Problem 5

Homogeneous transform

T[r_, p_] := <br />    Join[Join[Transpose[r], {p}] // Transpose, {{0, 0, 0, 1}}]

Tinv[r_, p_] := <br />    T[Transpose[r], -Transpose[r] . p] ;

Part (a)

aRb = RZ[Pi] ; aPborg = {3, 0, 0} ; aTb = T[aRb, aPborg] ; aTb// TraditionalForm

( -1 0 0 3 ) 0 -1 0 0 0 0 1 0 0 0 0 1

Part (b)

bRc = RY[Pi/2] . RX[Pi/2 + Pi/3] ; bRc // TraditionalForm

( 1 Sqrt[3] ) - -------- ... -------- -- 0 2 2 -1 0 0

aPcorg = {3, 0, 2} ; aTc = T[aRb . bRc, aPcorg] ; aTc // TraditionalForm

( 1 Sqrt[3] ) -- ------ ... 0 -1 0 0 2 0 0 0 1

Part (c)

bRc = RY[Pi] . RX[-Pi/2] . RZ[-Pi/6] ; bPcorg = {3, 0, 0} ; bTc = T[bRc, bPcorg] ; bTc // TraditionalForm

( Sqrt[3] 1 ) -------- -- ... ----- 2 2 0 0 0 0 0 1

Part (d)

aRb = RZ[Pi] . RX[-Pi/2] ; aRb //TraditionalForm

( -1 0 0 ) 0 0 -1 0 -1 0

aRc = aRb . bRc ; aPcorg = {-3, 3Sqrt[3], 2} ; aTc = T[aRc, aPcorg] ; aTc // TraditionalForm

( Sqrt[3] 1 ) ------- - ... 0 0 -1 2 0 0 0 1

cTa = Tinv[aRc, aPcorg] ; cTa // TraditionalForm

( Sqrt[3] 1 ) ------- - ... 6 0 0 -1 2 0 0 0 1

Created by Mathematica  (September 29, 2003)