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

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?

May 20, 2017 · Then di erentiation by the vector is straightforward, as fis linear in x: @y @x = R (27) Di erentiation by the rotation parameters is performed by implicitly left multiplying the rotation by the exponential of a tangent vector and di erentiating the resulting expression around the zero perturbation.

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  Di erentiation, Erentiation

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