# PARAMETER SPACE & DISPLAYING GEOMETRY

Within this project, and concerning the methodology, literally joining the dots will take a little getting your head around. What we plan to do from here on out is finding the best possible solution to the data we are given from the research task. We are drawing very close to giving out beacons to participants and actually ‘locating someone’ (accurately or not), can be done. However, drawing conclusions about the inferred interactions of people based on the analysis of those ‘positions’ is a little more tricky. Going back to what I was saying about time in my last post, time plays a major role in how I wish to calculate the ‘potential for interactions’.

I’ll come back to this in a second but just to clarify a few terms I’ll be using…

### Metric space

According to Wikipedia, In mathematics, a metric space is a set for which distances between all members of the set are defined. Basically, it represents the ‘actual’ distance of space between a ‘member’. What I mean by this equilateral triangle:

An equilateral triangle has three ‘members’(represented by the vertices’s(one at the top, one on the left and one on the right). Metrically the members are the same, as the distances are the same distance apart. This acute triangle however…

…is not metrically the same as the equilateral triangle above. This is where I introduce parametric…

### Parametric space

According to Wikipedia, In science, a parameter space is the set of all possible combinations of values for all the different parameters contained in a particular mathematical model. The ranges of values of the parameters may form the axes of a plot, and particular outcomes of the model may be plotted against these axes to illustrate how different regions of the parameter space produce different types of behavior in the model.

Often the parameters are inputs of a function, in which case the technical term for the parameter space is domain of a function. Parametrically, if we compare the top member of the equilateral triangle with the acute triangle ‘P’ member, the two members are the same because they fall on the same arrangement. The two members however are not metrically the same.

When this starts to get interesting is when we start looking at how people interact and use space, time and distance to calculate that. Naturally, humans move in a line within a 2 Dimensional space. (climbing over mountains and seas don’t count because we are in an office for this experiment, unless we fly) we are only really dealing with X, Y and time as a 3 Dimension.

### Concepts of Segments

In geometry, a line segment is a part of a line that is bounded by two distinct end points, and contains every point on the line between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints.

How we prepare to observe peoples movements within a space, I hypothesis there will be a very clear ‘start’ and ‘finish’ to the overall path that the participant moved over during the day(still working on how to handle stop and backtrack path mapping). Therefore, each ‘line segment’ can be determined by the split of time over the course of the day.

Two people interacting face-to-face require a set of physical rules. The first is they need to be positioned closely within space, and this needs to occur at the same segment in time. See for example, line A-B as one participants overall path, and now see line C-D as another participants path. At one point in time, both path’s lines intersect at E, however if the same parametric segment of time was not the same for both participants, then an interaction, (based on the rules I stated above(more like assumptions, but we’ll leave that discussion for later)) did not occur.

Lets say, for arguments sake that E occurred at the same point in time for one both participants(even though the line A-B is metrically different to the line C-D), then based on my rules, I can say that a interaction ‘may’ (still have to be bureaucratic) occurred.

### Definitives and infinitive space

This is when we start getting into the realm of definitives and infinitive space. Definitives are things that do not change even with alterations to (certain)parameters. Take the solar system for example.

No matter the distances even if it meant infinitive space, according to this image, the Earth is to the right of Mars and the Sun is to the left. I said ‘certain parameters because definitives still need structure, for example, if we were to change the position of the Sun or Mars or Earth, then that ‘definitive’ may not be valid anymore.

### Determining topology

Drawing assumptions and defining definitives are what will help shape how I go about measuring ‘Potential for interactions’. These variables help plot topology for measurement and understanding, analysis and conclusions and thats is what I will go through on my next post.