Calculate Rotation Matrix to align Vector A to Vector B in 3d?

Question

I have one triangle in 3d space that I am tracking in a simulation. Between time steps I have the the previous normal of the triangle and the current normal of the triangle along with both the current and previous 3d vertex positions of the triangles.

Using the normals of the triangular plane I would like to determine a rotation matrix that would align the normals of the triangles thereby setting the two triangles parallel to each other. I would then like to use a translation matrix to map the previous onto the current, however this is not my main concern right now.

I have found this website http://forums.cgsociety.org/archive/index.php/t-741227.html that says I must

• determine the cross product of these two vectors (to determine a rotation axis)
• determine the dot product ( to find rotation angle)
• build quaternion (not sure what this means)
• the transformation matrix is the quaternion as a 3 by 3 ( not sure)

Any help on how I can solve this problem would be appreciated.

Answer 1

The quaternion is a 4-dimensional complex number: http://en.wikipedia.org/wiki/Quaternion used to describe rotations in space. A quaternion (like a complex number) has a polar representation involving the exponential of the arguments (rotations), and a magnetude multiplier. Building the quaternion comes from the cross product (the product of the complex components), which will give you the argument in those 3 dimensions, you'll then get a number from that in the form A+Bi+Cj+Dk, and write it out in the matrix form described in the article there.

An easier way would be to simply fingure out what your original vectors are in the 4-space, and take the appropriate inverse operations to get your resultant quaternion (without going through the dot/cross product steps) but that requires a good foundation in hypercomplex algebra.

Answer 2

At the top of my head (do the checking yourself) Let the given vectors in $R^3$ be $A$ and $B$, for simplicity assume they have norm 1, and assume they are not identical. Define $C$ as the cross product of $A$ and $B$. We want an orthogonal matrix $U$ such that $UA=B$ and $UC=C$. First change bases, to the new base $(U_1,u_2,u_3)=(A,B,C)$. In this new basis the matrix doing the job is simply $G=\left(\begin{smallmatrix} 0&1&0\\1&0&0\\0&0&1\end{smallmatrix}\right)$. Then we need the basis shift matrix, to the new basis. Write the coordinates of the vectors in the old base as simply $A=(a_1,a_2,a_3), B=(b_1,b_2,b_3), C=(c_1,c_2,c_3)$. Then the basis shift matrix can be seen to be $\left( \begin{smallmatrix} a_1&b_1&c_1\\a_2&b_2&c_2\\a_3&b_3&c_3 \end{smallmatrix}\right)^{-1}$. The result is now simply $U=F^{-1} G F$, which is an orthogonal matrix rotating $A$ into $B$.

Answer 3

You can easily do all this operation using the following reusable library

Vector3 Lib

The following four steps worked for me.

         Vector3 axis = Vector3.CrossProduct(v1, v2);


        if (axis.Magnitude != 0)
        {
            axis.Normalize();
            Vector3D vAxis = new Vector3D(axis.X, axis.Y, axis.Z);
            Matrix3D m = Matrix3D.Identity;
            Quaternion q = new Quaternion(vAxis, AngleFromZaxis);

            m.RotateAt(q, centerPoint);
            MatrixTransform3D mT = new MatrixTransform3D(m);

            group.Children.Add(mT);

            myModel.Transform = group;
        }

Hope this helps !

With Regards, Shan