Geographical Analysis ISSN 0016-7363

A Measurement Theory for Time Geography Harvey J. Miller Department of Geography, University of Utah, Salt Lake City, UT

Ha¨gerstrand’s time geography is a powerful conceptual framework for understanding constraints on human activity participation in space and time. However, rigorous, analytical definitions of basic time geography entities and relationships do not exist. This limits abilities to make statements about error and uncertainty in time geographic measurement and analysis. It also compromises comparison among different time geographic analyses and the development of standard time geographic computational tools. The time geographic measurement theory in this article consists of analytical formulations for basic time geography entities and relations, specifically, the space– time path, prism, composite path-prisms, stations, bundling, and intersections. The definitions have arbitrary spatial and temporal resolutions and are explicit with respect to informational assumptions: there are clear distinctions between measured and inferred components of each entity or relation. They are also general to n-dimensional space rather than the strict two-dimensional space of classical time geography. Algebraic solutions are available for one or two spatial dimensions, while numeric (but tractable) solutions are required for some entities and relations in higher dimensional space.

Introduction Time geography is a powerful conceptual framework for understanding human spatial behavior, in particular, constraints and trade-offs in the allocation of limited time among activities in space (Ha¨gerstrand 1970). The last decade has witnessed a resurgence of time geography as researchers have improved the computational representations of basic time geographic entities such as the space–time path and prism (e.g., Miller 1991, 1999; Forer 1998; Kwan and Hong 1998). The recent development of location-aware technologies (LAT) and location-based services (LBS) has potential to create an even wider resurgence of time geography in social research and in geographic information services (the provision of geographic information to casual users; Shekhar and Chawla 2002). Correspondence: Harvey J. Miller, Department of Geography, University of Utah, 260 S. Central Campus Dr. Room 270, Salt Lake City, UT 84112-9155 e-mail: [email protected]

Submitted: June 11, 2003. Revised version accepted: April 20, 2004. Geographical Analysis 37 (2005) 17–45 r 2005 The Ohio State University

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LAT, such as the global positioning system (GPS) and radiolocation methods that piggyback on wireless communication systems, can allow measurement of basic time geographic entities and relations at spatio-temporal resolutions (and data volumes), hardly imaginable during time geography’s genesis in the mid-20th century. These data have potentially enormous scientific value to time geographers, as well as urban researchers, transportation analysts, and social scientists. LBS are the provision of location-specific content through wireless networks. LBS are widely expected to be the ‘‘killer app’’ of the wireless Internet. Many LBS queries are also time geographic queries (Miller and Shaw 2001, chap. 8), meaning that time geography can serve as a theoretical foundation for LBS. A problem is that time geography is not ready for the measurement tasks demanded by LAT and LBS: there are no formal and analytical statements of its basic entities and relations other than informal, geometric descriptions of Burns (1979) and Lenntorp (1976), and some recent but limited formulations by Miller (1991, 1999), Kwan and Hong (1998), and Hornsby and Egenhofer (2002). These are not effective for inferring time geographic entities and relationships from high-resolution measurement of mobile objects in space and time. They are not sufficient for analyzing the propagation of uncertainty when measuring these objects imperfectly. The looseness and incompleteness of these descriptions mean that comparisons of detailed data and time geographic analyses across different studies may be difficult since the entities and relations are not strictly comparable. A conceptual framework, while useful for generating ideas, does not provide the detailed specifications required for developing standard time geographic computational tools. Problems associated with representation and analysis of spatio-temporal entities also arise in geographic information science. In particular, the development of LAT and LBS is inspiring a growing literature on database design for storing information on moving objects. Since digital technologies can only sample an object’s location at discrete moments in time, a key problem is interpolating an object’s location at any arbitrary moment based on the sampled locations. Results indicate that this problem has elegant and tractable geometric solutions (e.g., Sistla et al. 1998; Moreira, Ribeiro, and Saglio 1999; Pfoser and Jensen 1999; Yanagisawa, Akahani, and Satoh 2003). This article addresses the lack of analytical rigor at the foundation of time geography by developing a measurement theory for its basic entities and relationships. Drawing from the literature on moving objects database design, this article shows that the location or spatial extent of time geographic entities at any moment in time can be solved as convex spatial sets, or the intersection of convex spatial sets, derived from the sampled locations and auxiliary information such as a maximum travel velocity. These sets are simple geometric objects that have algebraic solutions in one or two spatial dimensions. These simple objects support evaluation of time geographic relationships such as bundling and intersections. Numeric solutions are required for some entities and relations in higher dimensions, but these are relatively tractable since they involve simple surfaces. Measuring relationships 18

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such as bundling or intersections is also tractable since this involves simple geometry. The time geographic measurement theory consists of analytical statements of basic time geographic entities and relationships under perfect information. Although an ideal case, the framework supports imperfect measurement and the analysis of uncertainty by distinguishing between measured and inferred components and revealing the simple geometry required to construct the inferred from the measured. The geometric solutions provide functional requirements for software implementation of time geographic queries. The analytical statements of basic time geographic entities and relationships also provide precise definitions of previously informal concepts, potentially improving comparability among time geographic studies. This article first reviews classical time geography, more recent attempts to develop time geographic analytical tools, and research on data modeling for mobile objects. Next the measurement theory for time geography is provided; this includes analytical definitions of the space–time path, space–time prism, path–prism composites, and space–time stations. It also includes formal definitions of fundamental relationships between space–time paths and prisms, specifically bundles, and intersections. The final section concludes by identifying the research frontiers implied by the measurement theory.

Time geography and mobile objects Classical time geography Rather than attempting to predict human spatial behavior, Ha¨gerstrand’s (1970) time geography focuses on the constraints on human activities in space and time. Time geography views activities as occurring only at specific locations for limited time periods. Transportation allows individuals to increase the efficiency of trading time for space when traveling to participate in activities at these locations. Constraints that limit the ability of individuals to travel and participate in activities include: (i) the person’s capabilities for trading time for space in movement (e.g., access to private and public transport); (ii) the need to couple with others at particular locations for given durations (e.g., a meeting), thus limiting the ability to participate in activities at other locations; and (iii) the ability of public or private authorities to restrict physical presence from some locations in space and time (e.g., gated communities, shopping malls). Time geography distinguishes between fixed and flexible activities based on their degree of pliability in space and time over the short run. A fixed activity such as work cannot easily be rescheduled or relocated, while a flexible activity such as shopping is much easier to reschedule and/or relocate. The need to be present at a particular location during a specific time interval is a coupling constraint. Although the boundary between fixed and flexible activities can be vague, this distinction is 19

