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The geometry of consciousness
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Consciousness and Cognition journal homepage: www.elsevier.com/locate/concog
The geometry of consciousness ⁎
Michael K. McBeatha, , Ty Y. Tanga, Dennis M. Shafferb a b
Arizona State University, United States Ohio State University, Mansfield, United States
A R T IC LE I N F O
ABS TRA CT
Keywords: Analytic geometry Algebra Reference frame Egocentric Exocentric Allocentric Coordinate system Cartesian Euclidean Catching Intercepting Collision avoidance Baseball Outfielder Fly ball Consciousness
Conscious experience implies a reference-frame or vantage, which is often important in scientific models. Control models of ball-interception are used as an example. Models that use viewerdependent egocentric reference-frames are contrasted with viewer-independent allocentric ones. Allocentric reference-frames serve well for models like Newtonian physics, which utilize static coordinate-systems that allow forces and object-movements to be compartmentalized. In contrast, egocentric reference-frames are natural for modeling mobile organisms or robots when controlling perception-action behavior. Lower-level perception-action behavior is often characterized using egocentric coordinate-systems that optimize processing-speed, while higher-level cognitive-processes use allocentric frames that provide a stationary spatial reference. Brain-behavior models like the Ventral-Stream What System, and Dorsal-Stream Where-How System, also respectively utilize allocentric and egocentric reference-frames. Reference-frame clarification can resolve disputes about models of control-tasks like running to catch baseballs, and can provide insights for biomimetic-robots. Confusion regarding geometry and reference-frames contributes to a lack of clarity between how and when egocentric versus allocentric geometries are imposed, with perception-actions generally being more egocentric and conscious experience more allocentric.
1. Introduction On the evening of November 10, 1619, 23 year old gentleman soldier, Rene Descartes, had a lucid dream that revolutionized the history of science (Browne, 1977). The oldest, most complete extant record of Descartes’ dream depicts a series of notable events (Baillet, 1691), but the dream is often apocryphally described as him simply tracking a fly in a room and realizing that its ongoing position can be geometrically represented in an X, Y, Z coordinate system (Haven, 1998; Van Sickle, 2004). While this description of the dream is not technically accurate, it is known that Descartes awoke in an inspired state with a plan to revolutionize science, and soon after outlined the concept of Analytic Geometry. Descartes’ analytic geometry was a new method of representing information that combines algebraic equations and Euclidean geometric locations based on an X, Y, Z representation, since named after him as the Cartesian coordinate system. This breakthrough opened the door for a vast variety of information to be modeled and represented using geometric reference frames. It also can be viewed as an important advancement in the understanding of consciousness by articulating that there are mathematical ways to specify reference frames. Perceptual psychologists, physicists, and consciousness researchers now utilize geometric models with coordinate systems that are defined relative to a reference frame (Cuijpers, Kappers, & Koenderink, 2001). Egocentric reference frames are ones that define coordinates relative to an individual observer’s vantage or head orientation. For example, a Cartesian egocentric reference frame
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Corresponding author. E-mail address: [email protected] (M.K. McBeath).
https://doi.org/10.1016/j.concog.2018.04.015 Received 7 February 2018; Received in revised form 24 April 2018; Accepted 26 April 2018 1053-8100/ © 2018 Published by Elsevier Inc.
Please cite this article as: McBeath, M., Consciousness and Cognition (2018), https://doi.org/10.1016/j.concog.2018.04.015
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aligns with an observer’s head such that the orientation of the X-axis spans Left-Right, Y-axis spans Above-Below, and Z-axis spans Front-Behind. Alternatively, we can define cylindrical or spherical egocentric coordinate systems, the commonality being that the axes still move to remain aligned with the observer’s head at the origin. An egocentric reference frame is used on GPS devices that display a view from the perspective of the traveler. In contrast, allocentric reference frames are ones that define coordinates relative to external parameters, independent of observer vantage, and need not have an origin. These can be defined relative to another person or object, but, most commonly, the term is used to describe a stationary, gravitational, local world-based environment with axes such as East-West, North-South, and Higher-Lower elevation. An allocentric reference frame is used on GPS devices that display a stationary map or an unchanging top-view of the route of a traveler. In general, the history of science can be described as a march away from viewer-dependent egocentric characterizations, moving toward viewer-independent allocentric ones. Such allocentric models work well in physical domains of science, in which models like Newtonian physics utilize static coordinate-systems that allow forces and movements to be simplified into independent dimensions (de Mestre, 1990; Watts & Bahill, 1990). Physics models like those of Copernicus, Kepler, and Newton promoted heliocentric reference frames that enabled a superior allocentric understanding of physics compared to the more egocentric geocentric reference frames put forward by the ancient Greco-Romans like Ptolemy. One of the principal lessons promulgated by these early scientists and Descartes is for scientists to avoid the trap of favoring egocentric perspectives, and to instead consider more generalizable allocentric ones. However, this trend of moving toward using allocentric models in science has been challenged in the 20th century with the advent of relativity and quantum mechanics and their dependencies on observer vantage. Meanwhile, the field of Psychology has always favored egocentric explanations, in part because the goal of Psychology is to try to understand and model internal individual perspectives. In perceptual psychology individual perspectives are frequently described using egocentric variables like optical angle and perceived distance from an observer. In the realm of neuroscience, Goodale and Milner (1992, 2013) and researchers like Norman (2003) expanded on the work of Mishkin, Ungerleider and Macko (1983) finding physiological evidence supporting two separate visual streams that can allow for both allocentric and egocentric perspectives. One, the ventral stream, was characterized as the “what” system, and the other, the dorsal stream, was characterized as the “where” and later “how” system. Debates continue as to the exact functionality and independence of these two systems, and more recently brain researchers have produced monkey and human findings that promote a clearer distinction between a prefrontal cortical-based cognitive control system that processes abstract meaning and guides high-level conscious planning, and a sub-cortical basal ganglia-based system that processes rapid, automatized perception-motor behaviors and is typically largely unconscious (Colby & Duhamel, 1996; Hikosaka, & Isoda, 2010). This distinction between functions of the two systems is of interest to perception scientists and consciousness researchers because the psychophysical evidence also supports that the control algorithms of the dorsal perception-motor system operate largely within an egocentric reference frame in which observers are typically not consciously aware of how they control their actions (Vetter, Goodbody, & Wolpert, 1999). In contrast, shown in Table 1, the processes of the ventral cognitive control system appear to operate within a reconstructed allocentric reference frame in which observers are often consciously aware of the meanings and location attributes of objects and landmarks that are perceived or imagined (Norman, 2003). The emerging consensus among neuroscientists is that these two systems are not so distinct and largely work together in parallel rather than independently (McIntosh & Schenk, 2009). Construction of a representation with an allocentric, world-based reference frame allows observers to move and navigate about while maintaining layout and object constancy, as well as integration of top-down knowledge, but at the cost of being somewhat slower and more distortion-prone (Bridgeman & Hoover, 2008). Perceptual thresholds of the perception-motor system have been found to be five to ten times faster than those of the cognitive control system (Holloway, Dolgov, & McBeath, 2009). There is a similar parallel distinction in the arena of autonomous robot design, in which two levels of control are used. One level is comprised of rapid perception-action control systems that utilize simple feedback rules and emphasize speed over memory. The other is comprised of higher-level predictive control systems that compare scenery with memory and use slower, more complex reconstructive reasoning. Some robotic actions are guided primarily by rapid real-time control mechanisms (similar to those specified as the unconscious human perception-motor system), while others are guided more by slow matching and resetting of memory stores (similar to those specified as the human conscious cognitive planning system) (Sugar & McBeath, 2001; Suluh, Sugar, & McBeath, 2001; Sugar, McBeath, Suluh, & Mundhra, 2006).
