When students first meet transformations of co-ordinates, it tends to be a bit confusing and hard to make sense of. This isn't so much because transforming between different descriptions of a thing is difficult as because, up to that point, they've dealt with systems where there's an easy choice of description which, once you've learned to spot it, makes everything simple. The tricky thing is that not every system is amenable to such a choice of description; in particular, although its parts may each be easy to describe in terms that suit each part, the different parts are described in different terms, which complicates describing the interactions among the parts. However, given a suitable characterisation of the systems of description of the disparate parts, it is often possible to transform between the descriptions of the parts. We can model each part in terms of the description of it that makes that modelling simple; we can then characterise the behaviour of that part by some relations between the parameters describing it in its description, then transform those relations to express them in terms of the description that makes it easy to model some other part; the model of the first part might not be neatly expressed in this part's terms, but the conclusions of that model are now available as inputs to our model of the new part, using which we can now characterise the behaviour of this part by some relations among the parameters used to describe it and its response to and effects on the first part by relations between these and the parameters we used to cahracterise the first part (as viewed via our transformation). So transforming co-ordinates lets us build up models of more complex systems from models of simpler systems, each using its own co-ordinates.
In any case, our models express how things we experience (e.g. measure) relate to one another. The things we can measure aren't necessarily exactly the right things to measure, so we may end up using models in which the model parameter that we associate with some measurement doesn't exactly match it. For example, when folk first began studying heat, temperature and their relation to other (particularly mechanical, but also chemical) processes, they noticed that a column of mercury or water would expand as it was heated. The amount of expansion grew by roughly equal amounts upon adding roughly equal (in so far as the knew how to measure it) amounts of heat; so they used the expansion of columns of mercury in glass tubes to gauge temperature. Then they found that, for a given body of gas, the product P.V of pressure and volume varied only with temperature; and that, between different bodies of gas, increasing the amount increased P.V for any given temperature in proportion to the amount; when they divided P.V by a suitable measure of amount of gas, the result depended only on the temperature and was, roughly, just the temperature plus a constant. However, the offset from temperature did vary a little, if temperature was measured using the length of a column of mercury in a tube of glass; still, a wide variety of gasses gave (subject to suitable measure of amount) consistent P.V divided by amount senses of temperature; and so, in time, we switched to using this as our temperature parameter, across a broad range of temperatures. We then recalibrated our mercury thermometers so that the temperature markings aren't quite evenly spaced along the tube's length; mercury's rate of expansion with temeperature varies with (as defined by the P.V/amount of gasses) temperature. That change in how we measure temperature was a reparameterisation.
When we describe a body sitting on a flat surface and being pushed, partly parallel to the surface, partly perpendicularly, we find that the force parallel to the surface won't cause movement unless it exceeds some particular multiple of the force perpendicular to it; the multiplier is called the coefficient of friction for that body's contact with that surface. This uses co-ordinates parallel and perpendicular to the surface; and it works for surfaces in diverse orientations. We can use this to model a system made of several bodies connected by mechanisms (e.g. ropes and pulleys) that ensure particular relationships between the forces on the bodies; and typically there shall be gravity acting on the bodies, with the surfaces at various angles to the direction of gravity. So we use our model in each piece of surface's co-ordinates (parallel and perpendicular) and then relate the result to a common system of co-ordinates (vertical and horizontal) by transforms that rotate the model's co-ordinates; this is a fairly simple transform.
Written by Eddy.