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Tangent Space - PowerPoint PPT Presentation

Tangent Space. 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. x 3. x 2. x 1. Tangent Plane.

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Tangent Space

• 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

• 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

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

A tangent plane is independent of the coordinates. charts in a manifold.

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

Map from tangent space back to original manifold. charts in a 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

Tangent Map Composition charts in a manifold.

• 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

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