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2 1

Figure 1. A space–time path. 20

G

3

eo g Sp rap ac hic e al

Time

useful: fixed activities dictate strict coupling constraints while flexible activities allow more fluid coupling in space and time. Two entities are central to time geography, namely, the space–time path and prism. The space–time path traces the movement of an individual in space and time. Fig. 1 illustrates a space–time path in continuous two-dimensional space. New LAT such as wireless communication devices coupled with radiolocation methods or the GPS can allow recording of real-world paths at high levels of resolution (see Kwan 2000b). These paths can also be simulated using aggregate-level activity and time-use data (see Lenntorp 1976). Individual paths convey information about the individual’s activity space (the limited extent of the environment used by the individual) and the influence of fixed activities that comprise the anchor points of day-to-day existence (see Golledge and Stimson 1997). Collections of paths convey information on emergent space–time patterns and structures such as bundles (convergence of two or more paths for some shared activity), projects (space–time paths and activities required to complete an individual or organizational-level goal), and space–time activity systems (stable, multi-scale spatio-temporal patterns that emerge from intertwined allocation of time among activities in space; examples include traffic jams, popular nightclub districts, and urban sprawl) (Pred 1981; Golledge and Stimson 1997). The space–time prism is an extension of the path: this measures the ability to reach (be coincident with) locations in space and time given the location and durations of fixed activities. Fig. 2 illustrates a simple prism. In Fig. 2, the person must be at a given location (say, work) until time ti, must return again at time tj, and can travel with a known and finite maximum velocity. In this simple case, the prism comprises two cones: (i) a lower cone with an apex at the first activity location and oriented forward in time and (ii) an upper cone with an apex at the second activity location and oriented backward in time. The potential path space (PPS) is the interior of the prism: this shows all locations in space and time that the person can

Harvey J. Miller

Time Geography Measurement

tj f (maximum velocity)

Time

Potential Path Space

ti

Geographical Space Potential Path Area

Figure 2. A simple space–time prism (after Wu and Miller 2001).

occupy during the open time interval (ti, tj). A person can interact with another person only if the interiors of their prisms intersect, or if one person’s prism intersects with the other’s path. Projecting the PPS to the two-dimensional geographic plane delimits the potential path area (PPA): these are the set of geographic locations that the person can occupy during (ti, tj). Fig. 2 illustrates a simple space–time prism where the origin and destination are the same location. More generally, the origin or destination may be undefined, or they may be different locations. We can also consider the minimum time required for participating in the activity since this reduces the time available for travel. Fig. 3 illustrates a more general space–time prism and introduces some notation. The first fixed activity is located at xi and ends at ti, while the second fixed activity is located at xj and starts at time tj. This defines a time budget tj
ti for discretionary travel and activity participation. The maximum travel velocity during that time interval is vij. The activity time or minimum required time for participating in the activity is aij. This creates a cylinder of length aij between the lower and upper cones since the individual must be stationary in space for that length of time. The activity time consequently reduces the spatial extent of the PPS and PPA. The minimum time required to travel directly between the two fixed locations is tij . Another basic space–time entity is the station; this is a location in space where paths can bundle or cluster in space and time. Examples include retail outlets, offices, classrooms, and stadiums. These are traditionally conceptualized as vertical tubes with a finite temporal duration, and with space–time paths bundling inside 21

Geographical Analysis

t

tj aij tij

tij*

vij ti

x xi

xj

Figure 3. A general space–time prism (after Lenntorp 1976).

(see Golledge and Stimson 1997). The finite duration of the station reflects times when it is available (e.g., store operating hours, class time, scheduled sporting events). If a person wants to visit a station to conduct some activity, it must intersect with his or her prism. Time geographic measurement and analysis There seems to be no complete and consistent analytical statements of basic time geographic concepts, where ‘‘analytical’’ in this case refers to its strict mathematical sense: the ability to construct continuous mathematical functions that are defined for any neighborhood on a surface such as the plane (Weisstein 2002a). In other words, we should be able to make statements about variations in time geographic properties to an arbitrary level of spatio-temporal resolution. Time geographic analysis can benefit from a measurement theory that expresses how to map continuously varying relationships in the real world to the numeric domain in a way that preserves these relationships as fully as possible, as well as the consequences when these entities and relationships are measured imperfectly. The most complete time geographic systems are informal descriptions of constructible objects that support the geometric calculations of Burns (1979) and Lenntorp (1976). Although useful for summary calculations such as the prism volume, they are not sufficient as a measurement theory to support analytical statements about time geography: they cannot be reduced to functions that express the continuous variation in the prism across space and time. Other relations such as intersections, bundles, and stations are defined only informally in these and other sources (e.g., Pred 1981; Golledge and Stimson 1997). This means that 22

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comparisons of these entities and relationships across different studies may not be consistent and strictly comparable. For much of its history, a measurement theory for time geography was irrelevant since the technology did not exist for measuring its entities and relationships at high degrees of resolution in casual situations (as opposed to professional situations such as surveying engineering). Even if these data could be collected and stored, computational platforms were not sufficient to conduct analysis beyond coarse summaries. Motivated by advances in geographic information systems (GIS), researchers have recently developed formalisms to support computational implementation of time geographic entities. The network-based prisms developed by Miller (1991, 1999) and Kwan and Hong (1998) offer some analytical rigor, but only for a particular time geographic product (the prism) and a specific case (within a transportation network). Hornsby and Egenhofer (2002) develop a framework for multi-granularity representations of space–time paths, prisms, and composite paths/prisms to support space–time queries (although they invent a non-standard terminology). Their framework uses simultaneous inequalities to describe these entities. These are cumbersome for analytical statements about measurement and uncertainty propagation since they describe the entities implicitly. To date, attempts to integrate time geography and GIS do not achieve what Goodchild (2002) refers to as measurement-based GIS: this is software that provides access either to the original measurements or the functions used to infer the locations from the original measurements. This contrasts with traditional, coordinate-based GIS that provide access only to the locations of measured objects. Coordinate-based GIS limit the ability to perform spatial error analysis as well as attempt to reduce error and uncertainty, blunting the value of spatial data and software to users. If time geography is to be more tightly integrated into GIS, LAT, and LBS, it requires a rigorous framework that can support high resolution but imperfect measurement of the observable components used to infer its basic entities and relationships. Database design for mobile objects Traditional spatial databases and GIS software are static, typically representing spatial data at a given point in time. Integrating time into spatial databases and GIS is an active research frontier in geographic information science (see Langran 1992; Peuquet 2002). Some of the literature is directed specifically at socio-economic phenomena (see Frank, Raper, and Cheylan 2001). There are many overlaps among the questions and problems in this literature and time geography (Miller 2003). Particularly relevant for the problem in this article is the development of database designs to accommodate the mobile objects of concern in LAT and LBS. Since digital LAT can only sample an object’s location at discrete moments in time, a major concern in the moving objects database literature is interpolating an object’s location at an arbitrary moment given a finite set of sampled locations. One strategy is to determine spatial bounds on the movement possibilities between two 23

