vince_j_quaternions_for_computer_graphics

=1.

5.17Matrices

Matrices provide another way to express a quaternion product. For convenience, let’s repeat (5.8) again and show it in matrix form:

[sa , a][sb , b] = sa sb − xa xb − ya yb − za zb ,

sa (xb i + yb j + zb k) + sb (xa i + ya j + za k)

Let’s recompute the product qa qb using the above matrix:

= [−36, 6i + 12j + 12k].

5.17.1 Orthogonal Matrix

We can demonstrate that the unit-norm quaternion matrix is orthogonal by showing that the product with its transpose equals the identity matrix. As we are dealing with matrices, Q will represent the matrix for q :

For this to occur, QT = Q−1.

5.18 Quaternion Algebra

Ordered pairs provide a simple notation for representing quaternions, and allow us to represent the real unit 1 as [1, 0], and the imaginaries i, j, k as [0, i], [0, j], [0, k] respectively. A quaternion then becomes a linear combination of these elements with associated real coefficients. Under such conditions, the elements form the basis for an algebra over the field of reals.

Furthermore, because quaternion algebra supports division, and obeys the normal axioms of algebra, except that multiplication is non-commutative, it is called a division algebra. Ferdinand Georg Frobenius proved in 1878 that only three such real associative division algebras exist: real numbers, complex numbers and quaternions [1].

The “Cayley numbers” O, constitute a real division algebra, but the Cayley numbers are 8-dimensional and are not associative, i.e. a(bc) = (ab)c for all a, b, c O.

5.19 Summary

Quaternions are very similar to complex numbers, apart from the fact that they have three imaginary terms, rather than one. Consequently, they inherit some of the properties associated with complex numbers, such as norm, conjugate, unit norm and

Chapter 6

3D Rotation Transforms

6.1 Introduction

In this chapter we review the 3D Euler rotation transforms employed in computer graphics software. In particular, we identify their Achilles’ heel—gimbal lock—and the need to be able to rotate about an arbitrary axis. To this end, we will develop a matrix transform that achieves such a rotation, and in the next chapter develop a similar transform using quaternions.

6.2 3D Rotation Transforms

The traditional technique for rotating points and frames of reference is based upon Euler rotations, named after the Swiss mathematician, Leonhard Euler. They offer three ways to effect a rotation. The first is to rotate about one of the three Cartesian axes. The second combines any two of these rotations about different axes, and the third combines any three rotations.

Initially, this approach sounds appealing—and in many cases works well— however, there are problems associated with the technique. The first problem is that when two or more rotations are combined, it is difficult to visualise and predict how the final rotation will behave. The second is that it is complicated to effect a rotation about a specific axis, and the third, is that under some conditions, one loses access to one of the object’s rotational axes. This last problem is known as gimbal lock. To appreciate these issues we will construct a 3D rotation transform that suffers from gimbal lock.

6.3 Rotating About a Cartesian Axis

The transform for rotating a point about the origin in the plane is given by

Fig. 6.2 Rotating a point about an axis parallel with the z-axis

The matrices for rotating about an off-set axis parallel with the x-axis and parallel with the y-axis are:

6.5 Composite Rotations

Leaving aside the transforms for rotating about single and off-set axes, we have three transforms for rotating about the Cartesian axes: Rα,x , Rβ,y and Rγ ,z , which can be combined to produce families of double and triple transforms. As mentioned