# Easy examples of visualizing passive transformations in the plane?

I teach mostly physics and a little math at a community college in California. When I teach special relativity, my preferred pedagogy is to describe the Lorentz transformation using an almost purely geometrical and visual approach, with very little algebraic manipulation. My presentation uses what physicists would probably refer to as a passive transformation, in which we think of the points as staying fixed while the coordinate grid changes. Below is an example that shows the sort of presentation I use. (I actually use a slightly different set of graphical conventions so that it's intelligible in black and white.) In this example, the blue dot has coordinates (3,2) in the red coordinate system and (3,0) in the black coordinates.

I usually spend quite a bit of time introducing this type of graphical presentation, and I have a lesson worked out in which I initially use the Galilean transformation, which should be more familar, or at least less counterintuitive, before introducing the Lorentz transformation. But I'm wondering if my presentation could be improved or streamlined by presenting some totally different and more familiar mathematical example of this kind of change of variables. This seems like the kind of thing that people might have done more commonly back in the era of nomograms and slide rules. Basically I would like a pair of variables $$(x,y)$$ with some familiar, easy real-world interpretation, where there is some interesting notion of transforming to some new variables $$(x',y')$$.

The best examples I've been able to come up with are:

• $$(\rho,v)$$=density and velocity of water in a river; transformation is flying downstream or upstream in a plane

• $$(\text{lat},\text{lon})$$=latitude and longitude, transformation is changing to some other set of coordinates

I'm not really happy with either example. The $$(\rho,v)$$ example is not necessarily easy for students with weak physical intuition, and it's not very exciting because $$\rho$$ doesn't change. The $$(\text{lat},\text{lon})$$ requires projection to the plane of the page for purposes of drawing it, which is an extra complication, and there is not another set of coordinates that is necessarily familiar to most people.

A rotation could be a good example, but it fails to illustrate that angles are not necessarily preserved.

I would be happy with a good qualitative example. E.g., maybe something from economics or psychology, like $$x$$=happiness and $$y$$=wealth.

• (moderator, please convert to comments) 1. Can you share some student reaction to what you have done so far? Are they baffled or comprehending? How do they perform on tests of the material? [I think it is fine to think about what makes sense to design your teaching, but since you have experience of the actual performance in the field, share that also.] 2. I don't really know relativity that well so not sure if responsive. But if you want some charts that sort of look like what you are doing, you could consider Mollier diagram (steam) or psychrometric (humid air) chart. Also pump sizing diagram – guest Jan 11 '19 at 17:40