Coordinate systems and transformations

Mark one point in the plane. That is all there is. It has no numbers yet, because there is nothing to measure it against. A point is a place, not a pair of coordinates. Before we can compute with it, we have to choose something to compute with.

Now draw two axes through an origin. The same point acquires two numbers: how far along each axis it sits. That pair is a frame, and notice, the frame is something you imposed. The point did not gain a length or a direction. You only chose a ruler.

Here is the puzzle. Turn the axes by an angle. The point has not moved, yet its numbers change. But why should they change, if the point did nothing? Step through the figure. The blue frame swings, the green point stays put, and the two coordinates slide through a whole range on their own. The numbers changed because the ruler did.

So what is a coordinate, really? Drop a perpendicular from the point onto an axis. The foot of that perpendicular is the coordinate. The number is the length of a shadow. In symbols, $x_1' = P\cdot e_1'$ and $x_2' = P\cdot e_2'$: project the point onto each new axis and read the projection. Here is the key, the same point now casts two different shadows, because the light source moved.

To convert one frame's numbers into another's you only need to know where the new axes point, written in the old language. Each new axis becomes a column of the rotation matrix. Its entries are the cosines of the angles between old and new axes, so $\alpha_{ij} = \cos(\angle e_i, e_j')$. Pause on the minus sign: the first new axis leans away from $e_1$, so its second component is $-\sin\varphi$. A matrix is a table of directions.

A rotation is a change of description, never a deformation of the space. So two quantities are untouchable: the length of every vector, and the handedness of the frame. In matrix language, $\alpha\alpha^{\top} = \delta$ and $\det\alpha = 1$. Sweep the angle and try to break either. You cannot, and that impossibility is exactly what keeps physics the same for every observer.

That is the whole idea. A frame is an encoding. A transformation is a change of encoding. The rotation matrix is nothing more than the new axes written in the old ones. Displacements and velocities will inherit this same rule, and a tensor will need it twice, once for each index. But which quantities are allowed to follow this rule at all? That is the next question.