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«Quaia & Optican 1 Three-Dimensional Rotations of the Eye Christian Quaia, Ph.D. Lance M. Optican, Ph.D. Laboratory of Sensorimotor Research National ...»

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Quaia & Optican 1

Three-Dimensional Rotations

of the Eye

Christian Quaia, Ph.D.

Lance M. Optican, Ph.D.

Laboratory of Sensorimotor Research

National Eye Institute, NIH

Bethesda, MD, USA

Chapter for Adler’s Physiology of the Eye, 10th Edition. Kaufman & Alm, Eds.

Address correspondence to:

Dr. Lance M. Optican

Bldg. 49, Rm. 2A50

National Eye Institute, NIH

Bethesda, MD 20892-4435, USA Tel: 301-496-3549 Fax: 301-402-0511 E-mail: LOPTICAN@NIH.GOV FINAL DRAFT May 22, 2001 Quaia & Optican 2 Movements of the Eye Each eye is controlled by six extra-ocular muscles, and has six degrees of freedom: three for rotation and three for translation. However, the amount of translation possible is very limited, approximately 2 mm along the antero-posterior axis, and 0.5 mm in the frontal plane (Carpenter, 1977). Thus, the globe can be well approximated as a spherical joint with its center fixed in the head. With this approximation we only need to consider rotations around three orthogonal axes passing through the center of the eye.

These three axes define a system of coordinates for describing ocular rotations.

Unfortunately, the mathematical description of rotations of solid objects is much more complicated than that for translations. The final position reached after translations along the three space axes is independent of their order (e.g., x-axis followed by y-axis movement yields the same position as y followed by x). In contrast, the final orientation reached after a sequence of rotations around different axes depends on their order. For example, in the two panels of Fig. 1 a camera, starting from the same initial orientation (left column), is rotated around the same pair of axes (arrows in the figure), but in different order. Clearly, the final orientations (right column) are different for the two sequences of rotations. This would not be the case for translations. Thus, rotations are said to be non-commutative. The dependence of the final orientation on the sequence of rotations makes the study of eye movements less intuitive than one might hope.

However, careful attention to the definitions of eye orientation and rotation, and on the choice of mathematical tools used to quantify them, can greatly facilitate our thinking about eye rotations.

This latter element is particularly important. In fact, translations, and the resulting positions, can be described by simply specifying the three Cartesian coordinates of the center of the eye. This more familiar, Euclidean, space of translations is flat, and moving in only one direction will never result in getting back to the initial position. In contrast, rotations, and the resulting orientations, can not be described by any simple (i.e., intuitive) set of three coordinates. One of the fundamental reasons for this complexity is that the space of all rotations is curved. This can be easily noted by considering that if one keeps rotating an object around the same axis, eventually (after 360°) it will get back to FINAL DRAFT May 22, 2001 Quaia & Optican 3 Figure 1. Non-commutativity of rotations. The image on the right of each arrow is obtained by rotating the image on its left around an axis collinear with the arrow. A. The camera first rotates 90° around a vertical axis, and then 90° around a horizontal axis. B. The order of rotations is reversed. The final orientation of the camera is clearly different in the two cases. (Reprinted from Quaia and Optican, 1998) its initial orientation. To address the inherent complexity of rotations, several mathematical tools have been developed over the last 150 years, such as quaternions, sequences of rotations, rotation matrices, rotation vectors, etc. Although all these methods are equivalent (they all describe the same rotations), each method has both advantages and disadvantages in different applications (Tweed, 1997).

Quantifying Eye Rotations The first issue that must be addressed when describing rotations or orientations of the eye is whether to consider the three orthogonal axes of rotation as fixed in space, fixed in the head, or moving with the eye. Of course, keeping the axes fixed in space would be of little help, as the eye muscles move the eye relative to the head. The other two solutions have both advantages and disadvantages; the decision of which one to use depends on the specific oculomotor task under study.

FINAL DRAFT May 22, 2001 Quaia & Optican 4 Figure 2. Demonstration of head-fixed and eye-fixed coordinate systems. Left column shows positions reached by rotations around one of the six systems of eye-fixed axes (Fick). Because these axes move with the eye (torsion axis not shown), they can be represented by a gimbal system.

