Geographical Analysis ISSN 0016-7363

A Measurement Theory for Time Geography

Harvey J. Miller
Department of Geography, University of Utah, Salt Lake City, UT

Hägerstrand’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 ge-
ographic 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 in-
ferred 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. Alge-
braic 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 (Hägerstrand 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 de-
velopment of location-aware technologies (LAT) and location-based services (LBS)
has potential to create an even wider resurgence of time geography in social re-
search and in geographic information services (the provision of geographic infor-
mation 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: harvey.miller@geog.utah.edu

Submitted: June 11, 2003. Revised version accepted: April 20, 2004.

Geographical Analysis 37 (2005) 17–45 r 2005 The Ohio State University 17

Geographical Analysis

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 cen-
tury. 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 ge-
ography can serve as a theoretical foundation for LBS.
A problem is that time geography is not ready for the measurement tasks de-
manded 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-resolu-
tion measurement of mobile objects in space and time. They are not sufficient for
analyzing the propagation of uncertainty when measuring these objects imperfect-
ly. The looseness and incompleteness of these descriptions mean that comparisons
of detailed data and time geographic analyses across different studies may be dif-
ficult since the entities and relations are not strictly comparable. A conceptual
framework, while useful for generating ideas, does not provide the detailed spec-
ifications required for developing standard time geographic computational tools.
Problems associated with representation and analysis of spatio-temporal enti-
ties also arise in geographic information science. In particular, the development of
LAT and LBS is inspiring a growing literature on database design for storing infor-
mation 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 relation-
ships. 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 max-
imum 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 so-
lutions are required for some entities and relations in higher dimensions, but these
are relatively tractable since they involve simple surfaces. Measuring relationships

18
Harvey J. Miller Time Geography Measurement

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. Al-
though an ideal case, the framework supports imperfect measurement and the anal-
ysis 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 de-
velop 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 com-
posites, and space–time stations. It also includes formal definitions of fundamental
relationships between space–time paths and prisms, specifically bundles, and in-
tersections. 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, Hägerstrand’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. Con-
straints that limit the ability of individuals to travel and participate in activities in-
clude: (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 par-
ticular 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
Geographical Analysis

useful: fixed activities dictate strict coupling constraints while flexible activities al-
low 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 res-
olution (see Kwan 2000b). These paths can also be simulated using aggregate-level
activity and time-use data (see Lenntorp 1976). Individual paths convey informa-
tion 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 organiza-
tional-level goal), and space–time activity systems (stable, multi-scale spatio-tem-
poral 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
Time

3
e al
ac hic

2
Sp rap
g

1
eo
G

Figure 1. A space–time path.

20
Harvey J. Miller Time Geography Measurement

tj
f (maximum velocity)

Potential Path Space

Time

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 inter-
sects 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 loca-
tions 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 re-
quired 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 no-
tation. 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, of-
fices, 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

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 mathemat-
ical sense: the ability to construct continuous mathematical functions that are de-
fined 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 conse-
quences when these entities and relationships are measured imperfectly.
The most complete time geographic systems are informal descriptions of con-
structible 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
Harvey J. Miller Time Geography Measurement

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 irrel-
evant 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 imple-
mentation 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 pro-
vides access either to the original measurements or the functions used to infer the
locations from the original measurements. This contrasts with traditional, coordi-
nate-based GIS that provide access only to the locations of measured objects. Co-
ordinate-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 im-
perfect 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 da-
tabase 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 ob-
ject’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
Geographical Analysis

sample locations. The moving objects spatio-temporal (MOST) data model and re-
lated 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 inter-
section 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 statisti-
cally 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
Harvey J. Miller Time Geography Measurement

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 correspond-
ing 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 relation-
ships. The next section of this article extends these results to develop a measure-
ment 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 two-
dimensional Euclidean space of classical time geography (although we will most-
ly 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 one-
dimensional 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 real-
world 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, meas-
urement 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
Geographical Analysis

