Maths - Affine Conversion - Matrix to Euler/Vector

Definition of terms:

It you want to consider the rotation only then the code for this is shown here.

Alternatively if we want to consider the possibility that there is also a translation from the centre, and that the rotation may not be about the centre but may be about some arbitrary point, then we need to extend the notation as follows:


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. See this page for the conventions and standards used on this site.

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.












metadata block
see also:


Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.


This site may have errors. Don't use for critical systems.

Copyright (c) 1998-2016 Martin John Baker - All rights reserved - privacy policy.