Transcription of Lie Groups for 2D and 3D Transformations
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Lie Groups for 2D and 3D Transformations
LieGroupsfor2 Dand3 DTransformationsEthanEadeUp datedMay20,2017*1 Intro ductionThisdo ologicalgroupthatisalsoasmo othmanifold,withsomeotherniceprop ciatedwitheveryLiegroupisaLiealgebra,whi chisavectorspacediscussedb ortantly,aLiegroupanditsLiealgebraareint imatelyrelated,allowingcalculationsinone tob emapp cumentdo esnotgivearigorousintro ductiontoLiegroups,nordo esattempttoprovideenoughinformationthatt heLiegroupsrepresentingspatialtransforma tionscanb eemployedusefullyinrob (3)3 DRotations33 DrotationmatrixSE(3)3 DRigidtransformations6 Lineartransformationonhomogeneous4-vecto rsSO(2)2 DRotations12 DrotationmatrixSE(2)2 DRigidtransformations3 Lineartransformationonhomogeneous3-vecto rsSim(3)3 DSimilaritytransformations(rigidmotion+s cale)7 Lineartransformationonhomogeneous4-vecto rsForeachofthesegroups,therepresentation isdescrib ed,andtheexp oticsorcomputervision?
Lie Groups for 2D and 3D Transformations Ethan Eade Updated May 20, 2017 * 1 Introduction This document derives useful formulae for working with the Lie groups that represent transformations
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