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sample locations. The moving objects spatio-temporal (MOST) data model and related query language accommodates mobile objects with uncertain positions due to finite sampling (Sistla et al. 1998). MOST represents the object’s position based on its last recorded position and bounds (upper and lower) on its velocities in each spatial dimension. This defines a line interval in one dimension and a torus in two dimensions. Pfoser and Jensen (1999) determine the spatial limits on possible paths between sample locations using an elegant geometric argument. At any given time t between location samples at times ti and tj, the object can only be within the intersection of two circles centred on the sampled positions. Fig. 4 provides an illustration. The first circle encompasses the possible locations for the object based on traveling from xi in a straight line for the elapsed time at the maximum velocity. The second circle encompasses the possible locations for the object based on traveling to xj in a straight line at the maximum velocity for the remaining time interval. This intersection is a lens-shaped region at any given moment in time but traces an ellipse with foci xi, xj over the time interval between the two sample points. Moreira, Ribeiro, and Saglio (1999) interpolate the unknown path by statistically fitting a line to the observed locations. The estimated parameters define a vector anchored at a sample location that is valid over some time interval. The database stores the vector and an error estimate derived from the line fitting. The parameters are re-estimated after each new sample and the system generates a new vector if the newly sampled location is outside a tolerance distance from the

Error limits at time t

Error limits for time interval between samples

Figure 4. Mobile object location uncertainty at a point in time and over a time interval (after Pfoser and Jensen 1999). 24

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existing vector. The vectors form a polyline through a simple ‘‘snapping rule’’ based on a user-specified parameter. They also define a path superset entity corresponding to its upper bounds: this is also an ellipse that delimits the maximum possible travel extent based on the two sample points and their respective times. Although not recognized, Pfoser and Jensen (1999) and Moreira, Ribeiro, and Saglio (1999) use time geographic arguments to address the sampling problem in mobile objects database design and rediscover the PPA. These arguments can be expanded to encompass more of the basic time geographic entities and relationships. The next section of this article extends these results to develop a measurement theory for time geography.

A measurement theory for time geography This section develops an analytical measurement theory for basic time geographic entities such as the space–time path, prism, and station. It also develops rigorous definitions of relationships such as intersection and bundling. The definitions in this section distinguish between the components that are measured in space and time and those that are inferred from the measurements. The definitions are also general enough to handle n-dimensional metric space rather than just the strict twodimensional Euclidean space of classical time geography (although we will mostly be concerned with n 5 1, 2, or 3). The two- and three-dimensional cases correspond to spatial movement within a plane or natural space, while the onedimensional case could be mapped to a network structure based on the shortest path trees (the concluding section of this article comments in more detail on this research frontier). Three major assumptions underlie the definitions in this section. First, shortest path relations in the n-dimensional space exhibit the metric properties of identity, non-negativity, and triangular inequality. These properties may not hold in realworld settings such as congested urban areas. Extending these definitions to more general spaces such as quasi-metric or semi-metric (see Huriot, Smith, and Thisse 1989; Smith 1989) are worthwhile questions for additional research. Second, measurement is finite with respect to time: the data used to construct the time geographic entities or relationships are limited and recorded at specific points or moments in time. These are reasonable assumptions: while frequent temporal sampling can approximate continuous time, true continuous time sampling is impossible with digital technologies. As noted in the previous section, discrete time data recording is a standard assumption in the emerging literature on mobile objects databases, even when objects experience continuous change in the real world. Finally, we assume perfect information: the theory describes the ideal measurement case given a finite platform. Although this is unrealistic, it is appropriate for a measurement theory since it provides an ideal benchmark. Uncertainty and error occur when the ideal measurement case is not achieved in practice; this can be addressed as an 25

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extension of the fully developed theory (this is also discussed in more detail in the conclusion). Space–time path A space–time path consists of two major components: (i) a sequence of control points and (ii) a corresponding sequence of path segments connecting these points. Control points are measured locations in space and time. The minimum information available at each control point is its location and the time when the location was recorded: ci  cðti Þ ¼ xi

ð1Þ

where xi is a location in space and ti is a moment or instant in time. The set of control points that determine the path are a finite list of space–time observations strictly ordered by time: C ¼ fðcS ; . . . ; ci ; cj ; . . . ; cE ÞjtS o    oti otj o    otE g

ð2Þ

where tS, tE are the start time and end time (respectively) for the path, that is, the first and last observed locations. Control points can correspond to three types of locations in the real world: (i) known activity locations; (ii) locations where the path experiences a change in direction (e.g., a turn at a street intersection) or velocity; and (iii) arbitrary locations where a spatial reference and time stamp are recorded (e.g., from a GPS receiver). Measured is a general concept: a control point may also be assumed or derived from some source other than an LAT or simulation method (e.g., an activity diary). We assume that there is a control point at every change in path direction or velocity. This is unrealistic since these changes occur continuously in the real world, but is appropriate as the ideal case of perfect but finite measurement. Path segments are the unobserved locations in space that connect temporally adjacent control points. Given the information available at the control points, the simplest representation of the unknown observations is a straight line segment between observed points (Moreira, Ribeiro, and Saglio 1999; Pfoser and Jensen 1999). We can define the unobserved segment as an interpolation between adjacent control points using time as a parameter: Sij ðtÞ ¼ ð1
aÞxi þ axj

ð3Þ

where: a¼

t
ti tj
ti

ð4Þ

Fig. 5 illustrates the basic idea. Representing the paths as line segments is also consistent with the ideal case of perfect but finite measurement: a control point must exist at every direction and velocity change so that the locations between any temporally adjacent control point pair are completely determined. 26

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Time Geography Measurement

t P (t )

tj

cj

t

ci

ti

xi

S ij (t )

xj

x

Figure 5. Control points and segments in a space–time path.

Combining the segment definition with the list of control points provides the following time parametric equation for the space–time path:
ci ; t 2 ðtS ; . . . ; ti ; tj ; . . . ; tE Þ P ðtÞ ¼ ð5Þ Sij ðtÞ; ti ototj Equation (5) allows us to scroll through locations in the space–time path using time as an index. Time is a useful parameter since we can be assured that each location in time along the path is unique. In contrast, a path may occupy the same location in space repeatedly or for an extended amount of time. The travel velocity is not required to be constant for the overall path. Temporally adjacent control points imply travel velocities: vij ¼

kxj
xi k tj
ti

ð6Þ

where k k is the vector norm or distance between the locations. Equation (5) defines a polyline indexed with respect to time. This follows from the assumption of a finite number of perfect measurements (i.e., the control points) at given discrete points in time, and the information available from these measurements (the location and time stamp). It is also consistent with classical time geography. It is inconsistent with theories and models of physical movement. Physical theory dictates a smooth continuous curve, while a polyline can imply unnatural behaviors such as instantaneous changes in direction and velocity. An 27

