Measuring position and orientation (with no mathematical notations)

Wikipedia describes position as "the spatial location (rather than orientation) of an entity". In turn, an entity's orientation is described as "part of the description of how it is placed in the space it occupies". We intuitively understand and manipulate the position and orientation of objects on a daily basis. Writing is a good example. When writing, we continually change the position and orientation of the pen in relationship to a piece of paper to create letters.

In scientific/engineering endeavors, measuring the position and orientation of an object is important because it allows us to quantify movement characteristics. This post aims to provide a simple explanation of the process of measuring the position and orientation of an object in space. The goal is to not use any mathematical notation, although I presume that we all share a common intuitive understanding of the concept of a point and axis in the 3D space we inhabit. My failure to provide even a decent description of how to measure an object's position and orientation highlights the need for the precision introduced via mathematics (or my inability to explain a simple concepts succinctly - I pick the former). For simplicity, I will only consider measuring the position and orientation of rigid bodies, i.e. objects that do not deform.

All measurements must be made with respect to something, and this something is typically based out of convenience or necessity (you have no other options). For example, if you were writing on only one piece of paper, it might be convenient to measure the pen with respect to the paper. However, if you plan on writing on multiple pieces of paper laid out on your desk, it might make more sense to measure with respect to the desk.

On to the measuring process. Imagine a legal pad in front of you, with the writing lines running from your left to your right and the margin line (red) pointing ahead of you. Place the tip of a clicker pen in the center of the paper, with its button facing towards the ceiling, and its clip facing to your right. Where is the pen located in relationship to the paper? The answer would depend on whether you want to know the position of the button or the tip of the pen, and how you define your measurement system (the paper). Let's decide to measure the tip of the pen - this is a choice not a given. We can also make similar decisions regarding the paper. Let's decide that the lower left-hand corner of the paper is our reference point of interest (again, a choice). This is typically called the origin of the measurement system. Now imagine erecting a fence along the left edge of the paper. This fence provides the boundary between what we call left and right. Similarly, a fence running along the lower edge of the paper provides the boundary between front and back. Anything above the paper will be called up, and everything below the paper will be called down. Now we can make a statement that the pen is 4.25 inches to the right, 7 inches forward, and 0 inches up (a legal pad is 8.5 x 14 inches). That covers how to measure the position of the pen, what about orientation?

Measuring (or representing) orientation can be significantly trickier, but I present a method that's easy to visualize but has practical disadvantages (not discussed here). First, let's simplify the system and place the tip of the pen at the lower left-hand corner of the paper. It is then 0 inches to the right, 0 inches forward, and 0 inches up - so the tip of the pen is located at the origin of the measurement system (the paper). As before, the pen's button is facing towards the ceiling and its clip to the right. Now bend the clip so it forms a right angle with the pen's body. Some thought (or physical) experiments should suffice in showing that measuring the position of the clip's tip and the pen's button reveals the pen's orientation in space. For example, twist the pen around the long axis of its body. The position of the button stays the same but the position of the clip's tip changes (it traces a circle). Now tilt the pen forward, backwards, left and right. Both the position of the button and the clip's tip change. It seems that the position of the clip's tip tracks any orientation change, so it may be tempting to only measure this landmark to determine the orientation of the pen. But, consider rotating the pen about an axis that runs from the tip of the pen to the tip of the clip. Notice that the position of the clip's tip remains constant, but the position of the button does change. So, indeed, we do need to measure both the position of the button and the tip of the clip in order to ascertain orientation changes of the pen.

And this is why we make use of mathematics. Given the appropriate background and nomenclature, we can represent the position and orientation of the pen as measured from the perspective of a paper with two words and a symbol $^{Paper}T_{Pen}$.