Table 1 Qualities of ventral and dorsal brain systems (from Norman, 2003). Of particular interest to perception scientists and consciousness researchers are that the cognitive control system generally has an allocentric reference frame and high level of conscious awareness, while the perception-motor system generally has an egocentric reference frame and low level of conscious awareness. Factor
Ventral system
Dorsal system
Function Sensitivity Memory Speed Consciousness Frame of reference Visual input Monocular vision
Recognition/identification High spatial frequencies - details Long term stored representations Relatively slow Typically high Allocentric or object-centered Mainly foveal or parafoveal Generally reasonably small effects
Visually guided behavior High temporal frequencies - motion Only very short-term storage Relatively fast Typically low Egocentric or viewer-centered Across retina Often large effects e.g. motion parallax
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Fig. 1. The Optical Acceleration Cancelation (OAC) interception control strategy. The fielder runs along a path to keep the tangent of the ball image monotonically rising at a constant optical speed. If the ball image vertically decelerates, the ball will eventually head downward and, if not corrected, will land in front of the fielder. If the ball image vertically accelerates, the ball will travel beyond the fielder. When the fielder eliminates both types of optical acceleration, as shown here, the ball image climbs at a constant rate on a vertical projection plane, equivalent to an imaginary elevator rising at a constant speed from home plate, but tilted back by the distance that the fielder runs forward to catch the ball.
The importance of clarifying the use of egocentric versus allocentric reference frames and the geometry of consciousness can be elucidated with an example in the perception-motor task of ball interception, sometimes referred to as the outfielder problem. The next section briefly describes several recently debated control models of projectile interception by fielders running to catch balls, and shows how clarification of reference frame and coordinate geometry can help resolve model disagreements.
2. The geometry of catching balls Perception models of ball catching are typically based on control of optical variables that specify egocentric angles between the target ball, a moving fielder, and another reference point. This is consistent with ball catching being a fast-action perception-motor task, and a classic example of the dorsal perception-motor system discussed above. The various competing control models all usually utilize a principle of angular constancy in which fielders run along a path trying to maintain the image of the ball at a constant position or velocity relative to their reference frame (McBeath, 2018; Wang, McBeath, & Sugar, 2015a). Chapman (1968) introduced the first optical control model for baseball outfielders navigating to intercept a fly ball, in which fielders run along a path to keep the tangent of the vertical optical angle of the ball increasing at a constant rate (see Fig. 1). This is equivalent to keeping the image of the ball rising at a constant rate in egoocentric Cartesian coordinates or at a decelerating rate in egocentric spherical coordinates. Chapman’s model has been called the Optical Acceleration Cancelation (OAC) model of ball catching (McBeath, Shaffer, & Kaiser, 1995; 1996). Chapman and others speculated that fielders also maintain allocentric lateral alignment with the ball to solve how fielders determine how much to run to the side, a solution which specifies an egocentrically-defined vertical optical angle with an allocentrically-defined lateral alignment strategy (Babler & Dannemiller, 1993; McLeod & Dienes, 1996; Michaels & Oudejans,1992). A potential issue with this model of ball catching is that the way in which lateral optical alignment is defined implies that fielders formulate and maintain an accurate allocentric, world-based representation of ball position. Yet this conflicts with research confirming that observers, who do not run to catch the ball are poor at generalizing its position and deducing its destination (McBeath & Morgan, 1999; Saxberg, 1987). In 1995, we introduced the ball interception model called the Linear Optical Trajectory (LOT) strategy, which specifies fielders run 3
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Fig. 2. The Linear Optical Trajectory (LOT) interception control strategy. (a) Fielder runs along a path to keep the ball image monotonically rising along a constant projected optical angle, Ψ. (b) If fielder remains stationary or selects a path that does not converge upon the ball (as shown in the inset figure), Ψ will change, and the optical trajectory will curve towards the image of the ground. Continuously cancelling out optical curvature to keep the image of the ball rising along a straight optical line insures interception because the ball image never approaches the ground.
along a path to keep the image of the ball rising along a straight line at a constant egocentric angle in the picture plane (McBeath et al., 1995, 1996), see Fig. 2. This allows both the vertical and lateral optical angles to the ball to be specified egocentrically, so that they can be combined and operationalized under a single control mechanism. The LOT model essentially extended OAC into two dimensions in a manner such that fielders move to prevent the image of the ball from accelerating vertically relative to laterally, which also allows its position and motion to be expressed entirely within an egocentric reference frame. Unfortunately, in our original Science article, we included a diagram depicting an allocentric reference frame, which has led to some confusion about exactly how optical angles should be defined and specified. In short, one might ask, what is a linear trajectory within an egocentric reference frame, in particular when using an egocentric spherical coordinate system? Furthermore, one can ask how this differs from an allocentric representation of a straight line that observers may consciously perceive or construct, in short, what is the geometry of consciousness that is experienced when performing tasks like catching balls? In a series of studies over the decades following the introduction of the LOT model, research using our methodology of incrementally measuring the sum of changes in the vertical and lateral optical angles routinely produced statistical fits for the LOT model that accounted for between 95% and 99.5% of the variance in optical ball position (Dolgov, Birchfield, McBeath, Thornburg, & Todd, 2009; McBeath, Nathan, Bahill, & Baldwin, 2008; McBeath, et al., 1995; Shaffer, Dolgov, Maynor, & Reed, 2013; Shaffer, Krauchunas, Eddy, & McBeath, 2004; Shaffer, Marken, Dolgov, & Maynor, 2013; Shaffer & McBeath, 2001, 2005; Shaffer, McBeath, Roy, & Krauchunas, 2003; Shaffer, McBeath, Krauchunas, & Sugar, 2008; Sugar, McBeath & Wang, 2006; Wang, McBeath, & Sugar, 2015a, 2015b). Yet, other labs occasionally reported notably more curvature in their measures of optical ball position (e.g. McLeod, Reed & Dienes, 2001; Fink & Warren, 2009). This dispute came to a head when McLeod, Reed, and Dienes (2003; 2006) published a unified optical model that specified independent optical acceleration cancelation of the ball position in both the vertical and lateral direction, which we note seems like it should produce a straight line identical to the LOT model. If a visual stimulus object moves at a constant rate vertically, and a constant rate horizontally in a Euclidean space, then it must also be moving in a straight line. The same is true whenever movement within the two dimensions are proportional, such as when they slow down at the same rate. What has 4
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Fig. 3. Specifying the vertical and lateral optical angles, α, and β, in a spherical coordinate system. Depending upon the order of operations, the lateral ball angle can be defined in a number of different ways that produce different values. (i) If the lateral optical angle to the ball is measured by first sweeping along the ground plane, it will equal βP. (ii) If the lateral angle is measured by sweeping laterally after sweeping vertically, it will equal a somewhat smaller angle, β. (iii) A third way to define the lateral optical angle is the way we do so in measurements that reliably confirm the LOT model, to sum up all incremental changes in the lateral ball angle, so that β = ΣβI. The same holds true for the vertical ball angle, α = ΣαI. This summation definition produces a value of β intermediate the other two values.
