Game Mathematics

Official Week 9 Solutions

[Graphics:Images/solutions_gr_1.gif]

2.

[Graphics:Images/solutions_gr_2.gif]

Check [Graphics:Images/solutions_gr_3.gif] solution:

[Graphics:Images/solutions_gr_4.gif]

Check [Graphics:Images/solutions_gr_5.gif] solution:

[Graphics:Images/solutions_gr_6.gif]

3.

The article discusses a method of smoothly interpolating between two camera positions using quaternions.

4.

A multiplication of a complex number by [Graphics:Images/solutions_gr_7.gif] 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 [Graphics:Images/solutions_gr_8.gif] and use it as the axis for a quaternion that rotates [Graphics:Images/solutions_gr_9.gif] by twice the angle between [Graphics:Images/solutions_gr_10.gif] and [Graphics:Images/solutions_gr_11.gif] (as computed by the dot product).

6.

Yes, the matrix would be identical.

Proof:

Let [Graphics:Images/solutions_gr_12.gif] be the matrix associated with the quaternion rotation, and [Graphics:Images/solutions_gr_13.gif] be the [Graphics:Images/solutions_gr_14.gif]-axis rotation matrix. Suppose they both encode rotation about the [Graphics:Images/solutions_gr_15.gif]-axis by angle [Graphics:Images/solutions_gr_16.gif]. Then the inverse rotation (rotation by -[Graphics:Images/solutions_gr_17.gif]) is equal to the inverse of the rotation matrices. But since inverses are unique, this implies that [Graphics:Images/solutions_gr_18.gif], which, in turn, implies that [Graphics:Images/solutions_gr_19.gif].


Converted by Mathematica      January 17, 2002