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convexity is an interesting but difficult concept. Originally, when faced with a building with rooms and doorways it seems relatively uncomplicated and the 'simple' thing to do manually. However in urban space made up of mainly linear streets although there are convex deformations it becomes much harder to decide which convex space 'wins' over which others. When we first started to write algorithms to do this break up of spaces by computer we soon found something that mathematicians have known for a long time, and that is there is (in general terms) no single unique minimal convex partitioning of a two dimensional configuration. The test case is a completely symmetrical cross shaped building. It will be split into one 'nave' and two side chapels, but which axis should be the nave? The computer would need additional knowledge to decide. A grid city has this kind of situation all over the place. Interestingly, we were not put off and developed an algorithm that followed ru les and produced convex urban maps that were in agreement with what a trained person would do. It could do this for organic, unplanned cities - the first we tried were Gassin and Apt - and it worked perfectly, but when we tried Barcelona the grid problem happened, meaning that every time we ran the algorithm with a different random seed we obtained a different convex decomposition. This is when we realised we had a problem.

The answer we brought to this was that analysis of convexity did not require partitioning of space into unique point sets. The cross shape should not properly be represented by three convex spaces, but by two 'naves' overlapping at the middle. This property is lifelike in that if you stand in the middle you can see and be seen from every point in the building. The gave rise to the idea of the overlapping convex map. It is relatively straightforward to automate (so long as you do not have smooth convexly curved walls which give rise to infinite subdivisions), and makes very clear these 'I am both in this and that space' situations (the corner of the L shaped living room etc.). Now, start to think of what it means in urban space with long winding streets (represented as a finite number of flat faced buildings) - the resulting overlapping convex map is made up of an enormous number of different and multiply overlapped ling thin convex spaces. About as many of them as there are lines in the all line map. For all practical purposes this object is just too hard to see and understand to be useful. The fewest line axial map says most that needs to be said bout movement in urban space - and now we have elaborated this into the segment map with angular measures which gives a more fine grained representation.

The final part of this story relates to the VGA representation. Here we just throw a grid of points over the space and look at their graph of inter-visibility. In urban space this picks up much of the local deformation and interesting visibility effects that come from convex deformations of the boundary. So it is probably VGA that is now used to explore these dimensions in urban space. 

Alan