In the Fick system, the order of rotations is: horizontal, vertical, torsional. Middle column shows a rotation around a head-fixed axis (Euler). With an Euler axis, there is only one rotation, about an axis that is tilted appropriately. Right column shows another eye-fixed system of axes (Helmholtz).

In the Helmholtz system, the order of rotations is: vertical, horizontal, torsional. Gaze angles are referred to as primary (top row, looking straight ahead), secondary (middle row, on the horizontal or vertical meridian), or tertiary (bottom row, off both the horizontal and vertical meridians). As can be seen from the middle row, a rotation around one eye-fixed axis moves the eye into a secondary position. Note: all these tertiary orientations correspond to a 45° rotation up and to the left. In Fick coordinates, that corresponds to (31.8°, 31.8°, 0°), in Helmholtz coordinates to (31.8°,

31.8°, 0°), and in head-fixed coordinates to {(0, 1, 1)/Ö2, 45°}.

Eye-Fixed Coordinates

Eye-fixed reference frames are based on consideration of a mechanical mounting system for rotations, such as for a camera. The simplest way to make a camera mount is to have one axis for panning the camera left or right (yaw or horizontal axis), one for tilting it up or down (pitch or vertical axis), and one for twisting it clockwise or counterFINAL DRAFT May 22, 2001 Quaia & Optican 5 clockwise about the lens’s axis (roll or torsion axis). These axes are nested, one within the other, in a system of gimbals. (Note: the way the gimbals are nested specifies the mathematical order of the rotations, thus the order in which the gimbals are moved is irrelevant). Eye-fixed systems are defined by the order of their rotations (Haslwanter, 1995). There are three rotation axes so there are six possible sequences of rotations.

These eye-fixed coordinate systems are not very useful for the general treatment of rotations, because they favor one axis over the others (the first, which is independent of the other two). Nonetheless, two eye-fixed systems, due to Fick and Helmholtz, were commonly used in the past, and so are briefly mentioned here. The Fick system starts with a horizontal rotation around the vertical axis, followed by a vertical rotation around the new horizontal axis, and finally a torsional rotation about the new line of sight. The Helmholtz system starts with a vertical rotation around the horizontal axis, followed by a horizontal rotation around the new vertical axis, and finally a torsional rotation about the new line of sight. The left column of Fig. 2shows a Fick gimbal, and the right column shows a Helmholtz gimbal (torsional axes not shown). The movements in the eye-fixed axis cases have been decomposed into two rotations. The first position (top row) shows the eye looking straight ahead, in primary position. When the eye rotates from primary position around the head-fixed horizontal or vertical axis, it is said to move into a secondary position. This is shown in the middle row for Fick and Helmholtz gimbals.

Note that the first Fick rotation turns the eye to the left, whereas the first Helmholtz rotation turns the eye upward. The bottom row shows the eye rotated away from the horizontal or vertical meridian, into what is called a tertiary position.

Head-Fixed Coordinates

When using head-fixed axes, the description of rotations that we prefer (because it seems to us to be the most intuitive) is the so-called axis-angle form (Fig. 2, middle column), which follows from Euler's theorem. This theorem states that any orientation of a rigid body with one point fixed can be achieved, starting from a reference orientation, ˆ by a single rotation around an axis (through the fixed point) along a unit-length vector n by an angle F (Goldstein, 1980). Euler's theorem highlights an aspect common to all the FINAL DRAFT May 22, 2001 Quaia & Optican 6 Figure 3. Representing eye orientation using the axis-angle form. For each panel, the reference orientation is shown on the left. A. The camera is rotated 45° to the left, and the corresponding Euler’s axis points straight up, (0, 0, 1) in the XYZ coordinate system (see text). B. The camera is rotated 45° up and to the left around the axis (0, 1, 1)/Ö2. Note that the central cross on the camera’s lens appears twisted with respect to the vertical axis, even though the Euler axis has no torsional component. (Reprinted from Quaia and Optican, 1998) methods that can be used to represent rotary motion: the need to define a reference orientation. Although its choice is totally arbitrary, the one most commonly adopted in eye movement research is the orientation with the head upright and the eye looking straight ahead. The three main axes of rotation then point straight ahead (X-axis, roll or torsion rotations), straight to the left (Y-axis, pitch or vertical rotations) and straight up (Z-axis, yaw or horizontal rotations). The X, Y, and Z axes define a right-handed system of head-fixed coordinates, (x, y, z), that describe, for each eye orientation, Euler's axis of FINAL DRAFT May 22, 2001 Quaia & Optican 7 ˆ rotation, n. (Note: in a right-handed coordinate system, the direction that the eye turns for a positive angle is the direction that the fingers of the right hand curl when the thumb ˆ points along the axis n.) With this convention, for example, if the eye is rotated 45° to the left, its orientation is described by {(0, 0, 1), 45}, as that orientation is achieved by rotating the eye, starting from the reference orientation, by 45° around the vertical axis (0,0,1) (Fig. 3A, and Fig. 2 middle column, top row; note that we are looking at the camera from the front, so the X, Y, and Z axes point out of the page, to the right, and up, respectively). Similarly, if the eye where rotated 45° up and to the left, its orientation

