3D Navigation in architectural space

The use of Voronoi diagram for mapping information flows


     Mohammed Raafat Saleh MSc BSc, Christian Derix MSc DiplArch, Paul Coates AA Dipl

Center for Evolutionary Computing in Architecture (CECA)

School of Architecture and Visual Arts

University of East London



The spatial environment is regarded as a system that has its structure and configuration. The phenomena or manifestations of this system are present all over the space in the form of visual information bits or can also be referred to as ‘knowledge in world’; implicitly embedded in the elements of the system. Therefore, once our environment changes, the available knowledge and thus the phenomena also change. For Raubal and Worboys [Raubal and Worboys 1999], the knowledge in the world is present in many forms and on many cognitive levels. At a lowest level of awareness this knowledge can be regarded as being in the overall configuration and underlying structure of the spatial system.


We used the notion of ‘way-finding’ in attempt to uncover the phenomenal manifestations of an architectural space and so we could define it. The notion could show the underlying structure of the space throughout the people's behavior and navigation within this space, Way-finding is known as a combination of cognitive abilities with which spatial characteristics may be recognized and organized into coherent patterns. These abilities are collecting spatial information, information processing and decision making, as well as manipulating the space. We believe that these abilities require a spatial mapping system that quantifies the spatial aspects and represents its configuration.


We constructed an analytical software tool to study the impact of spatial configurations on people’s abilities to navigate and way-find. The structure of the analytical tool is based on computational geomtries that are  referred to as Voronoi diagrams and its dual concept Delaunay triangulation. These diagrams are generally regarded as structural  tessellations,  which define the optimum boundaries between elements in space (be that  on an urban, architectural or engineering scale). Here, the Voronoi has been applied as a representational  network that maps architectural space in three dimensions and simultaneously evolves an analytical structure of the intelligibility of the space it is embedded in.

The mechanism of this evolution depends on running the three mentioned way-finding abilities sequentially. The nodes of the voronoi diagram are a metaphorical representation of spatial sensory points that carry visual information about the environment, whereas its edges (a line connecting two nodes) represent the paths which the nodes employ as navigational routes towards a target location. After the routes are chosen, the whole structure reorganises itself and adjusts its geometric elements’ locations on the basis of the taken decisions and the weight values that are assigned to each node, indicating the number of times this node has been used. The adjusting stage in turn feeds back the new locations of the elements to the decision making stage before the sequence loops.


Eventually, the Voronoi diagram evolves into an emergent route pattern of paths chosen and omitted to navigate from a a start point to a target point within a given space. Comparing various evolved patterns – patterns vary by chosing different emitters and targets, or chaning the given space – we were able to understand some idiosyncrasies of the space the network was embedded in and assess their impacts on the process of way-finding.


The paper starts at section [1] by introducing parts of the body of literature concerning space cognition, cognitive maps and their relation to people’s way-finding. Section [2] will describe Passini’s definition of way-finding and some of his rules considered when selecting the next step. Section[3] shows the importance of defining the space via a visual representation and its relation to our spatial understanding and cognition. Section[4] briefly explain Voronoi diagrams and their dual concept Delaunay triangulation. The reasons for commonly using it as a visualising system will be explained in section[5]. The mechanical processes and the used algorithms of each of the way-finding abilities for this project are outlined in section[6]. Section[7] will fully describe one of the experiments that have been performed to test the feasibility of the tool in reading a 3D space. Finally, chapter [8] will conclude the paper and suggest further methods for additional analysis and space readings.


1.Space cognition and Navigation:


What are the cognitive mechanisms involved in understanding the spatial relationships between ourselves and other objects in the world?


The term ‘spatial cognition’ is used to describe those processes controlling behavior that must be directed to particular location, as well as those responses that depend on location or spatial arrangement of stimuli. And ‘Navigation’ refers to the process of strategic route planning and wayfinding, where ‘wayfinding’ is defined as dynamic step-by-step decision-making process which required negotiating a path to destination [Golledge].As a spatial behavior, negotiation demands a cognitive representation that distinguishes one place or spatial arrangement of stimuli from another. This representation is referred as cognitive map.


A Cognitive map is a structure which is an internal representation of an environment which its inhabitants use as a reference when navigating to a destination [Passini 1992].It is proposed that cognitive maps fall into two types. First linear or sequential maps which are based on movement through the space and sequential images of route and what is observed in the journey. Second, Spatial maps that don’t require this reference movement through the imagined space. Normally person’s cognitive map changes over time. When we are new to an area our cognitive map will be linear but this will usually evolve to become increasingly spatial [Piaget and Inhelder].


2.Wayfinding as behavioral abilities:


If ‘spatial cognition’ is defined as the process that governs people’s behaviour in a space, then it is vital to understand what is already known about this behavior; the act of what has come to be termed as ‘wayfinding’.Wayfinding has always been an interesting topic to researchers to investigate the relationship between space and its occupiers and also  an  important issue to professions like urban, interior or lighting practices in order to reduce the probability of problem occurrence due to poor way finding systems in their sites. Such problems could be: Retail sites lose business, visitors to museums miss exhibits, and patients arrive late and stressed at hospitals. Getting lost is a negative experience that affects people’s opinion of their organizations.


Among the different definitions of ‘wayfinding’, I found Passini’s definition is the most relevant to this paper. In his paper [Aurther1992] defined ‘finding a way’ as a process of spatial problem solving. It is regarded as a combination of cognitive and behavioral abilities such as processing, understanding and manipulating space. To be more precise, the abilities can be classified into  i) decision making ii) decision executing iii) information processing [Passini 1984:1998].People tend to base their decisions in wayfinding tasks upon the cognised information about their environment. For instance if they can see their target location certain criteria are used in choosing their route, whilst different ones are used if they can’t see it. Some of these criteria are:-

1.The shortest path towards the seen target location [Peponis].

2.Selection of longest-leg first[Peponis]

3.Avoidance of unnecessary back regression [Peponis].

4.Routes with minimum angle between itself and the target.

5.Routes with maximum capacities (for example main roads)


Then in order to reach a destination, decisions have to be transferred into actions. According to [Arthur and Passini, 1992] the execution of a decision involves the matching of a mental representation (cognitive map) of the environment with the real environment itself. Both decision making and execution are supported by information processing .Spatial perception (related to the process of acquiring knowledge from the environment) and spatial reasoning (related to the manipulation of spatial information) constitutes the main interrelated components [Passini].


3. Metric and Non metric Visual representation


Since the availability of relevant information about the environment is an important factor for decision making ,’wayfinding’ researchers regard the idea of visual mapping that could represents and quantifies aspects of the built environment is an essential knowledge instrument for supporting the processing of this information. In other words, the above mentioned behaviour abilities that are carried in the domain of cognitive representation is most understood by focusing on a visual mapping (representation) of the real environment.Unfortunately, the majority of the researchers who are concerned with this idea assume that the architectural space is only a Euclidean 2D space and thus they base their spatial experiments upon metric maps such as 2D route and tube maps. But does our understanding and cognition to the occupied space is a metric understanding? In fact we can argue that space is metric at least when we experience it on the local scale but,’ when we consider the way we know or understand configuration of space –let us call this cognitive space we seem to be faced with conundrum ‘[Alan Penn 2000].Nerlich in 1994 mentioned that there are four axioms that should be satisfied in the Euclidean 3d or 2d metric space. One of them that distance from x to y must be the same as the distance from y to x.Cognitively the distance of the route in one direction appears different in the opposite direction and thus cognitive space breaks up this axiom and appears not to be metric.


Many researchers have attempted to understand how the cognitive space is constructed and how it influences our decisions when navigation. Bill Hiller for example pointed out that the occupier knows about a space (whether urban or building interior space) via the difference in the succession of his visibility field when navigating this space. Upon this observation, he developed ‘axial line sight’ structure that has been used in many of the space syntax researches. The structure has based upon extending lines (fewest and longest) that are tangent to the vertices of blocks until they intersect and so could cover a  more global view of the spatial elements in order to explore the full limits of visibility and permeability within  an urban space.[Hiller and Penn at al,1998].


Another researcher is Conroy Dalton. She adopted the concept of analyisng the underlying structure of an architectural space using the traces of moving people in a virtual environment (VE).The traces were regarded as bottom-up spatial representation that helped her in noticing ‘Conservation of linearity’.Basically, the conservation of linearity leads a person to choose one route from A to B (starting point to a target point) and quite different route back from B to A. She also showed that this ‘asymmetry’ also holds if one chooses next step on basis of heading towards a destination; people tend to have good sense of direction and can compute the route with minimum angle heading from amongst those on offer.


Conroy and Bill Hillier concluded that people’s cognitive understanding to the space doesn’t follow the Euclidean metric space but a more ‘fundamental and rather simpler as topological or pre-topological space’. This might imply that we need to look for a different representing tool that essentially  respects the characteristic aspects of an architectural space and our cognitive understanding to there aspects.










Route from start to target could be different route back.


‘Axial line sight’ of alpha world





4. Voronoi diagram as a Visualizing system:


In order to describe a space we need to extract the characteristic aspects (phenomena) of this space. This might happen via analyzing people’s abilities in cognizing and manipulating the space. These abilities as said could be facilitated if we constructed a visual mapping system that manifest the space characteristics and structure in terms of a coherent topological patterns like the representing systems that have been used by Hillier; line sight network and Conroy; people’s traces.


In this paper, the visual representation that has been used, is based on mathematical computational geomtries that are  referred to as Voronoi diagram and its dual concept Deluanay triangulation.They were first discussed by a mathmatecian called Petter-Lejeune-Dirichle in 1850 then the diagrams were written in 1920 by Voronoi as a dual concept,hence the name Voronoi diagram.The  geometries could describe some of the aggreagting superstructures that are found in nature such as Crystaline and soap bubble aggreagtes.The diagrams have been always regarded as tesselating structures that represent proximity information about a set of objects or sites;it partitiones a space into subregions called voronoi cells and assign every space to its nearest subregion.


Unfortunately, the majority of Voronoi/Delaunay spatial applications have only concerned with their mathmatical metric and optimum geometric properties which  gave a very limited view about the vononoi potentials. Therefore we decided to explore more of the voronoi spatial properties by taking its geometric structure and use it in more abstract sense.In the experiments that we did,the voronoi was used as a visualising network instead of diagrams.











Generally in order to construct a voronoi diagram,the ‘voronoi cells’ or subregions require starting points which could be refered to  as the ‘nuclei’.By following Voronoi’s algorithm[mentioned in Dwyer’s paper 1991] and Convex hull[G.L.Miller,2002] in constructing the voronoi geometric system,an emerged pattern of polyhedra will be formed upon the basis of the distributed intial nuclei.The pattern is made out of intersecting planes,where each plane is bisecting the distance between 2 nuclei.These intersections form partitions that define the 3D convex polyhedra or 2D ploygons.Moreover the shape of the planes could vary from 3 edged planes(triangles) to multi edges(polygons).The variation depend on the pressure differences that are caused due to the  nuclei positions and their relation to their neighbors’s positions.In other words,if the locations are at the same distances,a regular polyhedron voronoi cell is produced with regular partitions..A very close analogy is  the patterns that are formed due to soap bubbles interaction.


Voronoi polyhedron due to equal influences of nuclei in 3D space.















5. Why Voronoi Diagram ?


1.In a real world, a spatial system such as a building space is covered with endless number of   sensory points that carry visual information about the objects in this space. From one sensory point we can see one or more of these objects and from another we can see different objects. , but there are points from which we can see the most of the object manifestations. The visual information that we collect from these points is important to navigate or search for our target. The voronoi can give these maximum points in the space. It has the ability of tessellating space by dividing it into convex sub regions (convex hull polyhedra in three dimensions).These sub regions are defined by bunch of intersecting points that constitute the partitioned boundaries of these volumes. The points could be shared by other cells and hence one point could be a strategic point from which we can see all the volumes that belong to this point. These points carry maximum information of the volumes relevant to them.


2.The positions of the sub regions are a result of relative computations to their nuclei positions with respect to their neighboring nuclei. The correlations between the volume structure aggregations produce the entire skeleton of the system. These computations (based on natural computation that emerge systems like crystals) could free us from the boundaries of the top-down Cartesian geometries and shapes that define our physical space volumes (cube or sphere).This free spatial structure suits the idea the phenomenal space which we form in our metal image has no Euclidean character.


3.The Voronoi diagram has both topographical and topological connecting structure. Therefore as a representing system it respects the topological concept of either our space cognition or spaces connectivity. Since our   perception to space is not ‘already’ constructed but due to human gradual construction via a step by step decision making and decision execution during our interaction with this space. Also, the space that we live as Bill Hillier has pointed out is organized in a very topological sense [space is the machine]. For example we can’t move from one space to next door just because its topographical proximity until there is a door in-between, but if this door doesn’t exist, then  we should pass through a mediating space (corridor) between the two spaces  following the topological proximity this time.


4.The voronoi give the opportunity to visualize a spatial system in three dimensions rather than two dimensions. In many wayfinding and space studies, for instance the studies that have been done by [Bill Hillier & Alan Penn 1998] although their analysis was intense and profound, it was incomplete as their focus was on two rather than the three dimensions spatial analysis. In fact people when they choose their next step, they choose it on the basis of what they perceive in 3D space not 2D. A two dimensional analysis could be more applicable to a building where all its elements are of same height, in this case we might assume that people are moving in 2D movement ,as all destinations will be seen at the same level of their field of vision, whereas if it contains a mezzanine for example and they can see some of their targets  at the mezzanine level(different level), their decisions criterion ill be based on  an extended Two-level vision instead of one level only, as they can see both spaces: the mezzanine and ground level space movement. Therefore it is essential to apply wayfinding experiments in a 3D domain in order to have a more global understanding of the people’s real perception to the space configuration and its influence on their behavior.














Voronoi tessellation of a 3D virtual building


A convex polyhedron voronoi polyhedron from inside





6. The mechanical process of the network:


By using a voronoi network as a representing (mapping) system, we worked on developing an analytical software tool that is capable to define an architectural space via analyzing the legibility of this space. In other words, it shows if the configuration of a spatial system could ease the navigation and finding the targets of its occupiers or not. The tool mainly interrogates the influence of the system’s elements on the people’s three common behaviour abilities: Collecting spatial information; availability to see space location, Information processing and decision making then manipulating the space.


Now I will describe each of the network mechanical processes of three abilities .First, we built a 3D virtual building model in which the reading function of the software tool can be tested. The mechanical processes of these abilities are carried out sequentially and by using AutoCAD it is graphically visualized in order to facilitate our analyzing observations to the tool’s outcomes.


6.1 Information collection ability


1.A set of starting space points (Voronoi nuclei) have been located in the middle of each space in the building. Every nucleus identifies the space and by using Voronoi’s algorithms, a voronoi network system is constructed where each volume cell in the system defines a space volume in the building. The nuclei have been visualized as cone shape objects.

2.Wayfinding is to choose a certain target and try to find the way towards it. Therefore, one or more targets locations can be chosen by the user of the software. Once they are specified their color changes to distinguish between them and the other nuclei points.

3.Then each node in the network collects its information about this target by checking its visibility to it. For that A ray trace algorithm has been used by letting each node send a ray towards the target and see if it’s blocked or not.


Voronoi Network in a virtual building model











6.2 Information processing and decision making


In a real world, the person chooses topologically his next position either towards a seen target or another location with a possible view to the target. The first choice is to choose as the next  visual sensory point in the space from which he can either have the best vision of the target(as he approaches it for example) or hope to have this vision. Accordingly, we let the network nodes (which are regarded as abstract representations to the sensory points) to decide their next nodes by themselves in order to find their way towards the target location. The wayfinding rules which were mentioned in section 2 have been used as the nodes’ decision making criteria.




1.Normally way finders start their searching journey from a starting space (ex: Building entrance).Again the user can choose the starting spaces by choosing the required voronoi starting space point. They are referred to as ‘emitter nuclei’, once they are chosen their color changes to green. In the beginning of the program, only the emitter nodes (i.e. voronoi nodes that are related to the emitter nucleus) are the nodes that have a privilege to choose their next node.

2.The emitter chooses its next nodes among its connected and only seen network nodes (The mathematical ray trace algorithm was used to check the nodes vision to each other). Now the chosen node has the privilege to choose its next neighbor node. These decisions are taken in a procedural and linear way that is similar to the way finder’s linear decision making during navigation in the real world. Furthrmore, we choose a path or route to move from one place to another, likewise the network nodes decide its next point step and the connected edge as the required route for this step.

3.We assigned a weight value to each node, bywhich it could know if it is capable to make a decision or not (only if weight > 0) and indicates the no of times each node has been chosen. In other words the more a node is chosen the more its weight increases and vise versa. The nodes with weights > 0 are called ‘active nodes’, the ones less than or equal zero is called ‘passive nodes’.

4.Weights weren’t   only assigned to the nodes but also to the used or chosen routes which lead to those nodes. The more a route is used the more its flux increase (weight).For visual observation when the weight values of either the nodes or edges increase, their colours and radii increase as well.









The emitter is modeled as a green cone and its space volume is defined by four nodes (green nodes).


Indications of the network nodes and their initial weight values.








1.The wayfinding criteria that have been used to decide the next node were classified to two types:

Target rule: If ‘active’ nodes (weights > 0) can see the target, they search among their connected routes for the route with minimum heading between itself and ‘target’ location.


