become clear, and we would like to articulate in the present work is that there are different ways to define a straight line, and these depend upon the reference frame used to measure linearity. In addition, it is of interest to confirm and differentiate the geometry of perception-action control mechanisms and the geometry of consciousness that observers typically experience (Bridgeman & Hoover, 2008; Reed, McLeod, & Dienes, 2010). Here, we clarify that there are several ways in which the lateral ball angle can be defined, and suggest that this ambiguity can explain why McLeod et al. (2001) interpreted their data to not support the LOT interpretation. One way to describe the lateral optical angle to the ball is to project the angle under the ball along the ground plane (labeled βP in Fig. 3), which is in congruence with how McLeod et al. did so (2003, 2006). There is also a somewhat smaller angle, (labeled β) that corresponds to the elevated lateral angle that spans from the ball to the point spherically projected above its initial position at home plate. This angle subtends a smaller lateral distance across the projection plane compared to βP, because the horizontally-level circle diameters on a spherical projection surface will decrease as a function of the upward tilt of the vertical optical angle to the ball, α. The shrinkage in the effective lateral angle, β, due to this upward tilt, is calibrated such that the rate of change of the diagonal demarking the ball trajectory locally produces a straight line along a vertical Cartesian plane in the background, and a great circle along the spherical projection surface in front of it. A third way of defining the lateral optical angle, β, is by adding up all of the instantaneous changes in the lateral angle as the ball ascends, β = Σβi. In other words, from the perspective of the moving fielder maintaining a LOT, the image of the ball is controlled so that instantaneous changes in the vertical optical ball angle, α, are kept proportional to the instantaneous changes in the lateral optical ball angle, β. (i.e. α = Σαi and β = Σβi). This is how we define β, and how we have always measured β in the LOT model. The difference between the three methods of defining β can be illustrated by imagining a ball image from a fielders perspective that starts immediately in front and climbs along a Cartesian straight line, or the great circle on the spherical projection that corresponds to that line, and travels, slanting upward and leftward, to a vertical optical angle of α = 89°. In this example, the ball image terminates directly leftward, but only one degree to the side from straight above. If we define and measure βP along the ground plane before raising the vertical angle, α, then βP = 90°. But if we measure it near the top, after raising the vertical angle, α, then it is less than 2°. Or, if we assign it the ongoing sum, β = ΣβI, this lateral angle will be some value in between, closer to the middle. In addition, if we consider the case of a ball image traveling up on a great circle along a spherical background surface, such a trajectory will produce continuous proportional changes in α and β, when we define them to be equal to the respective sums of ongoing changes in the vertical and lateral optical angles, i.e. α = ΣαI and β = ΣβI. 5
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What is interesting and sometimes confusing about these different methods of defining the lateral optical angle to the ball, is that the sum-of-instantaneous-changes definition is most consistent with a spherical egocentric reference frame that is typically presumed for the rapid perception-motor system, while the conscious experience of fielders is more consistent with a world-view, allocentric reference frame. The reliability with which fielders maintain a LOT (running to keep the image of the ball monotonically rising along a straight line), strongly confirms that the perception-motor control strategy operates in an egocentric reference frame. Yet, at the same time, the reported experience of fielders is that the ball appears to rise and fall along a roughly parabolic trajectory, which is a fairly accurate reconstruction of the actual physical trajectory within a Cartesian allocentric reference frame (McBeath & Morgan, 1999; Shaffer, Maynor, Madden, Utt, & Adamrovich, 2015; Shaffer, Maynor, Utt, & Briley, 2009; Shaffer & McBeath, 2005; Wang et al., 2015a). In other words, the pattern of findings for baseball catching supports that fielders have an unconscious perceptionmotor system that is egocentric, and a conscious cognitive reconstruction that is allocentric. 3. Baseball example: clarifying reference frames disambiguates geometric discrepancies As an example of how clarifying reference frames can disambiguate geometric disputes, we reanalyze the data from one of the most extreme tests of ball-catching model fit that has been used to argue against the LOT strategy, a high pop-up study by McLeod et al. (2001). Here we show how data collected by McLeod and his colleagues actually do converge with the LOT model when the lateral optical angle is defined in the manner we are specifying, and is re-analyzed and replotted. In their study fielders each ran multiple times to one of four locations, each located 2–5 m away. In a re-analysis of this data, we apply best-fit optical lines to all of their trajectories after defining the lateral angle as specified, β = ΣβI, on a plot of the ongoing β vs α. We then determined the amount of variance accounted for by a linear fit and ordered, from most to least, each of their five individual’s trajectories at each catching position. Fig. 4 is a replot of the middle three of each set of five trajectories, ordered by variance accounted for with a linear fit. The resultant median variance accounted for with a linear fit for all trajectories at each position is as follows: Position 1 = 97.2%, Position 2 = 98.9%, Position 3 = 95.9%, and Position 4 = 98.5%. On average overall, the LOT model accounted for 98% of the variance in optical ball movement for the group of 20 trajectories shown in McLeod et al.’s Fig. 5. This reanalysis disconfirms the assertion that their data do not conform to optical linearity, and shows how carefully defining optical variables can help resolve such disputes. Close inspection reveals that in addition to the highly reliable optical linearity, there are large slope differences for the optical trajectory lines of different fielders headed to the same spot, which confirms that the LOT model is robust over large individual differences in fielder behavior. Aggressive fielders run ahead of the ball and slightly overshoot it, while lackadaisical ones lag behind it and need to hustle more at the end to catch up. This type of variance is expected in a normally distributed population of fielders using a LOT control strategy. In addition, some individuals do appear to systematically produce optical curvature near the end of the trajectory in some instances. We have shown that the amount of optical curvature that occurs near the end of these catches is in the range expected due to the offset between a fielder’s head and outstretched hand used to catch the ball (Shaffer et al., 2008). In that study we collected data replicating the pop fly ball set-up of McLeod et al. (2001), with fielders each catching 10 pop ups at each of four locations 2–5 m away. Our findings revealed two geometric factors that can lead to the appearance of optical ball curvature. First, fielders typically catch balls with an outstretched gloved left hand. This can lead to a small, but reliable increase in curvature at the very end of the catch, as the ball trajectory diverges toward the hand. When running to their left with their mitt leading, the terminal optical curvature typically tails forward, flattening the optical slope. When running to their right, with their mitt trailing, the terminal optical curvature can backtrack as they run somewhat beyond the ball. These positional corrections are consistent with other ball interception research that supports that additional spatial cues beyond just the optical trajectory are used in the final phase of catching (Regan, 1997; Regan & Gray, 2001). A second factor that can lead to apparent optical curvature near the end of a trajectory is geometric distortion due to egocentric projective geometry and the increase in optical ball size as a fielder approaches the ball. Modelers typically plot data using implied allocentric Cartesian coordinates, but when our data is egocentric, we can lose sight of the implications of perspective effects like the changing optical size of a ball with distance. This is illustrated in Fig. 5, which shows data of the egocentric size and position of ground balls as a fielder runs to catch them. We verified that, as with fly balls, fielders intercepting ground balls also maintain a LOT, but one that is inverted, with a continuously downward linear trajectory (Sugar et al., 2006). Of note is that the optical ball size increases up to many degrees near the end of the trajectory, which may be interpreted as curvature of the ball centroid. The pattern of findings remains consistent with the perception-motor system maintaining a LOT while catching, and with the trajectory linearity being defined egocentrically. Since fielders generally do not seem aware that they are maintaining a LOT, this use of an egocentric reference frame appears to remain largely unconscious (Shaffer & McBeath, 2005). In contrast, the geometry of which fielders are consciously aware appears to be a reconstructed allocentric reference frame that they experience during activities like catching balls. In this conscious reference frame, the ball is perceived to rise and fall, and the fielder’s spatial environment is experienced to remain relatively constant as he or she navigates through it (Shaffer et al., 2015, 2009; Shaffer & McBeath, 2005). 4. Summary In conclusion, this paper discusses the importance of articulating reference frames in models of science and consciousness, and reviews how physics models tend to be allocentric, while psychological perceptual models tend to be egocentric. The rapid human perception-motor system appears to largely utilize an unconscious egocentric reference frame, while the slower reconstructive 6
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β
.
