Given three vectors representing an x direction, y direction, and z direction, is there a way to uniquely determine 3 angular rotations around each of the x,y,z axes. I have no experience with differential geometry. I'm just trying to convert a "frame" I have of an object in 3-dimensional space to a set of rotations I can use to rotate a 3D model into it's proper orientation. It's for making a thing walk on the surface of a sphere and this is the only thing I have left to do. Otherwise, I'd be left with a very bizarre looking thing (it would appear to roll on the surface of the sphere).
Once again, reiterating:
is there any way to take 3 vectors and determine rotations that produce them assuming we know which axis each one began as pointing in. Assume they are in regular cartesian coordinates (where x points out, y points right, and z points up).
There is no unique way. Say you have a model plane, pointing in the $x$ direction, $y$ direction is horizontal, perpendicular to $x$, and $z$ is vertical. Now you want to flip the plane upside down, pointing forward. That means $x'=x$, $y'=-y$, and $z'=-z$. I can get to this state in at least two different way. The most obvious one is a rotation around $x$ axis by 180$^\circ$. I can also rotate 180$^\circ$ around $y$ axis (plane is upside down, but pointing backwards in the $-x$ direction), and then rotate 180$^\circ$ around the $z$ axis.
In computer graphics programs what people are usually doing is deciding a simple rotation (usually the least amount of rotation) and decompose the path into equally spaced rotation angles, to make the motion seem smooth.