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
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