(lateral ball position in degrees visual angle)
β
(lateral ball position in degrees visual angle)
β
[160 msec between points]
β
(lateral ball position in degrees visual angle)
(lateral ball position in degrees visual angle)
Fig. 4. Replotted data of McLeod et al. (2001) graphically confirming that the results fit the LOT model well. Shown are the frame-by-frame optical ball positions of three fielders catching pop flies at each of four destination positions that were all 2–5 m distant. The sampling rate in each plot is 160 msec per ball position point. Note that there are some dramatic differences in the slopes of LOT solutions to the same destinations. Some fielders are more aggressive and have steeper slopes, but these produce trajectories that also systematically curve back near the end because the fielder reliably overshoots the ball. Similarly, some fielders are more lackadaisical, and have shallower slopes, but these produce trajectories that systematically curve forward near the end because the fielder reliably undershoots the ball. The principal trend is that the LOT model of maintaining a linear optical trajectory consistently applies across participants and conditions.
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Fig. 5. Optical size and position of four typical ground balls intercepted by a running fielder. As with fly balls, when fielders approach ground balls they select running paths that maintain a Linear Optical Trajectory (LOT), but in this case the trajectory continually descends. Here, data from head cams are shown for four trials in which the fielder intercepts bouncing ground balls launched from an initial distance of 75 feet. This diagram illustrates how the optical size of an approaching ball increases dramatically up beyond several degrees visual angle near the end of each catch. This increase in size allows for some curvature in the centroid of ball position while still maintaining a linear fit relative to the ball diameter.
cognitive executive-control system largely utilizes a conscious allocentric reference frame. The different reference frames can lead to some confusion in interpreting interception control models, in part because geometric properties like angles, sizes and what defines a straight line can differ. Sometimes we scientists may overgeneralize from our conscious experience and presume to treat and analyze the perception-motor geometry as if it was based on a Cartesian allocentric reference frame, because that is how we consciously experience our environment. Coming full circle, when we think about and act upon the world around us, we clearly operate at multiple levels. Many automatic perception-action tasks (such as the outfielder problem reviewed in this paper) appear to be driven by a lower-level perception-motor system, sometimes described as the dorsal stream or “where-how” system. These processes are largely unconscious and appear to operate in an egocentric reference frame. Meanwhile, other higher-level cognitive processes appear to simultaneously operate at a conscious level, described as the ventral stream or “what” system. These operate in a reconstructed allocentric reference frame. When fielders run to catch a ball, they reliably maintain a LOT, but are not consciously aware of doing so. Rather, they have an allocentric reconstruction that is typically a relatively accurate representation of the physical world, with balls appearing to rise and fall much like they actually do in the physical world. In short, specifying the geometry of consciousness appears to require a vantage and reference frame. We typically construct one that is consciously allocentric, and at the same time operate another that controls and updates our actions from an egocentric perspective. Articulating this distinction can help disambiguate geometric discrepancies, and clarify the geometry of consciousness. 5. Postscript Perhaps the most well-known contribution to the study of consciousness by Descartes is his famous quote, “Cogito ergo sum” - I think therefore I am. What is not as well known is that 17 centuries earlier, the Roman Poet, Horace, lyricized, “Non sum qualis eram” I am not what I was (Ferry, 1998). Though the earlier quote may not be as prophetic, the historical precedence makes it clear that we should not put Descartes before the Horace. References Baillet, A. (1691). La Vie de Monsieur Des-cartes. Paris: Daniel Horthemels. Part I, Book 2, Chapter 180–86. Browne, A. (1977). Descartes’s dreams. Journal of the Warburg and Courtauld Institutes, 40, 256–273. Babler, T. G., & Dannemiller, J. L. (1993). Role of image acceleration in judging landing location of free-falling projectiles. Journal of Experimental Psychology: Human
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McLeod, P., & Dienes, Z. (1996). Do fielders know where to go to catch the ball or only how to get there? Journal of Experimental Psychology: Human Perception and Performance, 22, 531–543. McLeod, P., Reed, N., & Dienes, Z. (2001). Toward a unified fielder theory: What we do not yet know about how people run to catch a ball. Journal of Experimental Psychology: Human Perception and Performance, 27(6), 1347–1355. McLeod, P., Reed, N., & Dienes, Z. (2003). How fielders arrive in time to catch the ball. Nature, 426, 244–245. McLeod, P., Reed, N., & Dienes, Z. (2006). The generalized optic acceleration cancellation theory of catching. Journal of Experimental Psychology: Human Perception and Performance, 32(1), 139–148. Michaels, C. F., & Oudejans, R. R. D. (1992). The optics and actions of catching fly balls: Zeroing out optical acceleration. Ecological Psychology, 4, 199–222. Mishkin, M., Ungerleider, L. G., & Macko, K. A. (1983). Object vision and spatial vision: Two cortical pathways. Trends in Neurosciences, 6, 414–417. Norman, J. (2003). Two visual systems and two theories of perception: An attempt to reconcile the constructivist and ecological approaches. Behavior & Brain Sciences, 25, 73–144. Reed, N., McLeod, P., & Dienes, Z. (2010). Implicit knowledge and motor skill: What people who know how to catch don’t know. Consciousness and Cognition, 19(1), 63–76. Regan, D. (1997). Visual factors in hitting and catching. Journal of Sport Sciences, 15, 533–558. Regan, D., & Gray, R. (2001). Hitting what one wants to hit and missing what one wants to miss. Vision Research, 41, 3321–3329. Saxberg, B. V. H. (1987). Projected free fall trajectories. Biological Cybernetics, 56(2–3), 159–175. Shaffer, D. M., Dolgov, I., Maynor, A. B., & Reed, C. (2013). Receivers in American football use a linear optical trajectory (LOT) to pursue and catch thrown footballs. Perception, 42, 813–827. Shaffer, D. M., Krauchunas, S. M., Eddy, M., & McBeath, M. K. (2004). How dogs navigate to catch Frisbees. Psychological Science, 15(7), 437–441. Shaffer, D. M., Marken, R. S., Dolgov, I., & Maynor, A. B. (2013). Chasin' choppers: Using unpredictable target trajectories to test theories of object interception. Attention, Perception, & Psychophysics, 