Statement of Interest

The structure of space in the world, the structure represented in mental models of space, systems of spatial concepts, their integration in spatial terms in natural language, and the formalization of these structures and concepts constitute my main research interest. This interest has led to defining a project on the formalization of spatial concepts by means of the axiomatic method together with Christopher Habel. The project is founded by the DFG (German SF) in the priority program on Spatial Cognition since 1996. In this project we work on questions concerning spatial frames of reference, shapes and spatial dynamics. In this context we developed formalisms for describing curves as shapes of objects and for oriented curves as trajectories of objects.

The context of the project also integrates my work on my habilitation thesis, which I expect to complete in 1999. This work includes a systematic approach on kinds of motion and the parameters defining them. This work is meant not only to look at the motion of material entities with respect to space occupation but also at other kinds of motion like the change of habitation. The interest on this topic was inspired by my participation in the GISDATA specialist meeting on 'spatio-temporal change in socio-economic units' in 1996, organized by A. Frank, J. Raper and J.-P. Cheylan. Thus, I hope to contribute also on the questions of which kinds of changes we ascribe to non-material entities. The research abstract gives an impression on my work on this topic.

What I want to contribute to the meeting

• classification of spatial changes and our categorization of spatial changes,
• preconditions for assigning certain kinds of motion to objects with different kinds of location,
• formalization of trajectories as integrating spatial and temporal order.

• a better grasp of application aspects of conceptual models of spatial dynamics in geographic contexts,
• discussions about different kinds of spatial dynamics,
• information about the contrast between and when to use 'movement' and 'motion'.

1993	Dissertation at the University of Hamburg, Germany. Department for Informatics. Title of the thesis: Struktur- und Quantitätsbezug. Zähl- und Maßangaben in Wissens- und Sprachverarbeitung (Reference to structure and quantity. Expressions of counting and measuring in knowledge and language processing)
1992 –1994	Doctoral student in the doctoral program on Cognitive Science of the University of Hamburg, Germany
1982 –1988	Studies at the University of Hamburg: Major: Informatik (Computer Science); Minor: Mathematics; Diploma 1988.

Profession

1. On Change and Motion

The conception of time and of temporal structure are strongly influenced by our perception and conception of changes in the world. If the world did not change, there would not be a reason to distinguish and talk about different times. Changes in and to the world can be conceived of as discrete or continuous. The discrete conception of change focuses on the difference between the situation before and the situation after the change or the boundaries of the change. The continuous conception of change focuses on the course of change, the development of the situation while the change goes on, and the internal structure of the change. I will call the discrete conception the event view and the continuous conception the process view of change. The event view of motion, for example, relates the starting position and the final position of the moving object. Direction and velocity of movement can in this view only be described on a global level, relating starting position and final position and the time of its beginning and its end. In contrast, the process view emphasizes the path or trajectory and the process of change along it. Direction and velocity can be ascribed locally with respect to any sub-period of the time of the process.

As a basic characterization of the event view of simple change I take that a proposition is true at one time and false at another time. More complex cases of change involve a periodical re-establishment of states.

In the informal uses of the term 'continuity', developments are said to be continuous if they are smooth, which can be specified by: no interruptions are noticeable (they do not stop in between) and no jumps are noticeable (there is some constancy to the change). This informal characterization allows two uses that are observable: one that allows for periods of constancy in the underlying structure (e.g., continuity in leadership) and one that asks for a constancy in change (continuity in progress).

In the event view, motion can be characterized by the contrast of a starting position and a final position. Discrete motion can be a jump between these two positions not involving any position in between. An example of this kind of motion is the change of habitation.

The process view of motion needs to refer to more places than just the starting position and final position. One reading of continuity (monotonicity in change or constancy in development) can be characterized based on a binary relation. It expresses the condition that places traced in a given order by a moving object are correspondingly related.

For a more specific account to continuity of motion we must be able to specify that immediate 'jumps' in space are not noticeable. That is: small changes in time result in small spatial changes. The main problem for giving a more systematic account is to define a measure for difference of regions and the topology for spatial traces of extended bodies (cf. Galton, 1997)

Another problem of continuity can be observed if we consider the material bodies that move through space not only as wholes but also regarding their part-structure. The continuity in body-movement or the rotation of a ball also includes that the parts of the whole change their positions relative to each other in a smooth manner. My left arm and my right arm do not immediately change their relative positions and neighboring segments of the ball stay neighbors.

2. Characteristics and Classes of Motion Based on Spatial Structure

One characteristic class of spatial changes of material bodies is the movement through space along a way, path or trajectory. Our movement from home to work or back again is easily considered as such motion, but also the movement of the earth around the sun or the movement of the moon around the earth, i.e., rotation around a center outside the body. In contrast to this, bodily movements like waving my arms or turning my head and rotations around inner centers like the earth's rotation around its center are not as clearly motions along trajectories.

