Complexity & Coordinate Systems

Complexity theorists often quote the non-linearity of dynamical complex systems as a reason why we should stop even trying to predict their future states and just “go with the flow” or “dance” with it. I find all such suggestions ill-advised.

Before we give up all attempts to simplify a complex process we are trying to figure out, I would like to share my experience from dealing with complex “aerial target tracking” problems, and discuss the importance of selecting the appropriate coordinate system in which to do the tracking.

Sensors (visual, IR radar, laser) and weapons, both work in the spherical coordinate system using the radial distance r (distance from the target to the origin point of the sensor or weapon), polar angle (or elevation) θ (angle with respect to the vertical “polar” axis or the orthogonal complement elevation angle from the horizontal surface), and azimuth φ (angle of rotation from some referential “meridian” plane or centre-line).

Source: Wikipedia

This coordinate system is very useful when collecting (measuring) positional data about aerial targets or for pointing weapons that will engage that target. However, the spherical coordinate system is less useful for target tracking purposes because the data describing a target trajectory is non-linear and looks something like this:

However, when the above data is transformed from spherical to cartesian coordinates (x, y and z), the situation is quite different:

Just by changing the coordinate system the trajectory data becomes linear, so predicting the future position of the aerial target is now much easier then if the tracking (prediction) would have been done in the original (spherical) coordinate system.

So, the lesson to take home here is that just because something appears to be complex, it may not be so after giving it another look from a different perspective because from that point of view it may present itself as much a simpler problem to solve.

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