75, 1486–1506. Shaffer, D. M., Maynor, A. B., Madden, B. K., Utt, A. L., & Adamrovich, C. N. (2015). Why what comes down must first look like it’s going up: A perceptual cognitive regularity dominates online perception. eBook: Nova Science Publishers141–156. Shaffer, D. M., Maynor, A. B., Utt, A. L., & Briley, B. A. (2009). Lack of conscious awareness of how we navigate to catch baseballs. In E. B. Hartonek (Ed.). Experimental psychology research trends. Hauppauge: Nova Science Publishers Inc. Shaffer, D. M., & McBeath, M. K. (2002). Baseball outfielders maintain a linear optical trajectory when tracking uncatchable fly balls. Journal of Experimental Psychology: Human Perception and Performance, 28, 335–348. Shaffer, D. M., & McBeath, M. K. (2005). Naïve beliefs in baseball: Systematic distortion in perceived time of apex for fly balls. Journal of Experimental Psychology: Learning Memory & Cognition, 31(6), 1492–1501. Shaffer, D. M., McBeath, M. K., Krauchunas, S. M., & Sugar, T. G. (2008). Evidence for a generic interceptive strategy. Perception & Psychophysics, 70(1), 145–157. Shaffer, D. M., McBeath, M. K., Roy, W. L., & Krauchunas, S. M. (2003). A linear optical trajectory informs fielders where to run to the side to catch fly balls. Journal of Experimental Psychology: Human Perception and Performance, 29(6), 1244–1250. Suluh, A., Sugar, T. G., & McBeath, M. K. (2001). Spatial navigational principles: applications to mobile robotics. In Proceedings of the 2001 IEEE international conference on robotics and automation (pp. 1689–1694). Seoul, Korea. Sugar, D. G., & McBeath, M. K. (2001). Robotic modeling of mobile ball-catching as a tool for understanding biological interceptive behavior. Brain and Behavioral Sciences, 24(6), 1078–1080. Sugar, T. G., McBeath, M. K., Suluh, A., & Mundhra, K. (2006). Mobile robot interception using human navigational principles: Comparison of active versus passive tracking algorithms. Autonomous Robots, 21(1), 43–54. Sugar, T. G., McBeath, M. K., & Wang, Z. (2006). A unified fielder theory for interception of moving objects either above or below the horizon. Psychonomic Bulletin & Review, 13(5), 908–917. Van Sickle, J. (2004). Foundation of a coordinate system. Basic GIS coordinates (pp. 2). Boca Raton: CRC Press. Vetter, P., Goodbody, S. J., & Wolpert, D. M. (1999). Evidence for an eye-centered spherical representation of the visuomotor map. Journal of Neurophysiology, 81, 935–939. Wang, W., McBeath, M. K., & Sugar, T. (2015a). Optical angular constancy is maintained as a navigational control strategy when pursuing robots moving along complex pathways. Journal of Vision, 15(3) 16+. Wang, W., McBeath, M. K., & Sugar, T. (2015). Navigational strategy used to intercept fly balls under real-world conditions with moving visual back-ground fields. Attention, Perception, & Psychophysics, 77(2), 613–625. Watts, R. G., & Bahill, T. A. (1990). The flight of the ball: The kinematics of fly balls. Keep your eye on the ball (pp. 133–149). New York: W.H. Freeman & Company.
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Contents lists available at ScienceDirect
Consciousness and Cognition journal homepage: www.elsevier.com/locate/concog
The geometry of consciousness ⁎
Michael K. McBeatha, , Ty Y. Tanga, Dennis M. Shafferb a b
Arizona State University, United States Ohio State University, Mansfield, United States
A R T IC LE I N F O
ABS TRA CT
Keywords: Analytic geometry Algebra Reference frame Egocentric Exocentric Allocentric Coordinate system Cartesian Euclidean Catching Intercepting Collision avoidance Baseball Outfielder Fly ball Consciousness
Conscious experience implies a reference-frame or vantage, which is often important in scientific models. Control models of ball-interception are used as an example. Models that use viewerdependent egocentric reference-frames are contrasted with viewer-independent allocentric ones. Allocentric reference-frames serve well for models like Newtonian physics, which utilize static coordinate-systems that allow forces and object-movements to be compartmentalized. In contrast, egocentric reference-frames are natural for modeling mobile organisms or robots when controlling perception-action behavior. Lower-level perception-action behavior is often characterized using egocentric coordinate-systems that optimize processing-speed, while higher-level cognitive-processes use allocentric frames that provide a stationary spatial reference. Brain-behavior models like the Ventral-Stream What System, and Dorsal-Stream Where-How System, also respectively utilize allocentric and egocentric reference-frames. Reference-frame clarification can resolve disputes about models of control-tasks like running to catch baseballs, and can provide insights for biomimetic-robots. Confusion regarding geometry and reference-frames contributes to a lack of clarity between how and when egocentric versus allocentric geometries are imposed, with perception-actions generally being more egocentric and conscious experience more allocentric.
1. Introduction On the evening of November 10, 1619, 23 year old gentleman soldier, Rene Descartes, had a lucid dream that revolutionized the history of science (Browne, 1977). The oldest, most complete extant record of Descartes’ dream depicts a series of notable events (Baillet, 1691), but the dream is often apocryphally described as him simply tracking a fly in a room and realizing that its ongoing position can be geometrically represented in an X, Y, Z coordinate system (Haven, 1998; Van Sickle, 2004). While this description of the dream is not technically accurate, it is known that Descartes awoke in an inspired state with a plan to revolutionize science, and soon after outlined the concept of Analytic Geometry. Descartes’ analytic geometry was a new method of representing information that combines algebraic equations and Euclidean geometric locations based on an X, Y, Z representation, since named after him as the Cartesian coordinate system. This breakthrough opened the door for a vast variety of information to be modeled and represented using geometric reference frames. It also can be viewed as an important advancement in the understanding of consciousness by articulating that there are mathematical ways to specify reference frames. Perceptual psychologists, physicists, and consciousness researchers now utilize geometric models with coordinate systems that are defined relative to a reference frame (Cuijpers, Kappers, & Koenderink, 2001). Egocentric reference frames are ones that define coordinates relative to an individual observer’s vantage or head orientation. For example, a Cartesian egocentric reference frame
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Corresponding author. E-mail address: [email protected] (M.K. McBeath).
https://doi.org/10.1016/j.concog.2018.04.015 Received 7 February 2018; Received in revised form 24 April 2018; Accepted 26 April 2018 1053-8100/ © 2018 Published by Elsevier Inc.
