To Approach Cylindrical Coordinates to Represent
Multivariable Spatio-temporal Data
Phuoc Vinh Tran
University of Information Technology (UIT), Vietnam National University - HCMC
Phuoc.gis@gmail.com; Phuoc.gis@uit.edu.vn
Abstract
Data representing a moving object include the data of time, position, and attributes. The data of positions and attributes of a moving object, which change over
time may be recorded asynchronously because of the difference of sampling
methods. Mathematically, these data may be synchronized over time by spacetime conversions to constitute the data tuples at various time moments. In this
article, we proposed the concept of data plane to represent data according to
each tuple at each time moment. Subsequently, we integrated the data planes into the dimensions of a cylindrical coordinate system to represent the movement
of objects in a space-time cylinder (STCy). In a space-time cylinder, positions
of moving objects are indicated on the data planes which are constituted by the
cylinder axis employed as the cylindrical axis of the cylindrical coordinate system, and the polar vectors of the cylindrical coordinate system. Each data plane
indicates the data of objects at a time moment. The position of a moving object
at a time moment is indicated by its coordinates on the data plane and the time
moment by the angular coordinate of this plane. The attributes of moving objects are represented on data planes as the attribute bars parallel to the cylinder
axis. The space-time path of a moving object surrounds the cylinder axis.
Hence, the space-time cylinder is consistent with the representation of cyclic
movements.
Keywords: space-time cylinder; spatio-temporal data; movement data; visualization.
1
Introduction
Three main components of the real world, object, space, and time are described in the
triad of “what”, “where”, “when” by Peuquet [16],[17], and analyzed further by Andrienko in the triad of “objects”, “locations”, and “times” [1],[2]. These analyses
mentioned the individual characteristics of sets of objects, locations, and times, the
relations between elements of a set and the relations between elements of different
sets. These relations classify objects as spatial objects, temporal objects, spatiotemporal objects, or moving objects according to the relations of objects with locations, objects with times, objects with locations and times, objects with locations,
times, and trajectories, respectively.
The movement of an object is depicted by the continuous change of the position of
the object through space. Proposed by Hargertrand in 1970 [10], the Cartesian coordinate system of three dimensions is employed as a space-time cube to represent the
data of positions of moving objects over time. In the coordinate system, the data of
positions of moving objects are indicated by their coordinates ( x, y) at each time
moment t . The space-time cube has been employed to represent movement data because it visualizes the change of the moving objects’ positions over time. Space-time
paths or temporal trajectories are the curves representing the relations between space
and time of moving objects [2-4],[6],[9-10],[16]. A challenge is how to represent the
attributes of moving objects over time in a space-time cube. Some authors have expanded the space-time cube to represent the attributes of moving objects over time.
The expansions integrated the parallel coordinates into a cube to represent the attributes of moving objects. For unmoving objects, it is possible to represent the positions
and attributes on only one cube [18],[19]. For moving object, it is possible to represent the positions and attributes on two cubes [14], or integrate the positions and attributes on one cube [20],[22].
The main idea of this article is to represent the data of positions and attributes of
moving objects at each time moment on the same plane, called data plane. The methods recording data provide with the data of position and the data of attributes of a
moving object at each time moment [7],[15]. Each data tuple indicates the data of the
positions and the attributes of moving objects at the same time moment. On a data
plane, the positions of moving objects are referred to their coordinates ( x, y) on the
axes of the plane, and the attributes are indicated by the bars parallel to one of the
axes of the plane.
A subsequent idea is to approach the cylindrical coordinates to representing the data planes as a spatio-temporal cylinder. In a cylindrical coordinate system, angular
coordinates indicate the time data of the data tuples, positions on the cylindrical axis
and magnitudes of polar vectors indicate the position data of the data tuples, the bars
parallel to the cylindrical axis indicate the attribute data.
