# symmetry-hyperbolic-rotation.mp4

Chapter section: Invariants and Metrics

Caption (verbatim):

Hyperbolic rotations

Image description (verbatim):

Hyperbolic rotation preserving the interval

## Before the animation

Chapter lines 351–355:

```math
\mathbf u\cdot\mathbf v
=
u_xv_x+u_yv_y.
```

Chapter lines 357–357:

An invariant of this sort, that fixes some notion of placing reliable rulers on the space, is a **metric**. For normal rotations and translations, these relationships are intuitive, but a symmetry may preserve a less intuitive metric. If we define invariant length not as \(x^2+y^2\) but as \(t^2-\mathbf{x}^2\), we have a "hyperbolic rotation." Separation between points stretches out to infinity as the rotation approaches its asymptotes, leaving the invariant interval unchanged even though the separation appears to grow using our everyday notion of distance. Such a hyperbolic metric, as we will see, encodes the asymptotic limit of light speed in the relativistic geometry **spacetime**.

## After the animation