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alternative is to assume that the unobserved locations between control points form a smooth curve. If we also record the instantaneous velocities (direction and magnitude) at each control point, we can define the path segment as a Be´zier curve anchored by the control points (Casselman 1998): Sij ðtÞ ¼ ð1
aÞ3 xi þ 3að1
aÞ2 vi þ 3a2 ð1
aÞvj þ a3 xj

ð7Þ

where vi and vj are the velocity vectors at the control points. Although this generates a more natural-appearing curve for representing physical movement, it has some disadvantages. Although the Be´zier curve is simple, it requires making additional assumptions about movement behavior (specifically, that it corresponds to a cubic polynomial function of time). This involves behavioral questions that are outside the domain of this measurement theory. Also, allowing path segments to be curves rather than line segments can complicate analyses based on the measurement theory, including the analysis of path bundling and intersections (see below). Also, line segments allow relatively simple error structures (see Zhang and Goodchild 2002). There is another way to relate the measurement theory to physical theory. Note that as the temporal sampling rate approaches the asymptotic limit of continuous time limtj
ti !0 P ðtÞ more closely approximates a continuous curve consistent with physical theory since the control points become arbitrarily close in time and space and the velocities become arbitrarily close to instantaneous. This case corresponds to urban field theory where continuous velocity fields condition travel and therefore spatial economic structure (Angel and Hyman 1976). We will consider this case further when discussing research frontiers in the conclusion. Space–time prism The space–time prism can exist between any pair of temporally adjacent control points. In this case, there is an open temporal interval (ti, tj) during which the individual can conduct discretionary travel and perhaps activity participation. Unlike a segment, however, the minimum travel time between xi and xj is small enough relative to the interval (ti, tj) that the individual can occupy locations in space other than the line segment between ci and cj. Rather than inferring the velocity from the locations of the control points in space–time, we assume a constant maximum velocity across space and during the prism time interval (ti, tj). This is consistent with classical time geography and is also convenient for practical applications since a maximum velocity generates an upper bound on an individual’s physical reach. However, it is unrealistic since this interval is relatively large and therefore velocity changes are likely in reality. There are at least two ways of resolving this. One is to assume an idealistic transportation environment in the perfect information case and treat real-world deviations from this ideal as imperfect information. Another possibility is to develop new definitions of the prism that allow velocity to vary during the prism’s existence in time. This case has been addressed in a limited manner through network-based time prisms 28

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(Miller 1991, 1999; Wu and Miller 2001). A more comprehensive treatment requires a field-based time geography as mentioned above and elaborated in the conclusion. Similar to path segments, we can state the prism as a parametric function of time. Although we will use set notation, it will suggest algebraic and numeric solutions. We will first consider the case of aij 5 0 (i.e., no stationary activity time) before the general case where aij may be positive. When activity time is zero, the prism at time t is the intersection of two sets: Zij ðtÞ ¼ fi ðtÞ \ pj ðtÞ

ð8Þ

fi ðtÞ ¼ fxj kx
xi k  ðt
ti Þvij g

ð9Þ

pj ðtÞ ¼ fxj kxj
xk  ðtj
tÞvij g

ð10Þ

where:

fi(t) is the set of locations that can be reached from xi by the elapsed time t
ti. pj(t) is the set of locations that can reach xj given the remaining time budget tj
t. Since these describe locations in space that are within a fixed distance of a point, fi(t) and pj(t) are closed and convex sets we will refer to as discs. The discs are line segments in one-dimensional space, circles in two dimensions, and spheres in three dimensions. Since we can interpret fi(t) and pj(t) as the possible futures of ci and the possible pasts of cj at time t (Hawking and Penrose 1996), we will refer to these sets as the future disc and past disc, respectively. We can evaluate Zij(t) by noting the topological relationships between the two discs during subintervals of (ti, tj). Assuming that the time budget is sufficiently large (see below), there is a temporal interval starting at ti when pj(t) completely encompasses fi(t), that is, ðtj
tÞvij 4ðt
ti Þvij þ kxj
xi k. Fig. 6a illustrates this case. The upper bound on this temporal region is: t0 ¼

ðti þ tj
tij Þ 2

ð11Þ

Figure 6. Topological relationships between future and past discs: (a) past disc encompasses the future disc; (b) future disc encompasses the past disc. 29

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where tij ¼ kxj
xi kvij
1 is the minimum travel time between ci and cj. Conversely, there is a temporal interval that ends at tj when fi(t) completely encompasses pj(t), that is, ðt
ti Þvij 4ðtj
tÞvij þ kxj
xi k. Fig. 6b illustrates this case. The lower bound on this temporal region is: t 00 ¼

ðti þ tj þ tij Þ

ð12Þ

2

The two discs overlap during the interval between these boundaries. Fig. 7 illustrates these temporal regions in one-dimensional space for the cases xi 5 xj and xi 6¼ xj. This suggests a strategy for evaluating Equation (8): (i) (ii) (iii)

t 2 ðti ; t 0  : Zij ðtÞ ¼ fi ðtÞ t 2 ½t 0 ; t 00  : Zij ðtÞ ¼ fi ðtÞ \ pj ðtÞ t 2 ½t 00 ; tj Þ : Zij ðtÞ ¼ pj ðtÞ

We can treat the interior subinterval boundaries as closed since at these instants the two alternative solutions on either side of a boundary are equivalent. In practice, we would default to the simpler of the two solutions. The intersection in Case (ii) is a line segment in one-dimensional space, a lens-shaped region formed from a circle–circle intersection in two-dimensional space and a lens-shaped

tj

t′ = t″

t′ ti

Z ij (t )

xi

tj

t″ t′

ti

xi

xj

Figure 7. Prism temporal boundaries where activity time is zero. Top half—origin and destination coincident. Bottom half—origin and destination different locations. 30

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volume resulting from a sphere–sphere intersection in three-dimensional space. (The intersection is a single point only when tj
ti ¼ tij , i.e., the time budget equals the minimum travel time between ci and cj; in this case, t 0 5 ti and t00 5 tj. We discuss this case in more detail below.) Analytical solutions exist for the locations of the intersection points and the intersection region size (length, area, volume) in one, two, or three dimensions (O’Rourke 1994; Weisstein 2002b, d). Querying whether a given location is within the prism at time t requires checking whether the inequalities in Equations (9) and (10) are simultaneously satisfied for the candidate location. We now consider the more general case where aij may be positive. Recall from the previous section that a cylinder of length aij separates the two cones comprising this prism. As Fig. 8 illustrates, this can be viewed as a restriction of the prism imposed by the need for stationary activity time. The prism at time t is the intersection of three sets: Zij ðtÞ ¼ fxj fi ðtÞ \ pj ðtÞ \ gij g

