Some notes on the process angular fractional configuration

Angular fractional configuration is really two linked concepts. The first core concept is fractional integration computation. This computation can be applied to a number of ways of describing linking between 0 and 1.

Its best to begin with a quick outline description of how normal configuration is computed. Let us begin with a simple building, a number of rooms are linked with a number of doorways. This is a simple logical layout,

Rooms either connect or do not connect to others. For clarity, we can label each room, we typically include the outside as the first space.

We can then remove the metric qualities of the building, reducing the building down to its topological ‘core’. In this case, we draw a line from each label where a direct pathway links a room to a room.

Removing the rooms, we are left with the topological skeleton of the space.

The essential discovery of configuration is that when we take this network (or more mathematically graph) representation and lay it out from different view points the shape of the graph (known as the J Graph) is different.

The J graph from A.

The J Graph from B

The J graph from E

Examining this J graphs the same system looks different. Traditional configuration analysis tries to describe this shape by counting steps of depth. So from E, D is at depth 1, C at depth 2, B is at depth 3, A, G,F is of depth 4( each ). This is a total of 18

From B, A,G,F and C are at depth 1 , D at depth 2, and E a depth 3. This is a total of 8. The important observation here is that each line is at a rational counting number( 1,2,3,4), what might it be if something had a depth of 0.5 or 0.1 or 0.8 ? For a system of rooms, this would not make sense- a doorway exists or does not exist. However, for the case of a convex overlapping space have interesting properties.

 

Case a large overlap. Case b narrow overlap.

Consider two overlapping convex spaces pictured above. With normal configurational analysis, we are expected that two shapes that overlap completely are identical interms of the J graph analysis as two shapes that hardly overlap. Does this matter, perhaps, if you are in the first case a moving from one space to another is trivial. If you can in case b, moving from a corridor like space to an open area is more likely than moving from a open area to the corridor space. In the J graph analysis, we are expected that both cases are identical. Perhaps moving towards a fractional analysis ( one where the J graph can contain fractional values ) would ‘pollute’ the pure topological measures. To explore this concept it is necessary to try it out in some cases.

The mainstay of urban level configurational analysis is the axial line, so the next question is there some way of introducing a fraction factor into the intersection of axial lines. One prevalent concept has been the introduction of distance into a J graph. This is more complicated that the initial view. Consider how a line A might intersect line B, two lines meet at by definition a point, how do we define a length between them?

A number of eminent authorities (Ruth,Alastir,Bill ) have been proposing for a wide range of reasons that angle of incidence may be an area worth further exploration. Let us begin by considering the angle of incidence between 2 overlapping lines.

shallow angle of incidence (distance = 0.1 )

Imagine walking down a slowly meandering high street. To model this geometrically we would be forced to model many axial lines which a low angle of incidence between them. Walking along these streets transferring from one line to the next would not be strong decision. While walking along the same high street, making a change in 90 degrees would be a stronger decision. We have this in everyday cases, we describe routes as follow the road the take the next right.

sharp angle of incidence ( distance = 1 )

 

Fractional Angular Analysis works by defining a fractional analysis where the angle of incidence is 1.0 where the axial lines are at right (90degree) angle. Lines that are parallel and intersect have fractional distance or 0.0.

table 1 angle of incidence against fraction

 

From this, lines that are nearly parallel have low fractional distances. We would expect this kind of analysis to make long meandering streets of many axial lines become stronger integrators. Equally making one right-angled turn might well mean a strong increase in ‘distance’. Intuitively Oxford Street is longer than a single axial line ( we cannot see all the way down Oxford Street ) , however there are rarely many shops on right angles to Oxford Street.

We now can build an network as in the first case, except here the J Graph will have steps which are not 'rational' numbers (1,2,3…) but irrational (or fractional numbers ) Such a fractional J Graph will look something like this.

 

It is still possible to do all the usual processing - we can find the average distance from the starting point. We can also have concepts like 'connectivity angle - co angle' this is similar to connectivity, it measures the total of all the angles of incidence from this line. Given we can build up an average depth we end up with a configurational like mesure which in this case is sensitive to angle.

This process has been implemented by a program called Meanda ( Mean Depth Angular ) images from which can be seen here.

 

More Notes

 

Going beyond 90 degrees

It would be interesting to go beyond Meanda’s fractional analysis, and try to include a 180 as 1 and 90 as 0.5 and 0 as 0.0. How ever, consider figure 9. An angle of 140 degrees Menada would consider this as an angle of 40 ( 180 - 140 ).

figure 9.

The intuitive view is that to include a backward angle line this must improve fractional analysis. After all going backon, your self is a very unusual process in pedestrian movement. How ever, by rearranging the lines we would find something-different ( fig 10)

by moving the line backward, we now sense the line, as being shallow again. To remove this problem Menada keeps lines between zero and 90 degrees.