Please cite this article as: McBeath, M., Consciousness and Cognition (2018), https://doi.org/10.1016/j.concog.2018.04.015
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aligns with an observer’s head such that the orientation of the X-axis spans Left-Right, Y-axis spans Above-Below, and Z-axis spans Front-Behind. Alternatively, we can define cylindrical or spherical egocentric coordinate systems, the commonality being that the axes still move to remain aligned with the observer’s head at the origin. An egocentric reference frame is used on GPS devices that display a view from the perspective of the traveler. In contrast, allocentric reference frames are ones that define coordinates relative to external parameters, independent of observer vantage, and need not have an origin. These can be defined relative to another person or object, but, most commonly, the term is used to describe a stationary, gravitational, local world-based environment with axes such as East-West, North-South, and Higher-Lower elevation. An allocentric reference frame is used on GPS devices that display a stationary map or an unchanging top-view of the route of a traveler. In general, the history of science can be described as a march away from viewer-dependent egocentric characterizations, moving toward viewer-independent allocentric ones. Such allocentric models work well in physical domains of science, in which models like Newtonian physics utilize static coordinate-systems that allow forces and movements to be simplified into independent dimensions (de Mestre, 1990; Watts & Bahill, 1990). Physics models like those of Copernicus, Kepler, and Newton promoted heliocentric reference frames that enabled a superior allocentric understanding of physics compared to the more egocentric geocentric reference frames put forward by the ancient Greco-Romans like Ptolemy. One of the principal lessons promulgated by these early scientists and Descartes is for scientists to avoid the trap of favoring egocentric perspectives, and to instead consider more generalizable allocentric ones. However, this trend of moving toward using allocentric models in science has been challenged in the 20th century with the advent of relativity and quantum mechanics and their dependencies on observer vantage. Meanwhile, the field of Psychology has always favored egocentric explanations, in part because the goal of Psychology is to try to understand and model internal individual perspectives. In perceptual psychology individual perspectives are frequently described using egocentric variables like optical angle and perceived distance from an observer. In the realm of neuroscience, Goodale and Milner (1992, 2013) and researchers like Norman (2003) expanded on the work of Mishkin, Ungerleider and Macko (1983) finding physiological evidence supporting two separate visual streams that can allow for both allocentric and egocentric perspectives. One, the ventral stream, was characterized as the “what” system, and the other, the dorsal stream, was characterized as the “where” and later “how” system. Debates continue as to the exact functionality and independence of these two systems, and more recently brain researchers have produced monkey and human findings that promote a clearer distinction between a prefrontal cortical-based cognitive control system that processes abstract meaning and guides high-level conscious planning, and a sub-cortical basal ganglia-based system that processes rapid, automatized perception-motor behaviors and is typically largely unconscious (Colby & Duhamel, 1996; Hikosaka, & Isoda, 2010). This distinction between functions of the two systems is of interest to perception scientists and consciousness researchers because the psychophysical evidence also supports that the control algorithms of the dorsal perception-motor system operate largely within an egocentric reference frame in which observers are typically not consciously aware of how they control their actions (Vetter, Goodbody, & Wolpert, 1999). In contrast, shown in Table 1, the processes of the ventral cognitive control system appear to operate within a reconstructed allocentric reference frame in which observers are often consciously aware of the meanings and location attributes of objects and landmarks that are perceived or imagined (Norman, 2003). The emerging consensus among neuroscientists is that these two systems are not so distinct and largely work together in parallel rather than independently (McIntosh & Schenk, 2009). Construction of a representation with an allocentric, world-based reference frame allows observers to move and navigate about while maintaining layout and object constancy, as well as integration of top-down knowledge, but at the cost of being somewhat slower and more distortion-prone (Bridgeman & Hoover, 2008). Perceptual thresholds of the perception-motor system have been found to be five to ten times faster than those of the cognitive control system (Holloway, Dolgov, & McBeath, 2009). There is a similar parallel distinction in the arena of autonomous robot design, in which two levels of control are used. One level is comprised of rapid perception-action control systems that utilize simple feedback rules and emphasize speed over memory. The other is comprised of higher-level predictive control systems that compare scenery with memory and use slower, more complex reconstructive reasoning. Some robotic actions are guided primarily by rapid real-time control mechanisms (similar to those specified as the unconscious human perception-motor system), while others are guided more by slow matching and resetting of memory stores (similar to those specified as the human conscious cognitive planning system) (Sugar & McBeath, 2001; Suluh, Sugar, & McBeath, 2001; Sugar, McBeath, Suluh, & Mundhra, 2006).
Table 1 Qualities of ventral and dorsal brain systems (from Norman, 2003). Of particular interest to perception scientists and consciousness researchers are that the cognitive control system generally has an allocentric reference frame and high level of conscious awareness, while the perception-motor system generally has an egocentric reference frame and low level of conscious awareness. Factor
Ventral system
Dorsal system
Function Sensitivity Memory Speed Consciousness Frame of reference Visual input Monocular vision
Recognition/identification High spatial frequencies - details Long term stored representations Relatively slow Typically high Allocentric or object-centered Mainly foveal or parafoveal Generally reasonably small effects
Visually guided behavior High temporal frequencies - motion Only very short-term storage Relatively fast Typically low Egocentric or viewer-centered Across retina Often large effects e.g. motion parallax
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Fig. 1. The Optical Acceleration Cancelation (OAC) interception control strategy. The fielder runs along a path to keep the tangent of the ball image monotonically rising at a constant optical speed. If the ball image vertically decelerates, the ball will eventually head downward and, if not corrected, will land in front of the fielder. If the ball image vertically accelerates, the ball will travel beyond the fielder. When the fielder eliminates both types of optical acceleration, as shown here, the ball image climbs at a constant rate on a vertical projection plane, equivalent to an imaginary elevator rising at a constant speed from home plate, but tilted back by the distance that the fielder runs forward to catch the ball.
The importance of clarifying the use of egocentric versus allocentric reference frames and the geometry of consciousness can be elucidated with an example in the perception-motor task of ball interception, sometimes referred to as the outfielder problem. The next section briefly describes several recently debated control models of projectile interception by fielders running to catch balls, and shows how clarification of reference frame and coordinate geometry can help resolve model disagreements.
2. The geometry of catching balls Perception models of ball catching are typically based on control of optical variables that specify egocentric angles between the target ball, a moving fielder, and another reference point. This is consistent with ball catching being a fast-action perception-motor task, and a classic example of the dorsal perception-motor system discussed above. The various competing control models all usually utilize a principle of angular constancy in which fielders run along a path trying to maintain the image of the ball at a constant position or velocity relative to their reference frame (McBeath, 2018; Wang, McBeath, & Sugar, 2015a). Chapman (1968) introduced the first optical control model for baseball outfielders navigating to intercept a fly ball, in which fielders run along a path to keep the tangent of the vertical optical angle of the ball increasing at a constant rate (see Fig. 1). This is equivalent to keeping the image of the ball rising at a constant rate in egoocentric Cartesian coordinates or at a decelerating rate in egocentric spherical coordinates. Chapman’s model has been called the Optical Acceleration Cancelation (OAC) model of ball catching (McBeath, Shaffer, & Kaiser, 1995; 1996). Chapman and others speculated that fielders also maintain allocentric lateral alignment with the ball to solve how fielders determine how much to run to the side, a solution which specifies an egocentrically-defined vertical optical angle with an allocentrically-defined lateral alignment strategy (Babler & Dannemiller, 1993; McLeod & Dienes, 1996; Michaels & Oudejans,1992). A potential issue with this model of ball catching is that the way in which lateral optical alignment is defined implies that fielders formulate and maintain an accurate allocentric, world-based representation of ball position. Yet this conflicts with research confirming that observers, who do not run to catch the ball are poor at generalizing its position and deducing its destination (McBeath & Morgan, 1999; Saxberg, 1987). In 1995, we introduced the ball interception model called the Linear Optical Trajectory (LOT) strategy, which specifies fielders run 3
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Fig. 2. The Linear Optical Trajectory (LOT) interception control strategy. (a) Fielder runs along a path to keep the ball image monotonically rising along a constant projected optical angle, Ψ. (b) If fielder remains stationary or selects a path that does not converge upon the ball (as shown in the inset figure), Ψ will change, and the optical trajectory will curve towards the image of the ground. Continuously cancelling out optical curvature to keep the image of the ball rising along a straight optical line insures interception because the ball image never approaches the ground.