ð13Þ

gij ¼ fxj kx
xi k þ kxj
xk  ðtj
ti
aij Þvij g

ð14Þ

where

gij is the set of locations that an individual can reach and still meet the coupling constraint at xj by tj, including the stationary time for the activity. This is the PPA from classical time geography. gij is a closed and convex set in x comprising locations within a fixed distance of two points. Consistent with well-known results from classical time geography (see Lenntorp 1976; Burns 1979), gij is an ellipse in two-dimensional space with foci xi, xj, major axis of length ðtj
ti
aij Þvij ; and minor axis of length ½ððtj
ti
aij Þvij Þ2
kxj
xi k2 1=2 . This collapses to a circle when xi 5 xj. gij is a line segment in one-dimensional space and an ellipsoid in

tj t ″′ t″ t′ t0 gij

ti xi

xj

Figure 8. Prism temporal boundaries when activity time is positive. 31

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three-dimensional space. We will refer to gij as the geo-ellipse of the space–time prism to highlight its requirement for two distance evaluations. In practice, we need to consider the intersection of gij with the future disc and the past disc only for some subintervals of (ti, tj). Using a similar argument as above, there is an interval starting at ti when fi (t) is small and encompassed by both pj (t) and gij. This boundary is: t0 ¼

ðti þ tj
tij
aij Þ 2

ð15Þ

Similarly, there is an interval ending at tj when pj(t) is small and encompassed by both fi(t) and gij. This boundary is: t 000 ¼

ðti þ tj þ tij þ aij Þ 2

ð16Þ

Clearly, ti ot 0 ot 0 and t 00 ot 000 otj ; see Fig. 8. This suggests a strategy for evaluating Equation (13): (i) (ii) (iii) (iv) (v)

t t t t t

2 ðti ; t 0  : Zij ðtÞ ¼ f i ðtÞ; 2 ½t 0 ; t 0  : Zij ðtÞ ¼ fi ðtÞ \ gij ; 2 ½t 0 ; t 00  : Zij ðtÞ ¼ fi ðtÞ \ pj ðtÞ \ gij ¼ gij ðtÞ; 2 ½t 00 ; t 000  : Zij ðtÞ ¼ pj ðtÞ \ gij ; 2 ½t 000 ; tj Þ : Zij ðtÞ ¼ pj ðtÞ:

where gij(t) is the geo-ellipse projected to time t, that is, ðxÞ ! ðx; tÞ 8x 2 gij . Although Case (iii) involves the intersection of three sets, we know that gij encompasses fi (t) and pj (t); hence we can disregard the intersection for the simpler set gij (t). If xi 5 xj, t0, t 000 partition (ti, tj) into only three subintervals, with Zij (t) evaluating to fi (t), gij (t), and pj (t), respectively, in the intervals. Evaluating Zij (t) in Cases (ii) and (iv) requires finding the intersection of two line segments in one-dimensional space, a circle and an ellipse in two dimensions, and a sphere and an ellipsoid in three dimensions. Analytical solutions exist for these intersection problems in one- and two-dimensional space (see Weisstein 2002c). The sphere–ellipsoid intersection problem in three dimensions is a special case of the quadric surface intersection problem. There is an analytical solution to this problem (see Levin 1976, 1979), although it may not be robust since it is highly sensitive to small perturbations in the polynomial parameters; this is an issue in finite computational platforms. Efficient geometric strategies have emerged as alternatives (see Miller 1987). It is also possible to treat each surface as a differential equation and numerically search along one of the surfaces to find the intersection with the other surface. This can be solved exactly subject to machine precision (Hosaka 1992). A special case occurs when tj
ti ¼ tij . In this case, aij 5 0; otherwise, the prism is infeasible. As can be easily confirmed, t 0 5 ti and t00 5 tj, implying that Zij(t) can be described by the intersection of fi(t) and pj(t) for all tA(ti, tj). Since tj
ti ¼ tij , 32

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  the radii of these two discs must sum exactly to xj
xi  at every tA(ti, tj), implying that the two discs intersect at a single point along the line segment xj xi . The location of this intersection point along the line segment is: xðtÞ ¼

t
ti ðxj
xi Þ ¼ aðxj
xi Þ tj
ti

ð17Þ

which is equivalent to Equation (3). In other words, we can view the prism Zij(t) and segment Sij(t) as special cases of each other: the segment is a prism where tj
ti ¼ tij ; while the prism is a segment with excess travel time ðtj
ti 4tij Þ between its two defining control points. Space–time lifelines Analyzing an individual’s space–time activities and accessibility often requires a combination of several space–time paths and prisms into a composite object. We will refer to this object as a space–time lifeline: this is a space–time path with prisms defined between some temporally adjacent control points. The segments can represent required travel and activity participation while the prisms can correspond to time intervals with discretionary travel and activity participation. They can also represent observed and unobserved behavior, respectively (e.g., a space–time path with some location tracking black-out periods, either accidental or intentional). Since the segment and prisms are special cases of each other depending on the relation between the control points and the time budget, we can define a space– time lifeline as: 8 q q q q q c ; t 2 ðtS ; . . . ti ; tj ; . . . ; tE Þ > > < i q q q q q q R q ðtÞ ¼ Sij ðtÞ; ti ototj ^ tj
ti ¼ tij ð18Þ > > : Z q ðtÞ; t q otot q ^ t q
t q 4t q ij

i

j

j

i

ij

where ^ indicates the logical predicate AND, all other terms as defined previously and superscripts added to indicate membership in a particular lifeline. Fig. 9 illustrates a space–time lifeline and a space–time station (see below). Note that the prism fits naturally with the segments simply by making explicit the path superscript suppressed in the earlier discussion: a prism now belongs to a particular path. The superscript also helps to distinguish among different time geographic entities; this will be important when discussing bundling and intersection relationships (see below). Stations Recall from the previous section that a station is a location where paths can bundle for some activity. This usually corresponds to a designated activity location such as a retail outlet, office, home, etc. Conceptualizing activity locations as stations allows the analyst to incorporate temporal components of activity location such as a retail outlet’s operating hours. Since it never changes location, a station can be designated by a spatial location x and a list of ordered pairs of time points 33

Geographical Analysis

Figure 9. Space–time lifelines and stations.

indicating the start and end times for operation ððtSr ; tEr Þ; ðtSr 0 ; tEr 0 Þ; . . .Þ where tSr otEr otSr 0 otEr 0 . The parametric equation for a station is therefore: Q r ðtÞ ¼ xr ;

t 2 ½tir ; tjr  _ t 2 ½tkr ; tlr  _   

ð19Þ

where _ indicates the logical predicate OR. Fig. 9 provides an illustration. In classical time geography, a station is typically represented as a vertical tube that can accommodate space–time paths inside. Although the tube conceptualization is intuitive for graphical depiction, Equation (19) instead defines a station as a special type of space–time path. This path has two unique properties: (i) it is always vertical, that is, it never changes location and (ii) it may have temporal gaps corresponding to interruptions in operating hours (e.g., closing hours during a day, closing days during a week). This definition has analytical advantages since we can treat bundling between a path and a station similar to bundling between paths. Bundles and intersections As noted in the previous section, two key relationships in classical time geography are bundles and intersections. Bundling of space–time paths refers to the convergence of two or more paths for some shared activity. Path bundling is evidence of individuals meshing their space–time activities to participate in projects. Path bundling is also a necessary condition for the emergence of many space–time activity systems. Bundling can also occur at stations, implying that the paths are vertical 34

