how to Rotate about an arbitrary axis?

Givens

1- X,y,and Z the world co-ordinate system 2-i,j,k another co-ordinate system. 3-the cosines in which each of i,j, and k make with the X,Y,开发者_如何学编程Z.

problem

• how to rotate the i,j,k system about i or j or k??

If you have the cosines of the angles formed by pairing each of i,j,k with each of xhat, yhat, and zhat (nine angles altogether), you have the makings for the direction cosine matrix. For example, see http://www.ae.illinois.edu/~tbretl/ae403/handouts/06-dcm.pdf (or just google direction cosine matrix). The direction cosine matrix is just another name for a transformation or rotation matrix.

Be careful, though!
There is no single standard scheme. You need to know that this is the case and read the literature carefully.

• Are you rotating the object or transforming coordinates? Rotation and transformation are conjugate operations. Some people (many people!) use the term 'rotation matrix' when they mean 'transformation matrix', and vice versa.
• Do you represent vectors as column vectors or row vectors? Here there is a lot more consistency; most people use column vectors rather than row vectors for things like positions, velocities, etc. BUT there are very good reasons to use row vectors (or column vectors if you are one of those contrarians) for things that properly belong in the dual space.
• Quaternions have even more ambiguity of representation than matrices. There's nothing wrong with that (I use quaternions all the time), but you do have to beware of these ambiguities when you read a paper or book, look at someone else's code, or exchange data.
• Finally, matrices and quaternions are only two of many charts on SO(3). There are lots of ways to represent rotations in 3-space.

You can first create either a rotation matrix or a quaternion. Then you use that to transform your vectors.

You can find the code to create a rotation matrix or a quaternion in pretty much any 3d maths library.

If I recall correctly you calculated the rotation quaternion as(assuming normalized axis):

q.x=axis.x*sin(alpha)
q.y=axis.y*sin(alpha)
q.y=axis.z*sin(alpha)
q.w=cos(alpha)