chuckb@milton.u.washington.edu (Chuck Bass) writes:

Graphics People:

I have the following problem. I have two unit vectors and I need to find the transformation matrix to go from one to the other. (I need to align an object to a normal vector)

    [x y z 1]      [T]    = [ x1 y1 z1 1]

I know that there is no translation in this transform so it is really the 3X3 problem.

    [x y z ]      [T']    = [ x1 y1 z1]

I have three equations here. Properties of [T'] give me three more equations from trigonometric relationships. Problem is, I need 3 more equations.

What am I missing?? Thanks for any insight.

Several weeks ago in a computer graphics class here, Blinn had us calculate the rotation matrix to go from EME50 to Voyager space so we could use actual JPL data and rotate Voyager's cameras so we could see Jupiter and its moons. Anyway, EME50 is this coordinate system defined based on something that happened in the solar system in 1950, and basically it was off from Voyager's coordinate system. The problem is the same as yours: Voyager had to keep its main antenna aimed at Earth all the time and its cameras were on a boom that was some weird angle off of the body. We were given EME50 coordinates, and to aim the camera, we had to first do the rotation matrix from EME50 to align vectors. In a long-winded way, here's what you do:

Find another vectors perpendicular to both vectors (x has xp, and x1 has x1p). Now form a third vector by taking x X xp (that's a cross-product of x and xp). Same deal with x1 and x1p. For notational simplicity, do this:

    x -- first vector
    m -- second vector perp. to x
    c -- the cross-product of x and m

    x1 -- the second vector
    m1 -- a vector perp to x1
    c1 -- the cross-product of x1 and m1

    T -- the transformation matrix you want to find.

Now write it out like this:

    |x|     |x1|
    |m| T = |m1| 
    |c|     |c1|,

Where [x, m, c] and [x1, m1, c1] are matrices. Now it should be obvious to the casual observer (:-), that T = [x, m, c]^-1 [x1, m1, c1]. I hope this is right and clear. Please, if I've made a mistake, correct me -- we wouldn't want to lose Voyager now.

A somewhat "nasty'' way to solve this problem:

Let V1 = [ x1, y1, z1 ], V2 = [ x2, y2, z2 ]. Assume V1 and V2 are already normalized.

    V3 = normalize(cross(V1, V2)). (the normalization here is mandatory.)
    V4 = cross(V3, V1).
             
         [ V1 ]
    M1 = [ V4 ]
         [ V3 ]

    cos = dot(V2, V1), sin = dot(V2, V4)
            
         [ cos   sin    0 ]
    M2 = [ -sin  cos    0 ]
         [ 0     0      1 ]

The sought transformation matrix is just M1^-1 * M2 * M1. This might well be a standard-text solution.