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and the individual is stationary in space. Paths can also bundle during movement; examples include public transportation and car-pooling. Intersection is the condition of two or more time geographic features sharing some locations in space and time. As noted in the previous section, conducting an activity at some station is not feasible unless the station intersects the individual’s space–time prism. Also, two (or more) people cannot physically meet unless their space–time prisms intersect. Note that an intersection may require coincidence of the objects for a minimum threshold time. Even if the station intersects with a person’s space–time prism this may be meaningless if the intersection is too short in time for the activity to be conducted (e.g., not enough time to shop at a retail outlet). Two space–time paths are bundled during an open time interval ðtB0 ; tB00 Þ if the following conditions are true: (1)

[temporal] Both paths cover the time interval. This means a path cannot start or stop during the interval ðtB0 ; tB00 Þ; it can start or stop only at the interval boundaries or outside the interval. Operationally, we require the following for a path involved in the bundle: cS  tB0 ^ cE  tB00

(2)

ð20Þ

If the path is a station with temporal gaps, the control points correspond to an ordered pair defining a time interval in the list of operating hours. Paths cannot bundle with stations if they are not available.   [spatial] Both paths are spatially proximal for the interval tB0 ; tB00 : ð21Þ kP q ðtÞ
P r ðtÞk  d
8t 2 ðt 0 ; t 00 Þ B

B

where d
40 is a user-defined threshold distance; we discuss d
¼ 0 or intersections below. We apply this condition to an open interval since the trajectories can change direction at the boundaries tB0 ; tB00 . In practice, we can first check for temporal covering and then for spatial proximity in a given time interval. We can also make the temporal covering condition implicit by defining the distance between the two entities as infinity if one or both does not exist at time t. Requiring temporal covering and spatial proximity are not perfect discriminators for bundling. Two paths can meet these conditions but still not share an activity: an example is two persons traveling in different directions through an intersection. This problem is more severe as the time interval becomes relatively smaller and the threshold distance is relatively large. A possible third condition is requiring the paths to exhibit concerted or tandem movement. One strategy is to treat time as a spatial dimension and evaluate the temporal distance in time between the paths, or the combined spatio-temporal distance (see Yanagisawa, Akahani, and Satoh 2003). Another method that is more consistent with time geographic theory is to check if the objects’ velocities are equal during the interval ðtB0 ; tB00 Þ: 35

Geographical Analysis

(3)

[equal velocity] The velocities of the two paths are equal for the interval ðtB0 ; tB00 Þ: v q ðtÞ ¼ v r ðtÞ

8t 2 ðtB0 ; tB00 Þ

ð22Þ

where vq(t) is the velocity of path q at time t. Again, we apply this condition to the open interval since velocities can change at the interval boundaries. If t falls on a path segment, we can calculate the velocity using the temporally adjacent control points that define that segment (see Equation (6)). If t corresponds to a control point for a path, we use the next control point in time to calculate velocity since the effect of a possible velocity change at this point is forward in time. Equal velocity is also not a perfect discriminator but will eliminate more coincidental bundles than the temporal and spatial conditions alone. There is a scale issue: while this definition is appropriate for geographic scales, at architectural scales individuals may not be moving at the same velocity when performing a shared activity (an example is an energetic person lecturing to a seated audience). In practice, we could set a small threshold value for differences in velocity rather than require strict equality. Under perfect information, the control points are sufficient to determine exactly the locations and velocity at all locations in the space–time path. Since any change in velocity or direction must occur at a control point, any change in bundling must also occur at a control point. Therefore, a temporal boundary on the bundle relation will always coincide (in theory) with at least one control point. Therefore, we can check for the temporal covering, spatial proximity, and equal velocity conditions by examining only at the control points rather than every location in both paths. Fig. 10 illustrates checking for equal velocity at control points: note that the control points that define the bundling interval boundaries can occur in either path. An alternative strategy is to normalize the paths by inserting synthetic control points at regular spatial or temporal intervals and then evaluate distance and velocities at the paired control points (Yanagisawa, Akahani, and Satoh 2003). A problem with evaluating bundling relations is the potentially large amount of distance evaluations required. Yanagisawa, Akahani, and Satoh (2003) use piecewise aggregation approximation to reduce the number of distance evaluations required when comparing discrete space–time paths. This technique reduces the number of control points in a path by generating synthetic control points that are averages of the control points within some defined neighborhood. The resulting distance evaluations are the lower bounds on the distances between the original paths. They also develop an indexing method based on R1 trees that supports the efficient retrieval of the required data from secondary memory. Intersections are special cases of bundles where d
¼ 0 for some open interval 0 00 ðtI ; tI Þ. The time interval can be a single point in time. There are three intersection cases to consider: (i) path–path; (ii) path–prism; and (iii) prism–prism. In the 36

Harvey J. Miller

Time Geography Measurement

t

tB″

t′B

x

Figure 10. Checking for bundling conditions at control points.

path–path case, the problem is equivalent to the polyline intersection problem in two-, three-, or four-dimensional space (treating time as an extra spatial dimension for this problem). See deBerg et al. (1997) and Preparata and Shamos (1985) for polyline intersection algorithms. We can solve the path–prism spatial intersection problem through an extension of solving the prism for some t. Recall that the prism at time t is a disc, the intersection of a disc and an ellipse, or an ellipse. Therefore, the path–prism intersection problem at any given t is equivalent to finding if a point lies within a disc, ellipse, or a disc–ellipse intersection, since the path is a point at any time t. Fig. 11 illustrates the tA[ti, t0], tA[t0, t 0 ], and tA[t 0 , t00 ] cases in two-dimensional space. The fourth and fifth cases are symmetric to the first two cases. These are straightforward geometric problems. Finding the prism–prism intersection at a given t is equivalent to finding the intersection between two, three, or four convex sets based on the prisms’ morphologies at that moment in time. Table 1 summarizes the possible cases. Fig. 12 illustrates the worst case in two-dimensional space, namely, a four-set intersection involving two discs and two ellipses. In three dimensions, this case would involve finding the intersection among two spheres and two ellipsoids. Determining whether a specific location lies within a prism–prism intersection requires determining if the location simultaneously satisfies the inequalities in Equations (9), (10), and (14) for both prisms. Extending the prism–prism intersection to an m prism intersection (where m is some integer) can be managed using Helly’s theorem. This theorem states that a 37

Geographical Analysis

(a)

ciq

c qj

pijq (t ) g ijq

(b) f i (t ) q

pijq (t ) g ijq

P r (t )

f i q (t )

P r (t )

f i q (t )

(c)

pijq (t )

g ijq

P r (t ) Figure 11. Path–prism intersection at time t in two-dimensional space. (a) (top) t 2 ½ti ; t 0 ; (b) (middle) t 2 ½t 0 ; t 0 ; (c) (bottom) t 2 ½t 0 ; t 00 .

