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Lie Groups for 2D and 3D Transformations - Ethan Eade

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LieGroupsfor2Dand3DTransformationsEthanE adeUp datedMay20,2017*1Intro ductionThisdo ologicalgroupthatisalsoasmo othmanifold,withsomeotherniceprop ciatedwitheveryLiegroupisaLiealgebra,whi chisavectorspacediscussedb ortantly,aLiegroupanditsLiealgebraareint imatelyrelated,allowingcalculationsinone tob emapp cumentdo esnotgivearigorousintro ductiontoLiegroups,nordo esattempttoprovideenoughinformationthatt heLiegroupsrepresentingspatialtransforma tionscanb eemployedusefullyinrob (3)3DRotations33DrotationmatrixSE(3)3DRi gidtransformations6Lineartransformationo nhomogeneous4-vectorsSO(2)2DRotations12D rotationmatrixSE(2)2DRigidtransformation s3Lineartransformationonhomogeneous3-vec torsSim(3)3DSimilaritytransformations(ri gidmotion+scale)7Lineartransformationonh omogeneous4-vectorsForeachofthesegroups, therepresentationisdescrib ed,andtheexp oticsorcomputervision?

A Lie group is a topological group that is also a smooth manifold, with some other nice properties. Associated with every Lie group is a Lie algebra, which is a vector space discussed below. Importantly, a Lie group and its Lie algebra are …

  Group, Lie group

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