along a path to keep the image of the ball rising along a straight line at a constant egocentric angle in the picture plane (McBeath et al., 1995, 1996), see Fig. 2. This allows both the vertical and lateral optical angles to the ball to be specified egocentrically, so that they can be combined and operationalized under a single control mechanism. The LOT model essentially extended OAC into two dimensions in a manner such that fielders move to prevent the image of the ball from accelerating vertically relative to laterally, which also allows its position and motion to be expressed entirely within an egocentric reference frame. Unfortunately, in our original Science article, we included a diagram depicting an allocentric reference frame, which has led to some confusion about exactly how optical angles should be defined and specified. In short, one might ask, what is a linear trajectory within an egocentric reference frame, in particular when using an egocentric spherical coordinate system? Furthermore, one can ask how this differs from an allocentric representation of a straight line that observers may consciously perceive or construct, in short, what is the geometry of consciousness that is experienced when performing tasks like catching balls? In a series of studies over the decades following the introduction of the LOT model, research using our methodology of incrementally measuring the sum of changes in the vertical and lateral optical angles routinely produced statistical fits for the LOT model that accounted for between 95% and 99.5% of the variance in optical ball position (Dolgov, Birchfield, McBeath, Thornburg, & Todd, 2009; McBeath, Nathan, Bahill, & Baldwin, 2008; McBeath, et al., 1995; Shaffer, Dolgov, Maynor, & Reed, 2013; Shaffer, Krauchunas, Eddy, & McBeath, 2004; Shaffer, Marken, Dolgov, & Maynor, 2013; Shaffer & McBeath, 2001, 2005; Shaffer, McBeath, Roy, & Krauchunas, 2003; Shaffer, McBeath, Krauchunas, & Sugar, 2008; Sugar, McBeath & Wang, 2006; Wang, McBeath, & Sugar, 2015a, 2015b). Yet, other labs occasionally reported notably more curvature in their measures of optical ball position (e.g. McLeod, Reed & Dienes, 2001; Fink & Warren, 2009). This dispute came to a head when McLeod, Reed, and Dienes (2003; 2006) published a unified optical model that specified independent optical acceleration cancelation of the ball position in both the vertical and lateral direction, which we note seems like it should produce a straight line identical to the LOT model. If a visual stimulus object moves at a constant rate vertically, and a constant rate horizontally in a Euclidean space, then it must also be moving in a straight line. The same is true whenever movement within the two dimensions are proportional, such as when they slow down at the same rate. What has 4
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Fig. 3. Specifying the vertical and lateral optical angles, α, and β, in a spherical coordinate system. Depending upon the order of operations, the lateral ball angle can be defined in a number of different ways that produce different values. (i) If the lateral optical angle to the ball is measured by first sweeping along the ground plane, it will equal βP. (ii) If the lateral angle is measured by sweeping laterally after sweeping vertically, it will equal a somewhat smaller angle, β. (iii) A third way to define the lateral optical angle is the way we do so in measurements that reliably confirm the LOT model, to sum up all incremental changes in the lateral ball angle, so that β = ΣβI. The same holds true for the vertical ball angle, α = ΣαI. This summation definition produces a value of β intermediate the other two values.
become clear, and we would like to articulate in the present work is that there are different ways to define a straight line, and these depend upon the reference frame used to measure linearity. In addition, it is of interest to confirm and differentiate the geometry of perception-action control mechanisms and the geometry of consciousness that observers typically experience (Bridgeman & Hoover, 2008; Reed, McLeod, & Dienes, 2010). Here, we clarify that there are several ways in which the lateral ball angle can be defined, and suggest that this ambiguity can explain why McLeod et al. (2001) interpreted their data to not support the LOT interpretation. One way to describe the lateral optical angle to the ball is to project the angle under the ball along the ground plane (labeled βP in Fig. 3), which is in congruence with how McLeod et al. did so (2003, 2006). There is also a somewhat smaller angle, (labeled β) that corresponds to the elevated lateral angle that spans from the ball to the point spherically projected above its initial position at home plate. This angle subtends a smaller lateral distance across the projection plane compared to βP, because the horizontally-level circle diameters on a spherical projection surface will decrease as a function of the upward tilt of the vertical optical angle to the ball, α. The shrinkage in the effective lateral angle, β, due to this upward tilt, is calibrated such that the rate of change of the diagonal demarking the ball trajectory locally produces a straight line along a vertical Cartesian plane in the background, and a great circle along the spherical projection surface in front of it. A third way of defining the lateral optical angle, β, is by adding up all of the instantaneous changes in the lateral angle as the ball ascends, β = Σβi. In other words, from the perspective of the moving fielder maintaining a LOT, the image of the ball is controlled so that instantaneous changes in the vertical optical ball angle, α, are kept proportional to the instantaneous changes in the lateral optical ball angle, β. (i.e. α = Σαi and β = Σβi). This is how we define β, and how we have always measured β in the LOT model. The difference between the three methods of defining β can be illustrated by imagining a ball image from a fielders perspective that starts immediately in front and climbs along a Cartesian straight line, or the great circle on the spherical projection that corresponds to that line, and travels, slanting upward and leftward, to a vertical optical angle of α = 89°. In this example, the ball image terminates directly leftward, but only one degree to the side from straight above. If we define and measure βP along the ground plane before raising the vertical angle, α, then βP = 90°. But if we measure it near the top, after raising the vertical angle, α, then it is less than 2°. Or, if we assign it the ongoing sum, β = ΣβI, this lateral angle will be some value in between, closer to the middle. In addition, if we consider the case of a ball image traveling up on a great circle along a spherical background surface, such a trajectory will produce continuous proportional changes in α and β, when we define them to be equal to the respective sums of ongoing changes in the vertical and lateral optical angles, i.e. α = ΣαI and β = ΣβI. 5
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What is interesting and sometimes confusing about these different methods of defining the lateral optical angle to the ball, is that the sum-of-instantaneous-changes definition is most consistent with a spherical egocentric reference frame that is typically presumed for the rapid perception-motor system, while the conscious experience of fielders is more consistent with a world-view, allocentric reference frame. The reliability with which fielders maintain a LOT (running to keep the image of the ball monotonically rising along a straight line), strongly confirms that the perception-motor control strategy operates in an egocentric reference frame. Yet, at the same time, the reported experience of fielders is that the ball appears to rise and fall along a roughly parabolic trajectory, which is a fairly accurate reconstruction of the actual physical trajectory within a Cartesian allocentric reference frame (McBeath & Morgan, 1999; Shaffer, Maynor, Madden, Utt, & Adamrovich, 2015; Shaffer, Maynor, Utt, & Briley, 2009; Shaffer & McBeath, 2005; Wang et al., 2015a). In other words, the pattern of findings for baseball catching supports that fielders have an unconscious perceptionmotor system that is egocentric, and a conscious cognitive reconstruction that is allocentric. 