Table 1 Type and Number of Spatial Sets Involved in Prism–Prism Intersections q

Zij ðtÞ

tA(ti, t0] tA[t0, t 0 ] tA[t 0 , t00 ] tA[t 0 , t 000 ] tA[t 000 , tj)

Zklr ðtÞ tA(tk, t0]

tA[t0, t 0 ]

tA[t 0 , t00 ]

tA[t00 , t00 0 ]

tA[t 000 , t1)

f/f/2 fg/f/3 g/f/2 pg/f/3 p/f/2

f/fg/3 fg/fg/4 g/fg/3 pg/fg/4 p/fg/3

f/g/2 fg/g/3 g/g/2 pg/g/3 p/g/2

f/pg/3 fg/pg/4 g/pg/3 pg/pg/4 p/pg/3

f/p/2 fg/p/3 g/p/2 pg/p/3 p/p/2

NOTES: (i) Table entries are: discs from q/discs from r/total number of discs; f, future-disc; p, past-disc; g, geo-ellipse; (ii) Temporal boundaries are specific to each prism although distinguishing superscripts are suppressed for clarity. 38

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Time Geography Measurement

ciq

c qj

ckr

clr

Figure 12. Prism–prism intersection at time t in two-dimensional space—two disc and two ellipse case.

finite collection of closed and convex sets in n-dimensional space has a non-empty intersection if and only if every choice of n11 of the sets has a non-empty intersection (Eckho 1993). Therefore, we must find the intersection of no more than two, three, or four sets (in one, two, and three-dimensional spaces, respectively) when determining the intersection of m prisms. In practice, this means that for n dimensions we choose n11 of the m sets at a time, stopping with the result that the intersection does not exist if we find a non-intersection among these sets. This implies an algorithm with an asymptotic complexity of O(mn11) since in the worst case all m sets must be examined n11 at a time. However, the worse case can be easily avoided by careful pre-sorting of the prisms to identify the extreme cases that are likely to result in an empty intersection (O’Kelly and Miller 1991). Determining if a given location is within an m-prism intersection requires in the worse case evaluating 6m distances and 4m inequalities. Conclusion Time geography is a powerful conceptual framework for understanding spatio-temporal constraints on human activity participation. It is less successful as an analytical framework since its fundamental components and relationships have never been stated in a rigorous and consistent manner. The rise of LAT and LBS means that these components can be measured to high levels of spatio-temporal resolution, potentially exposing the lack of analytical rigor at the foundation of time geography. The time geographic measurement theory in this article extends techniques in the moving objects database literature to develop rigorous definitions of 39

Geographical Analysis

fundamental time geographic concepts and relationships such as the space–time path, prism, stations, composite path-prisms, bundles, and intersections. Temporal disaggregation allows their solution as simple spatial objects, or as distance and intersection relationships between simple spatial objects. These objects and intersections have algebraic solutions in one or two spatial dimensions. Numeric solutions are required for some entities and relations in higher dimensions, but these are relatively tractable since they involve simple surfaces. Research frontiers implied by the time geographic measurement framework include query design, mapping the theory to networks, extending the theory to velocity fields, imperfect measurement, and incorporating virtual interaction. Query design The analytical definitions in this paper provide a foundation for building computational tools for time geographic querying and analysis. The time geographic measurement theory in this paper provides functional requirement statements for these computational tools (i.e., precise specifications of what the computational tools should be measuring but not necessarily how they should be computed). Although analytical simplicity does not necessarily imply computational efficiency, the elegant geometry revealed by temporally disaggregating time geographic entities suggests strategies for computational implementation. Analytical solutions are available in one and two spatial dimensions, and analytical solutions are highly desirable to the algorithm designer since they imply fast computation times. Numeric strategies are required for some solutions, particularly in three spatial dimensions, but these solutions are well-studied and tractable. The main issues within this research frontier are data modeling, computational and user interface problems associated with managing and processing the data in primary and secondary memory, and communicating the result to the user. Mapping the theory to networks The definitions in this article are general to any dimensional space, not just the twodimensional space of classical time geography. The two- or three-dimensional theory is appropriate if we are interested in movement possibilities in a plane or natural space. The one-dimensional theory can be mapped to networks based on shortest path distances. We can solve for the network analog of the PPA, the potential path tree (PPT), by computing the shortest path trees from the two anchoring locations and then testing for inclusion in the tree using the distance and velocity data attributed to the arcs (Miller 1991). This resolves only to the nodes in the network, leaving unresolved gaps in the network roughly approximating the true border within the network. Miller (1999) uses extended shortest path trees (Okabe and Kitamura 1996) to calculate the potential network area (PNA): a higher resolution network analog of the PPA that resolves to any location within the network. Wu and Miller (2001) extend the PNA to dynamic networks with discrete-time changes in flow and travel velocities. Extending the measurement theory to networks using 40

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Time Geography Measurement

these techniques is straightforward mathematically; as in query design, the challenges are computational and software design issues. Computing the network analogs of the time geographic discs and ellipses requires very efficient computational methods since these correspond to only one moment in time and there will likely be a large number of such time periods to be evaluated. If n is the number of nodes in a network, the extended shortest path tree calculation requires O(n log n) operations for each shortest path tree plus O(n2) for the breakpoint insertion (Okabe and Yamada 2001). The worse case can often be avoided in the time geography since time budgets and finite velocities limit the subnetwork relevant for calculations. These times are nevertheless daunting for detailed urban-scale applications, especially for real-time applications (such as LBS) or data mining and visualization of large space-time activity databases. More efficient data structures and processing methods are required. Extending the theory to velocity fields The movement behavior implied by the space–time path is unrealistic: a polyline admits unnatural turns and velocity changes. As the sampling rate approaches the asymptotic limit of continuous time, limtj
ti !0 P ðtÞ more closely approximates a continuous curve consistent with physical theory. This limiting case corresponds to urban field theory that treats movement, spatial interaction, and related concepts (such as retail market areas) as continuous phenomena occurring within a velocity field (see Angel and Hyman 1976; Puu and Beckmann 1999). Analytical solutions for the required minimum path relations in a velocity field are only available for a limited number of restrictive special cases such as the field being uniform or radially symmetric with respect to a single location, although tractable computational approximations are available (see Smith, Peng, and Gahinet 1989; Mitchell and Papadimitriou 1991; van Bemmelen et al. 1993). Although the finite measurement theory in this article is sufficient for many applications, explicitly extending the framework to develop a field-based time geography is an open and worthwhile research question that may generate new theoretical insights and analytical tools. For example, velocity fields have been used to measure the impact of a new highway on the optimal travel patterns in an urban area or to evaluate potential facility locations (Mayhew and Hyman 2000; Hyman and Mayhew 2001). A field-based time geography could generalize this approach as well as link it to individual activity schedules. Imperfect measurement Time geographic parameters such as control points and velocities are always measured with error and limited precision in reality. A critical research question is how error and uncertainty propagate through the inferred entities and relationships to degrade the quality of time geographic queries or variables used in social research. The immediate task is to connect this framework to the geographic information science literature on spatial data error and uncertainty in GIS databases and analytical operations. Relevant problems include line segments and polylines under 41