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 informa-
tion 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 contin-
uously 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 be-
tween observed points (Moreira, Ribeiro, and Saglio 1999; Pfoser and Jensen 1999).
We can define the unobserved segment as an interpolation between adjacent con-
trol points using time as a parameter:
Sij ðtÞ ¼ ð1
aÞxi þ axj ð3Þ

where:
t
ti
a¼ ð4Þ
tj
ti

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
Harvey J. Miller Time Geography Measurement

t
P (t )

tj cj

ti ci

x
xi S ij (t ) xj

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. Tempo-
rally adjacent control points imply travel velocities:
kxj
xi k
vij ¼ ð6Þ
tj
ti

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 meas-
urements (the location and time stamp). It is also consistent with classical time
geography. It is inconsistent with theories and models of physical movement. Phys-
ical theory dictates a smooth continuous curve, while a polyline can imply un-
natural behaviors such as instantaneous changes in direction and velocity. An

27
Geographical Analysis

alternative is to assume that the unobserved locations between control points form
a smooth curve. If we also record the instantaneous velocities (direction and mag-
nitude) at each control point, we can define the path segment as a Bézier 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 gen-
erates a more natural-appearing curve for representing physical movement, it has
some disadvantages. Although the Bézier curve is simple, it requires making ad-
ditional 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 measure-
ment theory, including the analysis of path bundling and intersections (see below).
Also, line segments allow relatively simple error structures (see Zhang and Good-
child 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 in-
dividual 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 in-
terval 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
Harvey J. Miller Time Geography Measurement

(Miller 1991, 1999; Wu and Miller 2001). A more comprehensive treatment re-
quires 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 so-
lutions. 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
where:
fi ðtÞ ¼ fxj kx
xi k  ðt
ti Þvij g ð9Þ
pj ðtÞ ¼ fxj kxj
xk  ðtj
tÞvij g ð10Þ
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 dimen-
sions. Since we can interpret fi(t) and pj(t) as the possible futures of ci and the pos-
sible 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 encom-
passes 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:
ðti þ tj
tij Þ
t0 ¼ ð11Þ
2

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

29
Geographical Analysis

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:

ðti þ tj þ tij Þ
t 00 ¼ ð12Þ
2

The two discs overlap during the interval between these boundaries. Fig. 7 illus-
trates 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) t 2 ðti ; t 0  : Zij ðtÞ ¼ fi ðtÞ
(ii) t 2 ½t 0 ; t 00  : Zij ðtÞ ¼ fi ðtÞ \ pj ðtÞ
(iii) t 2 ½t 00 ; tj Þ : Zij ðtÞ ¼ pj ðtÞ

We can treat the interior subinterval boundaries as closed since at these in-
stants 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″
Z ij (t )
t′
ti

xi

tj

t″
t′

ti

xi xj

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

30
Harvey J. Miller Time Geography Measurement

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 dis-
cuss 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 inter-
section of three sets:

Zij ðtÞ ¼ fxj fi ðtÞ \ pj ðtÞ \ gij g ð13Þ

where

gij ¼ fxj kx
xi k þ kxj
xk  ðtj
ti
aij Þvij g ð14Þ

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 lo-
cations 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
Geographical Analysis

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:
ðti þ tj
tij
aij Þ
t0 ¼ ð15Þ
2
Similarly, there is an interval ending at tj when pj(t) is small and encompassed by
both fi(t) and gij. This boundary is:
ðti þ tj þ tij þ aij Þ
t 000 ¼ ð16Þ
2
Clearly, ti ot 0 ot 0 and t 00 ot 000 otj ; see Fig. 8. This suggests a strategy for evaluating
Equation (13):
(i) t 2 ðti ; t 0  : Zij ðtÞ ¼ f i ðtÞ;
(ii) t 2 ½t 0 ; t 0  : Zij ðtÞ ¼ fi ðtÞ \ gij ;
(iii) t 2 ½t 0 ; t 00  : Zij ðtÞ ¼ fi ðtÞ \ pj ðtÞ \ gij ¼ gij ðtÞ;
(iv) t 2 ½t 00 ; t 000  : Zij ðtÞ ¼ pj ðtÞ \ gij ;
(v) t 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 . Al-
though Case (iii) involves the intersection of three sets, we know that gij encom-
passes 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) evalu-
ating 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 al-
ternatives (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
Harvey J. Miller Time Geography Measurement

