Trigonometry is based on the geometry of the unit circle *x*^{2}+*y*^{2 }=
1 in the Euclidean plane. Hyperbolic geometry is based on the geometry of the
pair of unit hyperbolas in the Lorentz plane: *x*^{2 }- *y*^{2 }=
1 with vertices on the horizonal axis and *-x*^{2}+*y*^{2 }=
1 with vertices on the vertical axis.

Here we compare an active counterclockwise rotation of the Euclidean plane by an increasing positive angle with an active hyperbolic rotation (pseudo-rotation = "boost") of the Lorentz plane by an increasing positive velocity parameter. This shows how the points on the blue triangles move under these tranformations. Animated GIFs courtesy of Maple.

a rotation of the Euclidean plane | a pseudo-rotation of the plane |