The paper is structured as follows. In the item 2, we briefly present related researches and conceptual framework employed in the article; in the item 3, we propose
the model of data plane to represent the data of objects at a time moment; in the item
4, we approach the cylindrical coordinate system to representing multivariable spatiotemporal data in spatio-temporal cylinders. The modes of cylinders represent data in
different cases. The static mode of cylinder represents the data of moving objects
during the entire movement period. The dynamic mode of the cylinder revives the
activities of objects implicit in data. The hide mode of the cylinder is employed to
stand out the data of movements.
2
Conceptual Framework and Related Works
Movement is the change of the position of an object over time [1][7]. An object of
which existing position changes continuously is called a moving object. The positions
of a moving object are indicated by its coordinates ( x, y) in the 2-D domain of the
observed area. The curve time-ordered connecting the positions of the coordinates
( x, y) where the moving object visited is called the trajectory [6].
The time is indicated on the time axis. Time moments are indicated as points on
the time axis t . Time intervals are indicated as segments on the time axis, from a
point ti to a point t j , where i, j {0,1, 2,..} , symbolized by t . The position of a
moving object is a function (mapping) from time to position: T : t  ( x, y) , or
T (t )  ( x, y) , and the tuple ( x, y, t ) is spatio-temporal data of the moving object. The
curve T  ( x, y, t ) time-ordered connecting the points ( x, y, t ) of a moving object in
the 3-D domain is the space-time path or the temporal trajectory of the object.
Each object has its thematic attributes [2],[7],[14],[15],[20]. The attributes of an
object can change over time. Some attributes of an unmoving object also change over
time (e.g. a gauge station is an unmoving object, the values recorded by the sensors at
the station are attributes changing over time [19]). Meanwhile, some attributes of a
moving object change over time (e.g. a bus is a moving object and its passengers are
an attribute changing on its route; a vehicle is a moving object and its goods is an
attribute changing on its route [20]). Attributes are recorded by different sampling
methods may be synchronized over time by inferring from the temporal trajectory
T  ( x, y, t ) of the moving object.
The data of a moving object include the data of positions and attributes changing
over time [1],[7],[13],[19],[20]. Movement data is a set of multivariable spatiotemporal data of moving objects including data of positions and attributes, which
change over time. The movement data are depicted by a table including several data
records of positions and attributes at various time moments.
3
Data Plane
Fig. 1. The data plane Pi at t i
A movement is a continuous activity over time. However, data of a movement are
recorded discretely at various time moments according to its sampling period. At each
sampling time, the data of positions and attributes of moving objects constitute a tuple
<identifiers, time, positions, attributes>. In the data table of moving objects, each
tuple is represented as a data record (o k , ti , xik , yik , aik .m ) , where ( xik . yik ) is the position
of the object ok at the time moment ti , and aik .m are the attributes a m of ok at ti
In this article, we employ planes of 2-D domain to represent the data tuples at various time moments, called data plane Pi (figure 1). A data plane refers to a plane representing data of positions and attributes of moving objects at a time moment. In a
data plane, the two axes of the plane indicate the positions of objects, the bars parallel
to an axis indicate the attributes of objects. The height of an attribute bar on the data
plane is in proportion to the value of the attribute at the time moment of the data
plane. Accordingly, all data concerning with moving objects at a time moment are
represented on a data plane. In other words, the data of each record on the data table
are converted into a data plane.
4
Space-Time Cylinder for Visualization
4.1
Space-Time Cylinder
Fig. 2. A cylindrical coordinate system to represent multivariable spatio-temporal data
We propose a novel approach to representing visually spatio-temporal data based on
cylindrical coordinates. This approach is called space-time cylinder (figure 2). A cylindrical coordinate system consists of three dimensions: the cylindrical axis, polar
vectors starting at and perpendicular to the cylindrical axis, and angular coordinates
constituted by different polar vectors and the original polar vector. For a space-time
cylinder, the dimensions of a cylindrical coordinate system are assigned to the cylinder as follows. The cylindrical axis is assigned to the axis of the cylinder, the position
coordinates x and y of moving objects are indicated by the magnitudes of polar vectors and the axial positions on the cylindrical axis, and the times t are indicated by
angular coordinates  , where 0    2 .