 
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 lo-
cation of this intersection point along the line segment is:
t
ti
xðtÞ ¼ ðxj
xi Þ ¼ aðxj
xi Þ ð17Þ
tj
ti

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 rep-
resent 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 illus-
trates 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 al-
lows 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 conceptual-
ization 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 al-
ways 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 advan-
tages 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 conver-
gence 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 bun-
dling 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
Harvey J. Miller Time Geography Measurement

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 per-
son’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 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.  
(2) [spatial] Both paths are spatially proximal for the interval tB0 ; tB00 :
kP q ðtÞ
P r ðtÞk  d
8t 2 ðt 0 ; t 00 Þ
B B ð21Þ
where d
40 is a user-defined threshold distance; we discuss d
¼ 0 or inter-
sections below. We apply this condition to an open interval since the trajec-
tories can change direction at the boundaries tB0 ; tB00 .
In practice, we can first check for temporal covering and then for spatial prox-
imity 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 discrimina-
tors for bundling. Two paths can meet these conditions but still not share an ac-
tivity: 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 corre-
sponds 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 co-
incidental 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 piece-
wise 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 intersec-
tion cases to consider: (i) path–path; (ii) path–prism; and (iii) prism–prism. In the

36
Harvey J. Miller Time Geography Measurement

tB″

t′B

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 inter-
section 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’ morph-
ologies 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 wheth-
er 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 )
(b)
g ijq pijq (t )
f i (t )
q
g ijq

P r (t )
f i q (t )

P r (t )

(c) f i q (t ) 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 Zklr ðtÞ

tA(tk, t0] tA[t0, t 0 ] tA[t 0 , t00 ] tA[t00 , t00 0 ] tA[t 000 , t1)

tA(ti, t0] f/f/2 f/fg/3 f/g/2 f/pg/3 f/p/2

tA[t0, t 0 ] fg/f/3 fg/fg/4 fg/g/3 fg/pg/4 fg/p/3
tA[t 0 , t00 ] g/f/2 g/fg/3 g/g/2 g/pg/3 g/p/2
tA[t 0 , t 000 ] pg/f/3 pg/fg/4 pg/g/3 pg/pg/4 pg/p/3
tA[t 000 , tj) p/f/2 p/fg/3 p/g/2 p/pg/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 dis-
tinguishing superscripts are suppressed for clarity.

38
Harvey J. Miller 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 inter-
section (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 dimen-
sions we choose n11 of the m sets at a time, stopping with the result that the in-
tersection 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 eval-
uating 6m distances and 4m inequalities.

Conclusion
Time geography is a powerful conceptual framework for understanding spatio-tem-
poral constraints on human activity participation. It is less successful as an analyt-
ical 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 inter-
sections have algebraic solutions in one or two spatial dimensions. Numeric solu-
tions 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 ve-
locity fields, imperfect measurement, and incorporating virtual interaction.

Query design
The analytical definitions in this paper provide a foundation for building compu-
tational 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). Al-
though analytical simplicity does not necessarily imply computational efficiency,
the elegant geometry revealed by temporally disaggregating time geographic en-
tities 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. Nu-
meric strategies are required for some solutions, particularly in three spatial di-
mensions, 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 mem-
ory, 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 two-
dimensional 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 po-
tential 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 net-
work, leaving unresolved gaps in the network roughly approximating the true bor-
der 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
Harvey J. Miller Time Geography Measurement

these techniques is straightforward mathematically; as in query design, the chal-

lenges are computational and software design issues.
Computing the network analogs of the time geographic discs and ellipses re-
quires 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 ra-
dially 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 high-
way 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 ac-
tivity schedules.
Imperfect measurement
Time geographic parameters such as control points and velocities are always meas-
ured 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 ana-
lytical 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 two-
dimensional 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-tem-
poral privacy that can be provided to an individual who still meets minimum ac-
curacy 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 prox-
imity through media such as telephony (see Hägerstrand 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 communi-
cations 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.

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