would be {(0, 1, 1)/Ö2, 45} (Fig. 3B, and Fig. 2 middle column, bottom row). (Note:

when eye orientations are discussed in the context of Listing’s Law, a slightly different reference orientation is chosen for convenience. See below.)

False Torsion

The examples of rotations about different axes shown in Fig. 2 demonstrate an interesting effect of the non-commutativity of rotations. Note that all three rotations were designed to point the eye 45° up and 45° to the left. In the Euler axis case (middle column) this entails a single rotation of amplitude 45° around an axis tipped 45° from the vertical, (0, 1, 1)/Ö2. In the eye-fixed axes cases, this entails two rotations of magnitude 45°/Ö2 (approximately 31.8°) around the first two gimbal axes. If rotations were commutative, like translations, then the final eye orientation in each case would be the same. However, it is clear from the bottom row that the final orientations are not the same, i.e., rotations are not commutative.

Figure 4 shows the three tertiary orientations from Fig. 2 plotted together. The graph has been rotated so that you are looking directly down the line of sight of the Fick coordinate system (white cross). Note that the Fick cross is upright, i.e., rotations in the Fick coordinate system preserve the gravitational vertical on the retina (as is obvious from the gimbals in Fig. 2). The eyeball itself is drawn rotated around an Euler axis (light gray cross). In this case, we see that both the eccentricity of the eye and its torsion are slightly different from the Fick case. A voluntary eye movement to this location would have this orientation (Listing’s Laws, see below). Finally, the dark gray cross shows the FINAL DRAFT May 22, 2001 Quaia & Optican 8 Figure 4. Non-commutativity of rotations and false torsion resulting from equivalent rotations around eye-fixed and head-fixed axes. Eye orientation is indicated by the crosses (Fick: white, Euler: light gray, Helmholtz: dark gray). In this figure, the eye has been rotated 45° around an axis tilted 45° ({(0, 1, 1)/Ö2, 45°} middle column, bottom row of Fig. 3). The view of the graph has been rotated around to Fick coordinates (31.8°, 31.8°, 0°). Thus, the Fick axis appears centered and upright, which follows because Fick rotations preserve the gravitational vertical on the retina. The Euler and Helmholtz crosses are progressively displaced and twisted from the Fick cross. This indicates the non-commutativity of rotations. Under normal circumstances, the eye assumes the orientation given by the Euler rotation (Listing’s Law). The Fick cross is rotated clockwise from there (from the eye’s point of view), and the Helmholtz cross is rotated counterclockwise. In all cases, this orientation was achieved without any rotations about the torsional axes. Thus, the twists of the local reference frame are referred to as false torsions.

final orientation reached by rotations around the Helmholtz axes. It’s cross is even further eccentric and twisted than the Euler cross. (Note: the distance between the crosses is a function of the size of the eye rotation; as the eye rotation shrinks in size, so does the difference between the eccentricity and twist of the crosses.) These twists are called false torsions, because they do not arise from rotations about a torsional axis. Obviously, the difference in eccentricity of these three crosses could be eliminated by adjusting the size of the horizontal and vertical rotations for the Fick and Helmholtz systems. However, the difference in the twists would persist, unless a non-zero torsional rotation was introduced.

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