Although our examples of movements along trajectories involve the other kind of spatial change, it seems that we can easily conceptualize the spatial change a material body undergoes as purely trajectory-based. This is, e.g., also reflected in the coding of spatial change in natural language, where differences between verbs that encode trajectory-based movement and modes of (internal or bodily) movement can be observed.

Trajectories of moving objects are generally conceived as linearly structured. Movements through space of individual bodies therefore provide diverse options to linearize space. Accordingly, the individual positions the material bodies assume in the course of motion are points in space. This basically means that the internal spatial structure, the shape and spatial extension of the body are considered irrelevant in the conceptualization of its movement through space. Since points do not have spatially distinguished parts, point-like objects can only move by changing their position completely. Thus, if other kinds of motion are accounted for, then the objects have to be conceptualized as spatially structured or extended. Motions that are not conceptualized as trajectory-based like body movement and rotation around an internal center always take the existence of an internal spatial structure of the objects into account.

Considering the trajectory and the motion along it means to consider the process view of spatial change. Additionally, this kind of movement is paradigmatic for our idea of continuous movement. Trajectories are spatially embedded linear structures and movement along trajectories is continuous if the trajectory is spatially continuous and the movement along it respects its spatial order (cf. Eschenbach, Habel & Kulik, 1998).

In contrast to trajectory-based motion, the main characteristic for pure internal motion is that the positions occupied by the body in the course of the motion do overlap. Most characteristic are movements in which a certain spatial region is occupied throughout the process. The spatial change responsible for our categorizing the process as motion in these cases are changes of the positions of parts of the objects (that can exhibit trajectories) and—connected to this—changes of the orientation of internally constituted spatial reference frames.

In the group of internal motion I want to include movements that cannot be characterized as change of location of the whole body. It shall include rotations of disks and spheres as well as the motion of fluids in a closed system or bodily movement. Therefore I do not postulate that internal motion is motion of the whole entity but just that it involves the motion of some part of the object.

The class of movements that keep some parts of space constant can be further subdivided. The classes of internal motion we discuss in some detail are growth, shrinkage, internal rotation, the movement of parts and short movements that can constitute trajectory movements.

Growth and shrinkage of entities can be defined by their monotone gain or loss of space of occupation throughout time. Growth is internal motion that results in the gain of space and is monotone in that it never loses any space in the process. Shrinkage is—accordingly—internal motion that results in the loss of space and shrinkage is monotone since the object does not gain any space in the process.

While growth and shrinkage imply that the region occupied in the beginning and in the end differ, some internal movements do not. Examples of such movements are perfect internal rotation of rigid symmetric bodies and the motion of fluids or gases in closed systems. In both cases, most of the parts of the object change place, but some parts—those that are symmetric with respect to the center of rotation—do not change their position. Still, all extended parts have some part that moves. Thus, internal rotation is characterized as internal motion of a body such that all extended parts include a moving part. Consequently, although all parts are in (internal) motion, stable centers can exist. This characterization only describes a relation between beginning and end. For the process view, the correspondence of trajectories of smaller parts of the rotating object can be considered.

The third class of internal motion to be discussed includes body movements and, in general, movements of parts of an object while other parts do not move. Bodily movements consist of movement of parts of the body relative to other parts. That means that those parts change their location while the other parts stay at a location. Parts of a body may change their location in different ways. We will have to consider the cases of parts that move along trajectories (as my finger tip when I wave my arm) and parts that move by rotating around its joint with the body (as my complete arm in this movement). This is taken care of by the formulation of, since it does not specify the character of motion of the moving part.

It is worth noting that partial growth and partial shrinkage is both, growth and partial motion or shrinkage and partial motion, respectively. Thus, the classes given here are not exclusive. Since change of form of objects needs to involve change of object parts in addition to the boundary or surface of the object, change of form is here subsumed under partial motion.

The last class of internal movements to be characterized is the class of short movements of larger rigid bodies that seem to be trajectory based but are too short to result in a complete shift of position of the larger body. In this case, all parts of the object that are small enough exhibit a trajectory based motion, thus, all parts have parts with a trajectory. This kind of internal movements cannot have stable centers.

There are mainly three connections between the two distinguished kinds of movement to be considered in the following. First, several movements are conceived of as combinations of internal movement and trajectory-based movement, like someone's walking or the rolling of a ball down a hill in contrast to a stones sliding. Second, parts of objects may exhibit trajectory based motion, while the body as a whole is moving internally. Thus, internal motion at a coarser grain of space can at a finer grain of space be trajectory-based motion. Third, internal motions in shorter time spans can be combined to trajectory-based motion over longer time spans.

3. References

Eschenbach, Carola, Christopher Habel & Lars Kulik (1998). Representing simple trajectories as oriented curves. Manuscript.

Galton, Antony (1997). Space, time, and movement. In O. Stock (ed.), Spatial and Temporal Reasoning(pp. 321–352.) Dordrecht: Kluwer.