Fig. 3. Space-time cylinder for representing multivariable spatio-temporal data
The data planes of moving objects in a space-time cylinder are made up by the cylindrical axis and polar vectors. The time moments of the data planes are in proportion to
angular coordinates  of the polar vectors. Each position of the data plane Pi at ti is
determined by an angle  i formed by the plane P0 at t0 and the plane Pi at ti . The
attribute bars on the data planes make up the surfaces of the cylinder. Accordingly,
each data plane in a space-time cylinder represents the positions and attributes of
moving objects at a time moment. In other words, each data plane represents all data
of a record of the data table. The curve time-ordered connecting the positions of a
moving object on the data planes is the temporal trajectory or the space-time path T
of the object. The temporal trajectories of moving objects surround the cylinder axis
(figure 3).
We considered that several movements are cyclic, a moving object departs from a
place to visit one or many places and turn back the departure place (e.g. buses depart
from their departure station to visit several bus stops to pick up and drop out their
passengers and return departure station, workers leave their home in the morning for
their offices and come back home in the evening). In a space-time cylinder, the temporal trajectory of a moving object is a curve time-ordered connecting the object positions on data planes. For a cyclic movement, the ending position of the route on the
plane of ti  2 fits in with its departure position on the first data plane of ti  0 .
Accordingly, the space-time cylinder is consistent with the representation of the multivariate data of cyclic movements.
4.2
Modes of Space-Time Cylinder for Data Geo-visualization
Fig. 4. The dynamic mode of a space-time cylinder
Mode of Static Visualization. For the static mode of a space-time cylinder, all data
planes of moving objects at all time moments are displayed (figure 3). In other words,
all data of the table are shown completely. To represent an available data table with
space-time cylinder, the data planes are designed so that the number of data planes is
equal to the number of data records of the table. Each data plane represents all data
fields of one record on the data table. The angular coordinate  of each data plane is
in proportion to the time moment of the record. The positions of data planes in the
cylinder are determined by their angular coordinates  , which are so calculated that
the entire movement period of moving objects fits in with 2 , the maximum of the
angle  .
Mode of Dynamic Visualization. In the dynamic mode of a space-time cylinder,
each data plane of moving objects at a time moment is shown one after another in
time line (figure 4). A cursor moves slowly with the automatic or manual control on a
time axis to display data planes. When the cursor moves from starting time to ending
time of the time axis, each data plane is shown each time the cursor reaches a time
point of the plane. On the contrary, when the cursor moves from ending time to starting time of the time axis, each data plane is hide each time the cursor reaches a time
point of the plane. In the dynamic mode, the data plane at t0 rotates around the cylinder axis each time the cursor moves from a time moment to another, the data plane
corresponding to the time moment of the cursor is always shown at the position perpendicular to the user’s view.
Mode of Hide Visualization. The hide mode of space-time cylinder is applied for the
cases of overcrowded data on screen. The goal of the hide mode is to only visualize
the data necessary for users. We consider that there are a lot of spatial data displaying
repeatedly on all data planes. Data of geographic area and frames of data planes are
shown on all data planes of the cylinder. In many cases, they are not really necessary
to be displayed on all data planes. Only spatial data different from the last plane
should be shown on each data plane. When the hide mode of a space-time cylinder is
turned on, the repeated data on data planes of i  0 are filtered and only the positions
and attributes of moving objects are displayed on data planes of i  0 .
5
Conclusion
In this article, we proposed the approach of the concept of data planes to represent
visually the data of positions and attributes of moving objects at different time moments. Movement data including the data of positions and attributes of moving objects are recorded discretely at various time moments. Each data tuple of the movement at a time moment is recorded as a record on a data table. Each data record, including data of positions and attributes, is represented on a data plane. We also proposed to employ cylindrical coordinates to represent the data of moving objects by
arranging the data planes around the axis of a cylinder, where the angular coordinates
of the data planes are in proportion to their times. The space-time cylinder is consistent with the representation of the multivariate data of cyclic movements.
Acknowledgements
I am very grateful to the Advanced Program of the University of Information Technology (UIT), Vietnam National University – HCMC, for its valuable grant to create
this article.
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