# visualize the underlying geometry and the tangent basis vectors defined by the metric demonstrating The tangent vectors ( $$E\_u, E\_v$$ ) rapidly rotate and stretch as $$u$$ changes, even though the Christoffel symbols that depend only on $$v\left(\Gamma\_{u u}^v, \Gamma\_{u v}^u\right)$$ remain constant. This demonstrates that the connection (Christoffel symbols) only captures one part of the basis vector change, while the rotation and stretching dependent on $u$ are governed by the other non-zero symbols and the metric components. As $$v$$ increases, the magnitude of $$E\_u$$ grows dramatically (proportional to $$v$$ ), causing the coordinate grid to expand away from the origin. This visual expansion directly confirms the functional dependencies of the key Christoffel symbols ( $$\Gamma\_{u u}^v=v$$ increases, $$\Gamma\_{u v}^u=1 / v$$ decreases), showing how the change in the scale factor $$v$$ controls the connection and local geometry. Geometric changes in the coordinate system, encoded by the non-zero Christoffel symbols, are highly dependent on the direction of motion. In the hyperbolic system, moving along the $u$-direction primarily causes the basis to rotate and stretch nonuniformly (governed by $$u$$-dependent terms), while moving along the $v$-direction causes a dramatic, proportional scaling of the grid (directly captured by the $$v$$-dependent terms like $$\Gamma\_{u u}^v= v)$$. {% embed url="" %} {% embed url="" %}