3D rotations of a triangle

[ste3e]

If I have two triangles A and B of the same dimensions, is it true that for any independent position in the world of both A and B there is a single quaternion that can map A to B? By "map" I mean rotate A so that it's plane and rotation of the plane lies parallel and same oriented as B's. If it is true, and if I know the positions of A and B, how would I calculate the quaternion that maps A to B? Let's say A faces south and B faces east and is also rotated through 45 degrees about the world x axis (it's own normal).

 

[mmm4444bot]

From the page http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/index.htm I found the following:


Use of quaternions to represent transformations in 3D.

The main practical application of this interesting algebra is to represent 3D rotations.

In fact quaternions can represent 3D reflections, rotations and scaling, however a single quaternion operation cannot include translations so if we want to rotate, reflect or scale around a point other than the origin, then we would have to handle the translation part separately (see affine translations).

 

[ste3e]

Translation and scale are not involved in the map as I have defined it, it is purely the rotation about the three axis that I wish to construct. It is simple enough to map two vectors but no vector can retain any rotation about which it is spun, thus the mapping of a triangle to a triangle is not simply a matter of mapping two vectors. Think of it like this: the two triangles (easily if they are right triangles) describe different base frames, constructing a quaternion from normal to normal of each frame is going to loose rotation information that a matrix between the base frames would contain. The question really is, is there a math way to generate the quaternion of that matrix without having to construct the base frames? Can the quaternion be calculate simply from the relative positions of the triangle vertices?

 

[mmm4444bot]

The question really is, is there a math way to generate the quaternion of that matrix without having to construct the base frames?

My first post responds to the following question.

is it true that for any independent position in the world of both A and B there is a single quaternion that can map A to B?

I hope that somebody responds to the real question because I do not know the answer. Cheers :cool: