No one has still given me a good reason why we can't just use any number of spatial partitions 'as is'. I don't see the Geographers worry about a formal definition of a road centre line or a land parcel. No one asks for a formal proof of road centre line being unique. Each time I ask for a response on how comparable two maps of road centre lines made by two different operators from the same photography all I get is silence. With out some base line for comparison making any statements about the reproducibility of the axial map is irrelevant. Still I think that my current work on finding community- neighbourhoods in axial lines could make a significant contribution to finding effectively rooms in buildings. The problem is that all the graph-neighbourhood finding algorithms tend to want to know how many rooms you want to find in advance. sheep On 8 Sep 2007, at 12:24, Alan Penn wrote: > Alasdair, > > you introduce an interesting, and I suspect a deep point here, in > the notion > of the need for the 'largest'. The pivot you are talking about > might be > thought of in the case of the L shaped room as a triangular convex > space > defined by pivoting a line across the inside corner of the elbow. > This space > is larger than the overlap, which it completely contains, and may be > metrically larger than either of the other two 'face hugging' > rectangular > spaces. In your/Maria's pragmatic VGA approach to computing this > through > looking for cliques, you are pretty much forced to look for the > largest > first, and so may often find this kind of space in preference to > the face > hugging kind. BUT... the point is that in the L shaped room, in > order to > completely cover the system, if you choose the triangular zone as a > convex > space first, you find that you have also to choose both face > hugging zones > as well, and end up with three convex spaces to cover something > that is > completely covered by just the two face huggers. In this sense the > 'largest' > rule seems to be less important than the 'fewest that cover the > system' > rule, but with the obvious caveat that smaller spaces that are > completely > contained by another are redundant. > > This may be one 'deep' part - we see this kind of thing happening > also in > axial line definition, where rules like 'longest' and 'fewest' > compete with > each other and where actually what is required is a more global > optimisation > of the representation to achieve (for example) depth or angle > minimisation, > coupled to removal of redundancy. > > I think that there is a second 'deep' part here as well. This is in > the > nature of 'pivoting'. Effectively what we do when we draw axial > lines is to > pivot on the convex projecting corners of the boundary, often > looking for > two opposing pivot points that allow a line just to get through, and > maximise its extension (note the subtle avoidance of the word > 'length':-). > Now I think it is possible, and can show examples, that the kind of > spatial > property the 'pivoted' space produces is something like the axial > line or > strip. It seems to me possible that our pairing of 'axial' and > 'convex' > representation are actually two complementary but interlinked kinds > of thing > that tell us quite different things about the more general isovist/VGA > properties of the world. > > SpaceBox, which doesn't work from the same VGA/clique premise, > starts with > face hugging: that is with the articulation of the boundary and its > effects > when the face is projected across open space at all convex corners. > This > gives rise to a finite subdivision (given a finite number of > straight faces > on the boundary in the original map). What John showed so elegantly > is that > these kind of lines across open space (plus the extended ends of > the axial > lines in an all axial line map) define partitions of relative > informational > stability, in which although the isovist changes at any point > within the > partition it does so smoothly and offers little in the way of 'new' > information on the geometry of the environment. When you cross one > of John's > E or S partition lines you gain new information (see a new wall > face or see > through between two corners) and often isovist properties change > substantially. > > It is easy to see that the pivot line that defines a maximal clique > in the > VGA approach may well not be related to these lines of > informational change, > (except in the extreme where it becomes the axial line), and may in > these > terms be a fairly irrelevant subdivision of space, experientially and > socially. So back to the initial point - why are we doing these > things? Well > in this field the representations are defined by our interest in human > experience of architecture and its relation to the social, and here > concepts > like John's of informational stability and change seem crucial. > > Alan > >> >> At the risk of throwing the cat amongst the pigeons, we probably >> ought >> to introduce some firm mathematics, even for fuzzily defined >> systems, as >> the problems of convex space coverage and isovist coverage are in >> fact >> different. ("Uniqueness" is an unnecessary distraction.) >> >> I'll use a visibility graph as the fuzzy representation, but note >> that >> what I am about to say is also true geometrically. >> >> Firstly note that the analogue of a isovist in a visibility graph >> is the >> neighbourhood of a node. For a certain resolution of graph, the >> set of >> isovists is easily computable. >> >> However, a minimal covering set of isovists is not easily computable, >> although a heuristic (such as, largest first) gives an easily >> computable >> set in approximately linear time. The reason a minimal covering >> set is >> not easily computable is that combinations of isovists together >> need to >> be tried. If you do this, you find it takes exponential time (all >> combinations may have to be evaluated) in order to find the minimal >> covering set. This does not mean the minimal covering set is not >> unique. It may or may not be. >> >> However, as pointed out by Maria Doxa, the analogue of a convex >> space in >> visibility graph terms is a maximal closed subgraph (clique). >> That is, >> to construct the space, we place John, Bill and Mike at various >> points >> in the graph and check that they are co-visible. This leads to >> something surprising: calculation of the cliques is again only >> possible >> in exponential time, as all combinations of nodes have to be >> calculated. >> So this is one step before completing a covering set: the >> individual >> spaces themselves cannot be computed easily. Again, a heuristic >> makes >> the construction of convex spaces in reality straightforward: >> tools such >> as SpaceBox construct spaces where each face of the convex space is >> aligned with a face. >> >> The limits of text mean I cannot easily demonstrate why this >> heuristic >> doesn't necessarily produce the largest convex polygon about a >> particular point. I'll try: imagine a pivot point and a line rested >> upon it. If the pivot is offset then the area above the line can be >> increased by tilting it. Now imagine the pivot broadened so it is a >> short face rather than a point, the area above the line can still be >> increased by tilting it. Rather than a single offset pivotal face, >> think of lots of pivotal faces arranged roughly into a circle. >> You have >> to "tilt" lines in opposition to each other in order to try to >> find the >> largest area. >> >> The important point in this is that this property (the convex space >> boundaries do not necessarily align with faces) may or may not matter >> socially -- it's not a matter of simply creating a fuzzy resolution. >> >> Alasdair >> >> Alan Penn wrote: >>> At the risk of awakening a previous discussion, it is for this >>> reason >>> (the impossibility of defining a unique convex partitioning) that we >>> invented the 'overlapping' convex space. In this an L shaped room is >>> divided into two convex elements, one for each leg of the L which >>> overlap at the elbow. Each convex space is considered as a node >>> in the >>> graph, and each overlap between two spaces is represented as a >>> link. For >>> a boundary consisting of only straight lines (no smooth convex >>> curves) >>> it is possible to compute all such convex spaces (curves can of >>> course >>> be approximated by straight lines, but that is bound to introduce >>> arbitrariness). This was first implemented in the Syntactica >>> software in >>> about 1989, and then written properly by Sheep in SpaceBox in >>> 1991. This >>> is not a partition in the conventional sense since a point in >>> space may >>> be a member of two (eg. in the elbow) or more, sometimes very >>> many, but >>> always a finite number of, convex spaces, and so points are not >>> 'partitioned'. John Peponis went on to show how by considering >>> each of >>> the spaces defined by different degrees of overlap as a partition >>> one >>> can derive a unique partitioning for this kind of boundary. Now, >>> I have >>> not been able to find anything in the mathematical literature >>> about this >>> kind of 'overlapping convex space' map, and I am aware of no formal >>> proof of its uniqueness or completeness, so although these >>> properties >>> seem pretty obvious they must rest as a conjecture - my math is >>> isn't up >>> to proving this kind of thing. >>> >>> >>> >>> That said, it is worth going back to what Bill and Julienne were >>> doing >>> in the SLS. First, inside buildings with rooms and doorways or just >>> archways through walls, the subdivision into 'unique' spaces for the >>> purposes of analysis of social relations is often not too >>> problematic. >>> The convexity of such spaces is a social property (as well as a >>> geometric one) - if Bill can see Mike and Mike can see John, then >>> John >>> can see Bill. This is why meetings generally happen in spaces of >>> this >>> sort of shape. This is of course a pretty metrically and >>> geometrically >>> fuzzy thing - minor deviations in the boundary wall (the alcove or >>> chimney breast) have little real effect on the 'fatness' of a >>> space for >>> these kind of social purposes. Anyway, the point is one doesn't >>> need to >>> get too mathematically prissy for the purposes of a social >>> analysis of >>> space of this kind, any more so than the housebuilder and their >>> client >>> needed to be consciously aware of this kind of mathematics when they >>> laid the walls out, or the users of the space are consciously >>> aware of >>> these maths when they appropriate space for particular social >>> behaviours >>> in use. The focus of the analysis tends to be on the relations in >>> space >>> between a whole set of different more or less well defined fat >>> spaces >>> which are appropriated for different social functions or meetings >>> (living room, dining, parlour, bedroom), and how these relate to >>> each >>> other and the outside world. >>> >>> >>> >>> Now, when one moves on to urban space things can get much less >>> clear. >>> Having said that, the class of French villages that Bill and >>> Julienne >>> were looking at do often seem to define 'fat' spaces with pretty >>> regular >>> associations with other things (like building entrances). Given >>> that in >>> those days everything was being done by hand, the use of a unique >>> convex >>> partitioning is not only sensible, but the only possible way to >>> get at >>> what seems like an important social property that characterised the >>> vernacular settlement, and appeared to distinguish it from the >>> kinds of >>> modern housing estate that these villages supposedly inspired in >>> the UK, >>> but which consistently created fat spaces with blank walls. Again >>> the >>> point is that the analysis has a specific purpose concerned with the >>> social, and its precision is appropriate to that. The axial >>> representation which has turned out to be the most useful for urban >>> space is of course not problematic in this sense - but that has been >>> discussed at length on this list before J >>> >>> >>> >>> Alan >>> >>> >>> >>> >>> >>> Ruth is absolutely correct. There is no solution to the problem of >>> identifying a unique partition of a space into convex subspaces. The >>> problem is not well defined and a corollary of this is that there >>> is no >>> unique set of isovists that define a space. >>> >>> There are various notes about this scattered around the various >>> papers >>> on isovists but I havent seen an exhaustive discussion of this. >>> My own >>> paper has a note - >>> >>> Batty M, 2001, "Exploring isovist fields: space and shape in >>> architectural and urban morphology" /Environment and Planning B: >>> Planning and Design/ *28*(1) 123 - 150 >>> http://www.envplan.com/abstract.cgi?id=b2725 >>> >>> and I reproduce the quote >>> >>> Emacs! >>> >>> Mike >>> >>> At 16:14 07/09/2007, Ruth Conroy Dalton wrote: >>> >>> Dear Ahmed, >>> >>> There is no software for discrete convex spaces, as there is no one >>> unique solution. Sheep (Nick Dalton) wrote some software, a long >>> time >>> ago, called space box, which produced overlapping convex spaces - >>> but >>> this only runs on very, very old Apple Macs. Unless, it's an >>> extremely >>> complicated layout or you have hundreds and hundreds of them, it >>> will >>> take you far less time to draw them by hand. >>> >>> Regards >>> >>> Ruth >>> >>> >>> >>> Hi all, >>> i was wondering if anybody has an idea if there is a software in >>> which i >>> can calculate the number of convex spaces in any layout and how >>> to get >>> this software >>> >>> thanks for you help >>> >>> >>> *Ahmed Mohamed Refaat Mostafa >>> * >>> School of Architecture >>> School of Environment & Development (SED) >>> Manchester University >>> mobile no: (+44) 07878185642 >>> >>> >>> Sick sense of humor? Visit Yahoo! TV's Comedy with an Edge >>> <http://us.rd.yahoo.com/evt=47093/*http:/tv.yahoo.com/collections/ >>> 222> >>> to see what's on, when. >>> >>> >>> >>> >>> >>> -------------------------------------------------------------------- >>> ---- >>> >>> *Michael Batty* | CASA | University College London | 1-19 Torrington >>> Place London WC1E 6BT UK | Tel 44 207 679 1782 | Mobile 44 7768 >>> 423 656 >> | >>> email: [log in to unmask] <mailto:[log in to unmask]> | web: >>> www.casa.ucl.ac.uk <http://www.casa.ucl.ac.uk/> >>> >>> >>> >>> >>> >> >> -- >> Course Director >> MSc Adaptive Architecture & Computation >> UCL Bartlett School of Graduate Studies >> >> http://www.vr.ucl.ac.uk/people/alasdair