single (combined) matrix is

cos(tz) cos(ty) + sin(tz) sin(tx) sin(ty)    sin(tz) cos(tx)    -cos(tz) sin(ty) + sin(tz) sin(tx) cos(ty) -sin(tz) cos(ty) + cos(tz) sin(tx) sin(ty)    cos (tz) cos(tx)    sin(tz) sin(ty) + cos(tz) sin(tx) cos(ty) cos(tx) sin(ty)    -sin(tx)    cos(tx) cos(ty)

One other requirement is given a new coordinate system how does one derive the corresponding three Euler angles. If the orthonormal vectors of the new coordinate system are X,Y,Z then the transformation matrix from (1,0,0), (0,1,0), (0,0,1) to the new coordinate system is

Xx Yx Zx Xy Yy Zy Xz Yz Zz

Also

And lastly

Note:

• The "programmers" function atan2() has been used above which uses the sign of the two arguments to calculate the correct quadrant of the result, this is in contrast to the mathematical tan() function.

• While the above gives particular values for t_x, t_y, and t_z there are a number of cases where the solution is not unique. That is, there are multiple combinations of Euler angles that will give the same coordinate transformation.