Tangent SpaceTangent VectorTangent PlaneTangent BundleTwisted BundlesTangent MapsTangent Map CompositionTangent SpaceTangent SpaceTangent VectorTangent VectorMotion along a trajectory is Motion along a trajectory is described by position and described by position and velocity.velocity.•Position uses an originPosition uses an origin•References the trajectoryReferences the trajectoryDisplacement points along Displacement points along the trajectory.the trajectory.•Tangent to the trajectoryTangent to the trajectory•Velocity is also tangentVelocity is also tangentx1x2x3rTangent PlaneTangent PlaneMotion may be constrainedMotion may be constrained•Configuration manifold Configuration manifold QQ•Velocities are not on the Velocities are not on the manifold.manifold.Set of all possible velocitiesSet of all possible velocities•Associate with a point Associate with a point xx QQ•N-dimensional set N-dimensional set VVnnTangent plane or fiberTangent plane or fiber•TTxxQQ xx VVnnV1S1S2xV2Tangent BundleTangent BundleFibers can be associated Fibers can be associated with all points in a chart, and with all points in a chart, and all charts in a manifold.all charts in a manifold.•This is a tangent bundle.This is a tangent bundle.•Set is Set is TTQQ QQ VVnn•Visualize for a 1-d manifold Visualize for a 1-d manifold and 1-d vector.and 1-d vector.V1S1Twisted BundlesTwisted BundlesA tangent plane is A tangent plane is independent of the independent of the coordinates.coordinates.Coordinates are local to a Coordinates are local to a neighborhood on a chart.neighborhood on a chart.Charts can align in different Charts can align in different ways.ways.•Locally the same bundleLocally the same bundle•Different manifold Different manifold TTQQV1S1Tangent MapsTangent MapsMap from tangent space Map from tangent space back to original manifold.back to original manifold.• = = TTQQ QQ; ; ((xx, , vv) () (xx))•Projection map Projection map Map from one tangent space Map from one tangent space to anotherto another•ff: : UU WW; ; UU, , WW open open•ff is differentiable is differentiable•TTff: : TUTU TWTW•((xx, , vv) ) (f( (f(xx), ), DfDf((xx))vv))•Tangent map Tangent map TTffDfDf((xx) ) is the derivative ofis the derivative of ffV1S1Tangent Map CompositionTangent Map CompositionThe tangent map of the composition of two maps is The tangent map of the composition of two maps is the composition of their tangent mapsthe composition of their tangent maps•TTff: : TUTU TW; TTW; Tgg: : TWTW TXTX•TT((gfgf) = ) = TTgg TTffEquivalent to the chain ruleEquivalent to the chain
View Full Document
Unlocking...