Non-Target rules: If ‘active’ nodes can’t see their target, finding the ‘optimum route’ towards the node with highest probability of seeing the target will be their aim.

This time the rules upon which they decide their next network nodes and edges are:

l        Searching for nodes with minimum distance.

l        Searching for nodes with highest no of connections which means more       routes connected.

l        Searching for routes with highest capacity (flux) and this means that many people have used it. People tend to follow the flow of others when they can’t determine their targets.


By using a three dimensional vector where each criterion represents dimension, the node with highest 3D vector magnitude is the one with highest probability








Distance 2 > distance 1





6.3 Decision execution and manipulating the space


The Voronoi system is divided to two parts: a network that represents or defines the whole building in three polar dimensions and space points (voronoi nuclei), upon which the network is constructed, represent each of the building spaces, respecting the Euclidean characteristics of the space. From a different view, the space points can be assumed to be people in different locations in the building and their movement are affected by the physical spatial boundaries (gravity and differences between the building stories).This could give the opportunity to find out if any person could reach the specified target location from any place in the building or not and hence could evaluate the degree of its intelligibility.




l        Every node in the network checks its relevant voronoi nucleus (space point) and feedback its weight value to it. The nucleus in turn adds up the value to the other values that are received by the rest of its voronoi nodes.

l        Then all nuclei simultaneously compare their sum weights with their network neighbors’ weights. Each nucleus or space point assimilates other points with higher weight values. Basically, the weight values indicate the number of usages of the relevant nodes and so the amount of information that these nodes know about the target location.Therefore, if one nucleus has low value means low chance to find the target location, but by assimilating the ones with high values it might has a better chance for finding its target. This mechanism is referred to as ‘Hill climbing mechanism’.


The emerged pattern from this mechanism can be described in terms of the different densities of the space points and directionality of these densties.In other words if the space is intelligible enough, most of the space points will aggregate towards the target(s) locations(high density),else they will scatter.




Casella di testo: Voronoi nodes (n) feedback their reading weights to their relevant space points (T)










6.4 Recursion and learning mechanism


Piaget mentioned that the cognition of space is a recursive human gradual construction and not given a priori [Piaget and Inhelder].Also Penn and Turner pointed out that the cognitive space required to support the representation of the more global understanding of configuration based on same form of learning form the experience For example, we all take the same route everyday when we go to work, school or club. Every time we go through it we see the same elements (houses, street posters, light posts...etc) and therefore every time we compute the same information .Hence, the more we learn about the knowledge that manifests the route, the  more this knowledge becomes apparent and comprehensive. This is referred as a leaning mechanism.




l        On the basis of the recursive and learning concept, the three behaviour mechanisms(information collection, processing and execution) are carried out recursively for number of times that we referred to as ‘generation period’. During each generation either both the same nodes and routes are chosen or different ones. When the same node/route is chosen their radius increase and become bigger; indicating that same amount of information is used and hence emphasized.Ofcourse, these increasing values are sent to the voronoi nuclei. So if the same nodes are chosen the space points (nuclei) move in the same direction, else they move in diverting directions.

l        The recursion give time for the network to complete its decision making process and either find the target(s) locations or loose its way (the routes that are used might drift of the target).

7.  Experiment


Casella di testo: We used the voronoi tool in showing the legibility of a three dimensional model with a simple configuration. The model contains two main nuclei: one emitter and one target nucleus. The emitter (which is located at the ground level)with its voronoi nodes can easily see the target location (at the 1rst floor).The aim is to see if the configuration of the model will allow the rest of space nuclei; which represent each space in the model, to find the target then   move towards it or not.
Although, the software tool was running in a three dimensional space, the virtual building is shown as wire frame format for simplification.









Above figure shows the virtual

model and space nuclei of cone shape bodies. The emitter is a green cone, target is yellow and the rest are red.






















In a procedural and linear way, the optimum routes from the emitter to the target are chosen




At t=1 the emitter nodes (green dots) start choosing their initial routes(red bars) towards the target location(yellow dot).Since the green dots could easily see the target-with no obstacles-the decision making criterion used was ‘target rule’[see sub-section 6.2].The chosen routes were of  minimum angle between itself and the yellow dot. At t=3 more red bars started to form, which means more nodes with high ability to collect information about the target were chosen. Eventually, the chosen routes started tended to become straighter and hence started to define the optimum directionality towards the target location










Every time the network adjusts their locations according to their weight values, different representation of the space is produced.