3. Baseball example: clarifying reference frames disambiguates geometric discrepancies As an example of how clarifying reference frames can disambiguate geometric disputes, we reanalyze the data from one of the most extreme tests of ball-catching model fit that has been used to argue against the LOT strategy, a high pop-up study by McLeod et al. (2001). Here we show how data collected by McLeod and his colleagues actually do converge with the LOT model when the lateral optical angle is defined in the manner we are specifying, and is re-analyzed and replotted. In their study fielders each ran multiple times to one of four locations, each located 2–5 m away. In a re-analysis of this data, we apply best-fit optical lines to all of their trajectories after defining the lateral angle as specified, β = ΣβI, on a plot of the ongoing β vs α. We then determined the amount of variance accounted for by a linear fit and ordered, from most to least, each of their five individual’s trajectories at each catching position. Fig. 4 is a replot of the middle three of each set of five trajectories, ordered by variance accounted for with a linear fit. The resultant median variance accounted for with a linear fit for all trajectories at each position is as follows: Position 1 = 97.2%, Position 2 = 98.9%, Position 3 = 95.9%, and Position 4 = 98.5%. On average overall, the LOT model accounted for 98% of the variance in optical ball movement for the group of 20 trajectories shown in McLeod et al.’s Fig. 5. This reanalysis disconfirms the assertion that their data do not conform to optical linearity, and shows how carefully defining optical variables can help resolve such disputes. Close inspection reveals that in addition to the highly reliable optical linearity, there are large slope differences for the optical trajectory lines of different fielders headed to the same spot, which confirms that the LOT model is robust over large individual differences in fielder behavior. Aggressive fielders run ahead of the ball and slightly overshoot it, while lackadaisical ones lag behind it and need to hustle more at the end to catch up. This type of variance is expected in a normally distributed population of fielders using a LOT control strategy. In addition, some individuals do appear to systematically produce optical curvature near the end of the trajectory in some instances. We have shown that the amount of optical curvature that occurs near the end of these catches is in the range expected due to the offset between a fielder’s head and outstretched hand used to catch the ball (Shaffer et al., 2008). In that study we collected data replicating the pop fly ball set-up of McLeod et al. (2001), with fielders each catching 10 pop ups at each of four locations 2–5 m away. Our findings revealed two geometric factors that can lead to the appearance of optical ball curvature. First, fielders typically catch balls with an outstretched gloved left hand. This can lead to a small, but reliable increase in curvature at the very end of the catch, as the ball trajectory diverges toward the hand. When running to their left with their mitt leading, the terminal optical curvature typically tails forward, flattening the optical slope. When running to their right, with their mitt trailing, the terminal optical curvature can backtrack as they run somewhat beyond the ball. These positional corrections are consistent with other ball interception research that supports that additional spatial cues beyond just the optical trajectory are used in the final phase of catching (Regan, 1997; Regan & Gray, 2001). A second factor that can lead to apparent optical curvature near the end of a trajectory is geometric distortion due to egocentric projective geometry and the increase in optical ball size as a fielder approaches the ball. Modelers typically plot data using implied allocentric Cartesian coordinates, but when our data is egocentric, we can lose sight of the implications of perspective effects like the changing optical size of a ball with distance. This is illustrated in Fig. 5, which shows data of the egocentric size and position of ground balls as a fielder runs to catch them. We verified that, as with fly balls, fielders intercepting ground balls also maintain a LOT, but one that is inverted, with a continuously downward linear trajectory (Sugar et al., 2006). Of note is that the optical ball size increases up to many degrees near the end of the trajectory, which may be interpreted as curvature of the ball centroid. The pattern of findings remains consistent with the perception-motor system maintaining a LOT while catching, and with the trajectory linearity being defined egocentrically. Since fielders generally do not seem aware that they are maintaining a LOT, this use of an egocentric reference frame appears to remain largely unconscious (Shaffer & McBeath, 2005). In contrast, the geometry of which fielders are consciously aware appears to be a reconstructed allocentric reference frame that they experience during activities like catching balls. In this conscious reference frame, the ball is perceived to rise and fall, and the fielder’s spatial environment is experienced to remain relatively constant as he or she navigates through it (Shaffer et al., 2015, 2009; Shaffer & McBeath, 2005). 4. Summary In conclusion, this paper discusses the importance of articulating reference frames in models of science and consciousness, and reviews how physics models tend to be allocentric, while psychological perceptual models tend to be egocentric. The rapid human perception-motor system appears to largely utilize an unconscious egocentric reference frame, while the slower reconstructive 6
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β
.
(lateral ball position in degrees visual angle)
β
(lateral ball position in degrees visual angle)
β
[160 msec between points]
β
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(lateral ball position in degrees visual angle)
Fig. 4. Replotted data of McLeod et al. (2001) graphically confirming that the results fit the LOT model well. Shown are the frame-by-frame optical ball positions of three fielders catching pop flies at each of four destination positions that were all 2–5 m distant. The sampling rate in each plot is 160 msec per ball position point. Note that there are some dramatic differences in the slopes of LOT solutions to the same destinations. Some fielders are more aggressive and have steeper slopes, but these produce trajectories that also systematically curve back near the end because the fielder reliably overshoots the ball. Similarly, some fielders are more lackadaisical, and have shallower slopes, but these produce trajectories that systematically curve forward near the end because the fielder reliably undershoots the ball. The principal trend is that the LOT model of maintaining a linear optical trajectory consistently applies across participants and conditions.
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Fig. 5. Optical size and position of four typical ground balls intercepted by a running fielder. As with fly balls, when fielders approach ground balls they select running paths that maintain a Linear Optical Trajectory (LOT), but in this case the trajectory continually descends. Here, data from head cams are shown for four trials in which the fielder intercepts bouncing ground balls launched from an initial distance of 75 feet. This diagram illustrates how the optical size of an approaching ball increases dramatically up beyond several degrees visual angle near the end of each catch. This increase in size allows for some curvature in the centroid of ball position while still maintaining a linear fit relative to the ball diameter.
cognitive executive-control system largely utilizes a conscious allocentric reference frame. The different reference frames can lead to some confusion in interpreting interception control models, in part because geometric properties like angles, sizes and what defines a straight line can differ. Sometimes we scientists may overgeneralize from our conscious experience and presume to treat and analyze the perception-motor geometry as if it was based on a Cartesian allocentric reference frame, because that is how we consciously experience our environment. Coming full circle, when we think about and act upon the world around us, we clearly operate at multiple levels. Many automatic perception-action tasks (such as the outfielder problem reviewed in this paper) appear to be driven by a lower-level perception-motor system, sometimes described as the dorsal stream or “where-how” system. These processes are largely unconscious and appear to operate in an egocentric reference frame. Meanwhile, other higher-level cognitive processes appear to simultaneously operate at a conscious level, described as the ventral stream or “what” system. These operate in a reconstructed allocentric reference frame. When fielders run to catch a ball, they reliably maintain a LOT, but are not consciously aware of doing so. Rather, they have an allocentric reconstruction that is typically a relatively accurate representation of the physical world, with balls appearing to rise and fall much like they actually do in the physical world. In short, specifying the geometry of consciousness appears to require a vantage and reference frame. We typically construct one that is consciously allocentric, and at the same time operate another that controls and updates our actions from an egocentric perspective. Articulating this distinction can help disambiguate geometric discrepancies, and clarify the geometry of consciousness. 5. Postscript Perhaps the most well-known contribution to the study of consciousness by Descartes is his famous quote, “Cogito ergo sum” - I think therefore I am. What is not as well known is that 17 centuries earlier, the Roman Poet, Horace, lyricized, “Non sum qualis eram” I am not what I was (Ferry, 1998). Though the earlier quote may not be as prophetic, the historical precedence makes it clear that we should not put Descartes before the Horace. References Baillet, A. (1691). La Vie de Monsieur Des-cartes. Paris: Daniel Horthemels. Part I, Book 2, Chapter 180–86. Browne, A. (1977). Descartes’s dreams. Journal of the Warburg and Courtauld Institutes, 40, 256–273. Babler, T. G., & Dannemiller, J. L. (1993). Role of image acceleration in judging landing location of free-falling projectiles. Journal of Experimental Psychology: Human
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