Tangent Space

1. Tangent Space

2. Tangent Vector • Motion along a trajectory is described by position and velocity. • Position uses an origin • References the trajectory • Displacement points along the trajectory. • Tangent to the trajectory • Velocity is also tangent x3 x2 x1

3. Tangent Plane • Motion may be constrained • Configuration manifold Q • Velocities are not on the manifold. • Set of all possible velocities • Associate with a point x Q • N-dimensional set Vn • Tangent plane or fiber • TxQxVn S1 V1 q S2 x V2

4. Fibers can be associated with all points in a chart, and all charts in a manifold. This is a tangent bundle. Set is TQQVn Visualize for a 1-d manifold and 1-d vector. Tangent Bundle V1 S1

5. A tangent plane is independent of the coordinates. Coordinates are local to a neighborhood on a chart. Charts can align in different ways. Locally the same bundle Different manifold TQ Twisted Bundles V1 S1

6. Map from tangent space back to original manifold. p = TQQ; (x, v) (x) Projection map p Map from one tangent space to another f: UW; U, W open f is differentiable Tf: TUTW (x, v)  (f(x), Df(x)v) Tangent map Tf Df(x) is the derivative off Tangent Maps V1 S1

7. Tangent Map Composition • The tangent map of the composition of two maps is the composition of their tangent maps • Tf: TUTW; Tg: TWTX • T(gf) = TgTf • Equivalent to the chain rule next