In the beginning, the voronoi network was constructed according to Voronoi/Delauany algorithms. The nodes locations were based on the building spaces locations and their configuration, giving an initial representation to this building. As explained all the nodes except the emitter nodes were with weight values equal zero. At t=1 each node started to check its privilege to start the decision making algorithm and its vision to the yellow dot location. As soon as the first route and nodes were chosen, their values incremented. After each choice, each node used ‘hill climbing’ method to adjust its locations by moving towards its neighbour nodes with high weight values.


Consequently, the global voronoi system evolved, giving a different representation and hence different routes decision were taken. As it is shown in the above figures, the network position has ended at the central region of the building where the nodes can see the most of the target location and so can be chosen and increase their values .Each step could be regarded as learning step that emphasized the decisions that were taken from the step before.









Due to the network evolution, the space points (nuclei) either find their way or get lost.



Casella di testo: The space points were distributed in the virtual model’s spaces (rooms). The densities of these points were roughly uniform .Slowly this uniformity started to change and the points started to become denser at the central region. As mentioned in sub-section 6.3, the network nodes feedback their values to their relevant space nuclei. According to these values, they moved towards the place with the highest weights. The space points were influenced by the chosen routes and nodes directionality and network evolution. As a result the space points formed a homogenous agglomeration along the invisible line that connects the emitter location with the target location.


8. Conclusion:


l        The spatial environment is regarded as a system that has its structure and configuration. The phenomena or characteristics of this system are present all over the space in the form of visual information bits, which can also be referred to as ‘knowledge in world’ that are implicitly embedded in the elements of the system.

l        We can know about an architectural space (for example building interior) and even compare it with others by understanding the spatial relationship between ourselves and the characteristics of this space. In other words by understanding the influence of the space on our behaviour. Wayfinding is a spatial issue that can explain this relationship.

l        The way-finding process involves three main cognitive and behavioral abilities such as processing the information or knowledge that is found in the space, which itself is a main factor for fulfilling the other two abilities: decision making and execution when navigating and finding a target in an architectural space.

l        The Voronoi diagram as a computational spatial geometry could be used to represent a space through its structure. It is used as a knowledge network that maps and quantifies the aspects of a built environment for supporting our information processing within this environment.

l        The main endeavor by writing  an analytical software tool is to visualize the three wayfinding behaviour abilities and study the effect of the configuration of any space on them. As a first step, the project is intended as a starting point to analyze and study the legibility of the structure of the space. It demonstrates how comprehensive it is to explain its manifestations and gives us precise information about them to act in an appropriate way. For further research we might work on a way to improve the structure of this space via may be feeding back analysis results of the project to ‘spatial swarm agents’ or by applying natural fitness functions that could be compared with the project results in order to modify the spatial configuration by using genetic algorithm.

l        As a computer aid technique we can use it creatively in architectural design, bringing in much new knowledge about space and function as we do so.

9. References:


Raubal ,M. and M. Worboys,’A Formal Model of the process of wayfinding in Built Environments ‘,Lecture Notes in Computer Science,p.381-400,1999


Peponis, J., Zimring, C. & Choi, Y.,’Finding the Building in wayfinding Environment and Behaviour’, 1990


Arthur. & Passini,R.,’wayfinding :people ,signs and architecture ‘,NewYork,MacGraw Hill Ryerson,1992


C.Derix and R.Thum,’SOS: Self-Organising Space’, Proceedings,  Generative Art 2000 Conference, Milan, 2000


Passini R. ‘wayfinding in architecture’NewYork, Van Nostrad Reinhold Company, 1984


J.Piaget & B.Inhelder,’the child’s conception of space’, London, 1956


Penn, A. & Turner, P.A ‘A system and method for intelligent modeling of public spaces’, UK Patent office application, October 2000


Penn, A., Hillier, B., Banister & D.Xu.J.,’configurational modeling of urban movement network’, Environment and Planning B, 1998


Hillier, B.,’Space is the Machine: A configurationally theory of Architecture’, Cambridge University Press, 1996


Conroy, R.A.,’Saptial Navigation in Immersive Virtual Environments’, PhD Thesis, submitted to the University of London, 2001


R.A.Dwyer,’Highr-dimensional voronoi diagrams in linear unexpected time’, Discrete comput. Geom, 1991


G.L.Miller,’3D Convex Hull’, Comput. Geom, 2002