4.1 Orthogonal Linear Transformations

Rotations are best approached via the concept of an orthogonal linear transformation (OLT), which is defined to be a LT that does not change lengths of vectors or angles between vectors. Put another way, OLTs preserve dot products:

O(v₁).O(v₂) = v₁.v₂ (4.1)

Clearly reflections and rotations are OLTs, because both are linear and neither changes the lengths of, or the angles between, vectors.

Matrices representing OLTs relative to orthonormal bases have special properties, and are called orthogonal matrices. The column vectors of an orthogonal matrix are mutually orthogonal and have magnitude 1, because they are the images, under the OLT, of the orthogonal unit vectors in the orthonormal basis. You can verify that the matrix computed in Example 3.1, representing a rotation relative to an orthonormal basis, has orthogonal column vectors of length 1.

This property can also be expressed as follows: If O is the matrix of an OLT relative to an orthonormal basis then

O^-1 = O^T (4.2)

Proof. By Formulas (2.2) and (3.5), the component in the i^th row and j^th column of O^TO is the dot product of the i^th row of O^T and the j^th column of O. That is, it is the dot product of the i^th and j^th columns of O. If i = j, then the dot product is the square of the magnitude of the i^th column vector of O, namely 1. If i = j, this dot product is 0 because the i^th and j^th columns of O are orthogonal to one another. Thus O^TO = I. A similar argument shows that OO^T = I. |||

A 3-D OLT can be either a reflections or a rotations. The distinction is that rotations preserve handedness (ie a rotation cannot transform a right hand into a left hand) while reflections reverse it. What if you want to know whether some OLT in 3-D space is a reflection or a rotation? (If you find it hard to imagine yourself wanting this, pretend that kidnappers have seized your dog and you must distinguish reflections from rotations to fulfil their ransom demands). One test is the following: both rotations and reflections preserve dot products, as we saw in the definition of an OLT in Equation (4.1), but it can be shown that rotations also preserve cross products while reflections reverse them:

Rota(v) × Rota(w) = Rota(v × w), but (4.3)
Refl(v) × Refl(w) = -Refl(v × w).

For example, Rota(e₁) × Rota(e₂) = Rota(e₁ × e₂) = Rota(e₃). But the more usual way, in linear algebra books, of distinguishing between 3-D reflections and rotations is to say that the determinant of a reflection is -1 while the determinant of a rotation is 1. It's not worth our while to learn much about determinants in general, but for a 3 by 3 matrix M, the formula is especially simple:

detM = M(e₁) . ( M(e₂) × M(e₃) ) (4.4)

That is, the first column, dotted with the cross product of the other two columns in order, gives the determinant. If the matrix represents an OLT, the determinant must be -1 or 1; the sign tells you whether the OLT is a reflection or a rotation.

A rotation, therefore, is an OLT that preserves handedness (ie an OLT with determinant 1). A rotation matrix R is a matrix corresponding to a rotation. If the bases for the matrix are orthonormal, then we have R^TR = I.

Matrices corresponding to rotations in 3-D space have nine components (3 × 3). The requirements of orthogonality and preservation of handedness reduce this set of matrices from nine degrees of freedom to three. This means that three numbers, such as the Bechert angles (see Section 4.4), suffice to describe 3-D rotations. But this does not mean that every (or even any) set of 3 elements from a rotation matrix determines that matrix. For example, the two different rotation matrices below:

1	0	0						-1	0	0
0	1	0						0	1	0
0	0	1						0	0	-1

have seven corresponding elements in common; thus knowing even seven elements of a rotation matrix does not always tell you what rotation matrix you have. But it can be shown that if you know six elements that form two complete rows or two complete columns, then the matrix is specified. This, essentially, is why two eye coils in two magnetic fields, yielding four components of the rotation matrix of the eye, do not suffice, in general, for the computation of eye position. Two coils in three fields, yielding two complete columns of the rotation matrix, do suffice.

