2.
Check solution:
Check solution:
3.
The article discusses a method of smoothly interpolating between two camera positions using quaternions.
4.
A multiplication of a complex number by corresponds to a π/2, counterclockwise rotation of the point represented by the complex number.
For example, the point (1, 0) can be represented as the complex number 1 + 0i. Multiplying this complex number by i, we get 0 + i, which represents the point (0, 1).
5.
Calculate the cross product and use it as the axis for a quaternion that rotates
by twice the angle between
and
(as computed by the dot product).
6.
Yes, the matrix would be identical.
Proof:
Let be the matrix associated with the quaternion rotation, and
be the
-axis rotation matrix. Suppose they both encode rotation about the
-axis by angle
. Then the inverse rotation (rotation by -
) is equal to the inverse of the rotation matrices. But since inverses are unique, this implies that
, which, in turn, implies that
.