Definition of terms:

• Euler Angles
• Matrix

This conversion is better avoided, since it is non-linier and has singularities at + & - 90 degrees, so if already working in terms of matricies its better to continue using matrices if possible.

This depends on what conventions are used for the Euler Angles. The following assumes NASA Standard Airplane

So it we look at the Euler to Matrix conversion we can see that:

m01/m00 = s1 / c1 = tan(heading)
m12/m22 = s3 / c3 = tan(bank)
m02 = -s2 = -sin(attitude)

so this gives:

heading = atan(m01/m00)
bank = atan(m12/m22)
attitude = asin(-m02)

Note this only applies to a martix which represents a pure rotation. The equations for heading and bank should be independent of uniform scaling as it will cancel out in the division. It would be better to find an expression for attitude which is also independent of scaling.

Since there are several ways to produce the same rotation using heading, bank and attitude then the solution is not unique. tan(0) is 0, tan(90 degrees) is infinity, tan(-90 degrees) is -infinity. So the results will depend on whether arctan processes a result between -90 and 90 or 0 and 180.

Issues

• It is better to use atan2 function than atan function as described here.
• For issues concerning asin and atan see here.
• Most trig functions (except openGL) use radians not degrees.

Example

we take the 90 degree rotation from this:

to this:

As shown here the matrix for this rotation is:

[R] =

1 0 0 0 0 -1 0 1 0

So using the above result:

heading = atan(m01/m00) = atan(0) = 0
bank = atan(m12/m22) = atan(infinity) = 90 degrees
attitude = asin(-m02) = asin(0) = 0

As you can see here, this gives the result that we are looking for.

We have to be very careful about the following issues:

• One of the intermediate steps involves infinity (division by 0) therefore most maths libraries have an atan2 function to avoid this step.
• Most maths libraries use radians instead of degrees (apart from OpenGL).