4.2 The Rotation Group

Note the following facts: 1. The composition of two rotations is always another rotation (it is never a reflection, say, or a two-headed mule). 2. The identity transformation I can be regarded as a rotation, namely the null rotation, which leaves everything fixed. 3. Every rotation has a unique inverse, namely the rotation of equal size in the opposite direction. A set of objects that compose associatively with one another, and that includes an identity element and an inverse for every element, is called a group. This is a much simpler algebraic notion than a vector space, and is very widely applicable. In particular, we have just shown that the set of rotations in an n-dimensional space is a group in which the combining operation is composition. The set of all rotation matrices for a given space is a group in which the combining operation is matrix multiplication. Ultimately, it is the correspondence between these two groups that makes rotation matrices useful in the study of rotations.

Problem 4.1 Is the set of all 3-D reflections a group? If not, which of the group axioms 1-3 from the last paragraph fail?

4.3 Application

For us, rotation matrices mainly provide a stepping stone to a more useful representation of rotations using quaternions (coming soon). However, for some purposes, rotation matrices are better than quaternions. As a rule, quaternions are useful for composing rotations and for questions involving rotation axes, but rotation matrices are better (simply because the formula is slightly simpler) for computing the effect of a given rotation on a given vector. In this section we'll look at examples of this last application.

Example 4.1 Rotation matrices are useful for computing how gaze direction changes with eye rotation. Suppose we represent the direction of gaze using a unit vector g (the gaze vector) pointing in the direction of regard. If the eye undergoes a rotation R, g will rotate with the eye, and the new gaze direction will be [R][g]. For example, suppose g is initially (.71, .5, .5) relative to an orthonormal head-fixed basis with e₁ forward, e₂ left and e₃ up; that is, g points mostly forward but also left and up. And suppose the eye rotates 30° up in the head. Then the matrix for this rotation relative the basis e_i is

cos30	0	-sin30				.87	0	-.5
0	1	0		=		0	1	0
sin30	0	cos30				.5	0	.87

and so [R][g] = ( (.87)(.71) - (.5)(.5), (1)(.5), (.5)(.71) + (.87)(.5) ) = (.36, .5, .79).

4.4 How We Compute the Rotation Matrix of the Eye

We now have enough math to explain our algorithm for computing the rotation matrix of the eye from coil signals.

1. As Problem 2.5 indicates, the signal coming from an eye coil in a magnetic field is the dot product c.m of a vector c orthogonal to the coil plane and a vector m in the direction of the field.

2. Now suppose the three magnetic field vectors m₁, m₂ and m₃ form an orthonormal basis for 3-D space, meaning that they all have magnitude 1 (ie all the fields have the same strength, which we regard as "1") and they are mutually orthogonal. Then the three signals from a coil vector c:

c.m₁ c.m₂ c.m₃ (4.5)

are the three coordinates of the vector c with respect to the basis formed by the m_i. (Would this still be so if the three magnetic fields were not mutually orthogonal?).

3. The six signals from two eye coils on one eye are therefore the coordinates of two unit vectors, c₁ and c₂, fixed on the eye. These two vectors suffice to uniquely determine the rotational position of the eye; (why would one vector not suffice?). Klaus Bechert's microprocessor uses these vector components to compute a 3-number representation of eye orientation, similar to the "Euler angles" used by some physicists, so that the data can be stored compactly in three channels on disk. The details of this 3-number representation are probably not vital for you, since the first thing that OMNIVOR does, on reading the data files, is to reconstruct the vectors c₁ and c₂. More precisely, the subprogram SAMPLE reads in the stored data and multiplies them by a conversion factor to get the "Bechert angles" in radians; the three Bechert angles of the eye are called av(i), and those of the head hv(i). The subprogram CALCOIL (meaning CALculate COIL vectors) then reconstructs the two unit vectors c₁ and c₂. Another subprogram then computes the cross product of c₁ and c₂ to get the coordinates, relative to the same m_i basis, of an imaginary third coil vector c₃ orthogonal to c₁ and c₂.

