Transformation Cookbook

Rigid body transformations can be a source for confusion and mistakes in theoretical and practical settings. Rotations alone can be represented in a variety of ways. Three fundamentally different ways that I found most useful to understand in research and application are: rotation matrices, unit quaternions and axis-angles. There are connections between all three representations that can be exploited if properly used. Another common problem is notation: mistakes are preprogrammed if one is not careful to explicitly denote from which coordinate system into which other coordinate system a transformation transforms. And finally it is not straight forward to derive the gradient or Hessian for a cost function with respect to a rotation or full rigid body transformation.

Since such transformations are fundamental to robotics, computer vision and 3D perception and hence to my research, I took the time to write up what I call the Transformation Cookbook. The naming follows the famous Matrix Cookbook which has helped me figure out some pretty hairy matrix derivatives. I hope it will help clarify aforementioned issues, prevent some mistakes, and condense all necessary information into one document.

This is work in progress. Please alert me to any problems you find, things you would like to have added or things that are unclear.