Geographical Analysis

uncertainty (Shi 1997, 1998; Zhang and Goodchild 2002), point-in-polygon under uncertainty (Leung and Yan 1997), and error propagation in buffer analysis (Shi, Cheung, and Zhu 2003). Several of the time geographic uncertainty problems identified in this article are more restricted and simpler than the general problems in the GIScience literature: for example, time geography measurement in twodimensional space involves circles and ellipses rather than more general polygons. A related but more focused research agenda concerns methods for protecting privacy with respect to LAT and LBS. As Dobson and Fisher (2003) persuasively argue, these technologies have the potential to create geo-slavery or the ability to track and control individual movements and positions in space and time. However, preserving privacy can also involve a trade-off with the accuracy of answers provided by an LBS. Since there is a close correspondence between LBS queries and time geographic queries, the framework developed in this article could be used to analyze this trade-off. For example, what is the maximum level of spatio-temporal privacy that can be provided to an individual who still meets minimum accuracy requirements for the particular query? Armstrong, Rushton, and Zimmerman (1999) develop the concept of geographic masking to preserve locational privacy. The framework in this article could be used to develop spatio-temporal masking for LBS and related technologies and services. Extending the theory to virtual interaction Although time geography recognizes the ability to interact without physical proximity through media such as telephony (see Ha¨gerstrand 1970), the classical theory nevertheless focuses on physical presence, movement, and interaction. It is increasingly difficult to maintain a conceptual separation between physical and virtual interaction given the increasing prevalence of information and communications technologies within high-mobility lifestyles. Extending time geography to include virtual interaction has been an active research frontier (e.g., Adams 1995, 2000; Couclelis and Getis 2000; Kwan 2000a). The time geographic measurement theory offers a potential strategy for encompassing virtual interaction that focuses on the measurable properties of these interactions. Rigorous analytical definitions of the time geographic objects and relationships that comprise virtual interaction and are consistent with the physical theory in this article are required. Acknowledgements We would like to thank Scott Bridwell, Mike Goodchild, Morton O’Kelly, Martin Raubal, Claus Rinner, and the referees for their helpful comments. References Adams, P. C. (1995). ‘‘A Reconsideration of Personal Boundaries in Space–Time.’’ Annals of the Association of American Geographers 85, 267–85. 42

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Adams, P. C. (2000). ‘‘Application of a CAD-Based Accessibility Model.’’ In Information, Place and Cyberspace: Issues in Accessibility, 217–39, edited by D. G. Janelle and D. C. Hodge. Berlin: Springer. Angel, S., and G. M. Hyman. (1976). Urban Fields: A Geometry of Movement for Regional Science. London: Pion. Armstrong, M. P., G. Rushton, and D. L. Zimmerman. (1999). ‘‘Geographically Masking Health Data to Preserve Confidentiality.’’ Statistics in Medicine 18, 497–525. Burns, L. D. (1979). Transportation, Temporal and Spatial Components of Accessibility. Lexington, MA: Lexington Books. Casselman, B. (1998). Geometry and Postscript; e-document available at Digital Math Archives (www.sunsite.ubc.ca/DigitalMathArchives). Couclelis, H., and A. Getis. (2000). ‘‘Conceptualizing and Measuring Accessibility within Physical and Virtual Spaces.’’ In Information, Place and Cyberspace: Issues in Accessibility, 15–20, edited by D. G. Janelle and D. C. Hodge. Berlin: Springer. deBerg, M., M. van Krevald, M. Overmars, and O. Schwarzkopf. (1997). Computational Geometry: Algorithms and Applications. Berlin: Springer. Dobson, J. E., and P. F. Fisher. (2003). ‘‘Geoslavery,’’ University Consortium for Geographic Information Science Research Brief; available at www.ucgis.org Eckhoff, J. (1993). ‘‘Helly, Radon and Carathe´odory Type Theorems.’’ In Handbook of Convex Geometry, 389–448, edited by P. M. Gruber and J. M. Wills. New York: NorthHolland. Forer, P. (1998). ‘‘Geometric Approaches to the Nexus of Time, Space and Microprocess: Implementing a Practical Model for Mundane Socio-Spatial Systems.’’ In Spatial and Temporal Reasoning in Geographic Information Systems, 171–90, edited by M. J. Egenhofer and R. G. Golledge. Oxford: Oxford University Press. Frank, A., J. Raper, and J. -P. Cheylan, eds. (2001). Life and Motion of Socio-Economic Units. GISDATA 8, 21–34. London: Taylor & Francis. Golledge, R. G., and R. J. Stimson. (1997). Spatial Behavior: A Geographic Perspective. New York: Guilford. Goodchild, M. F. (2002). ‘‘Measurement-based GIS.’’ In Spatial Data Quality, 5–17, edited by W. Shi, P. F. Fisher, and M. F. Goodchild. London: Taylor & Francis. Ha¨gerstrand, T. (1970). ‘‘What About People in Regional Science?’’ Papers of the Regional Science Association 24, 7–21. Hawking, S., and R. Penrose. (1996). The Nature of Space and Time. Princeton, NJ: Princeton University Press. Hornsby, K., and M. J. Egenhofer. (2002). ‘‘Modeling Moving Objects Over Multiple Granularities.’’ Annnals of Mathematics and Artificial Intelligence 36, 177–94. Hosaka, M. (1992). Modeling of Curves and Surfaces in CAD/CAM. Berlin: Springer-Verlag. Huriot, J.-M., T. E. Smith, and J.-F. Thisse. (1989). ‘‘Minimum-Cost Distances in Spatial Analysis.’’ Geographical Analysis 21, 294–315. Hyman, G., and L. Mayhew. (2001). ‘‘Market Area Analysis Under Orbital-Radial Routing with Applications to the Study of Airport Location.’’ Computers, Environment and Urban Systems 25, 195–222. Kwan, M.-P. (2000a). ‘‘Human Extensibility and Individual Hybrid-Accessibility in Space– Time: A Multiscale Representation Using GIS.’’ In Information, Place and Cyberspace: Issues in Accessibility, 241–56, edited by D. G. Janelle and D. C. Hodge. Berlin: Springer. 43

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