4. By running OMNIVOR in "coil" mode and hitting the key "d" , we choose a reference eye position, read the Bechert angles for this position and compute the nine coordinates of the three coil vectors, which in this position only we call c₁, c₂ and c₃.

5. Now suppose the eye has rotated away from reference position by a rotation represented, in the m_i basis, by the rotation matrix R. Then clearly

c_i = Rc_i for i = 1, 2 and 3 (4.6)

We can group the 3 c vectors together as the columns of a 3 by 3 matrix C; and similarly the c vectors form a 3 by 3 matrix C. Then we have

C = RC (4.7)

Or, multiplying both sides by the inverse of C -- one can show that as long as c₁ and c₂ are not parallel (they are actually orthogonal in Skalar annuli that we use), the matrix C in any eye position is invertible -- we obtain

R = CC^-1. (4.8)

In OMNIVOR, the matrix C is computed and inverted, and the inverse stored in a file, as soon as "i" is punched. C^-1 is called the D matrix in the program, and the file containing the D matrix from the first calibration for, say, subject 2 in experimental series eh is called something like zeh21.d. (Actually, zeh21.d contains two D matrices, one for the left eye and one for the head). The subprogram that CALculates the D matrices is called CALD.

If you want to look at coil vectors, you choose menu options "coils" in OMNIVOR; these variables are computed without using any D matrix. For almost any other menu choice (eg quaternions, velocity etc.), the rotation matrix of the eye (and that of the head) must be calculated, and therefore the D matrices are used. It is in subprogram CALQ (meaning CALculate Quaternion) that the third coil vector c₃ is calculated, the matrix C is constructed, and the matrix multiplication in Formula (4.8) is performed, though if you look at CALQ you may not recognize this because some shortcuts are taken to save time; for example, CALQ computes only the first two components of c₃, and it computes the third column of R by taking the cross product of the first two columns. Finally, CALQ converts the rotation matrix into a quaternion by a formula that we haven't covered yet.

4.5 Eigenvectors

Suppose that A is a LT (not necessarily an OLT) from a vector space V to itself, and that the equation

Av = sv (4.9)

holds for some vector v and some scalar s. In other words, the effect of A on some vector v is simply to multiply it by some scalar s. Then s is called an eigenvalue of A, and v is called an eigenvector of A corresponding to the eigenvalue s.

Most introductory texts on linear algebra describe techniques for calculating the eigenvalues and corresponding eigenvectors of a LT. We shall not cover these techniques because, while eigenvectors are very important in our work, the quaternion representation of rotations which we'll learn makes the computation of important eigenvectors trivially easy. In the rest of this section we'll examine the geometric significance of the eigenvectors of 3-D OLTs.

Any 3-D reflection has two eigenvalues: -1 and 1. Any vector orthogonal to the plane of the mirror is reversed in direction by the reflection, without its size being changed; that is, the reflected vector is -1 times the original, and so the vector is an eigenvector corresponding to eigenvalue -1. The set formed by all these eigenvectors is the line orthogonal to the plane of the mirror. On the other hand, any vector in the plane of the mirror is unchanged by the reflection: it is an eigenvector corresponding to eigenvalue 1. The set formed by all these eigenvectors is the plane of the mirror. Any vector that is neither in the plane of the mirror nor orthogonal to it is not an eigenvector of the reflection.

Any 3-D rotation has 1 for an eigenvalue: any vector lying along the axis of the rotation is unchanged by the rotation, and is therefore an eigenvector corresponding to eigenvalue 1. The line formed by these eigenvectors is the axis of the rotation. For the special case of the null rotation, every vector in 3-D space is an eigenvector corresponding to eigenvalue 1.

Problem 4.2. Can you find a 3-D rotation with an eigenvalue other than 1?