TRIANGULATION

MICHAEL SCHWAB

*Triangulation* is a method that allows for the reconstruction of a site when a grid or coordinate system for the mapping of a site does not exist.

When compared with the latter, triangulation engages space in a very different way. A coordinate system has already addressed all possible points in space and has, thus, covered space with an abstract, invisible layer. Triangulation, on the other hand, opens up space as it proceeds. Space that is not addressed remains unknown in a deeper sense.

Pixel versus vector drawings can illustrate this. In a pixel drawing, a surface is represented by x times y pixels. Even if an area is not important, it will always be represented by one or more pixels with an associated value. A vector drawing, on the other hand, references only those elements that are relevant. No information is stored for areas of no relevance, which usually results in smaller vector drawing files.

Triangulation *develops a space* producing figures.

Actual triangulation

When a real site is measured and triangulation is used to represent it, actual triangulation happens.

Historically, the best examples consist of geodesic surveys, such as the nineteenth-century ‘Principal Triangulation of Great Britain’. These surveys use a baseline between two initial points, which can be measured physically, using a chain, for example. Using this baseline, the location of a third point can be measured optically. The lengths of the baseline and the two angles at the beginning and end of the baseline, which are determined by sight, are sufficient for the construction of the third point in relation to the first two. Using the third point further triangulations can take place and a triangulation network can be constructed that covers whole countries, or, rather, has helped constitute those countries in the first place.

The figure above illustrates the way in which I have been reconstructing sites using distances only. This is easier for smaller sites where distances can be measured; it is impossible for larger sites where the angles are all that can be measured (last slide).

On the right is the tool that I built to measure distances in paces.

Below is a reconstruction using a drafting compass.

And here is a lines-only version:

Apparent triangulation

Not everything that looks like a triangulation actually is one. Often measurement systems return x, y, z coordinates from points.

Using this data, I can pretend to use triangulation, which will give me the directed line pattern as the basis for a figure, but which will have been constructed very differently.

In apparent triangulations, the body is somewhat detached from the drawing.

Below is a frame from *Rebody* that is made using apparent triangulation.

Triangulation in research methodology

The notion of ‘triangulation’ has been applied to research methodology. It is meant to indicate that more than one method may be used to identify an epistemic object.

However, for this to work, one needs to assume object 1 (resultant from method 1) to be identical with object 2 (resultant from method 2), and so on. This is highly problematic since the identity of the object is either an assumption or a higher view of the methods needed to make them comparable, conflating their potentially radical difference.

The point that is constructed when two lines meet or when two methods create an object is an idealisation.

PROVISIONAL GLOSSARY

→ Figure

Read more in: Michael Schwab, ‘Drawing the Trans-body’

Application of the Triangulation Drawing Algorithm for *Place Roger Priou-Valjean*, a drawing published in my book *Paris*:

Green = first set of three measurements; yellow = second set of three measurements; red = third set of three measurements, and so on, resulting in the drawing on the left.

Paris Drawings Method

The basic rules are:

- The arbitrary identification of a group of trees within a city space.
- A visual study of that group and the surrounding architectural space, which leads to an arbitrary decision about the tree from which to start the measurement of the group.
- A set of three measurements to the three trees standing closest to the first tree.
- Another set of three measurements to the trees standing closest to the second tree (the tree closest to the first tree) excluding the measurement to the first tree, as this distance has been measured before.
- The continual reapplication of rule 4 until all trees within the group have been measured against their three closest neighbours, providing that no distance was measured twice.
- The reconstruction of the group on paper.

The extended rules are:

- If a set of three measurements is not sufficient to develop the whole of the group from the initial tree, choose another starting tree.
- If no tree of the group as initial tree allows for the construction of the group within three measurements, extend the amount of measurements taken per tree to four.
- If the number of measurements per tree is not sufficient, repeat rule (a) with one more measurement and so on until the whole group can be developed using N measurements.

N is the minimum number of measurements per tree necessary for the development of the entire group. N designates the order in which a particular drawing has been executed.

Schwab, Michael, *Paris* (London: Copy Press, 2008), 16–17.