worksheet4.20100126.4b5f93ea568e15.47128035

worksheet4.20100126.4b5f93ea568e15.47128035 - Worksheet on...

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Unformatted text preview: Worksheet on Euler angles On the orientation of a rigid body A proper orthogonal matrix A general formula Let @ 0, D and j , @ 0, 2 L . The matrix In[1]:= H H = 88 Cos @ D , Sin @ D , 0 < , 8- Sin @ D , Cos @ D , 0 < , 8 0, 0, 1 << . 88 Cos @ D , 0,- Sin @ D< , 8 0, 1, 0 < , 8 Sin @ D , 0, Cos @ D<< . 88 Cos @ j D , Sin @ j D , 0 < , 8- Sin @ j D , Cos @ j D , 0 < , 8 0, 0, 1 <<L MatrixForm Out[1]//MatrixForm= Cos @ D Cos @ j D Cos @ D- Sin @ j D Sin @ D Cos @ D Cos @ D Sin @ j D + Cos @ j D Sin @ D- Cos @ D Sin @ D- Cos @ D Sin @ j D- Cos @ D Cos @ j D Sin @ D Cos @ j D Cos @ D- Cos @ D Sin @ j D Sin @ D Sin @ D Sin @ D Cos @ j D Sin @ D Sin @ D Sin @ j D Cos @ D is proper orthogonal, since In[2]:= Transpose @ H D .H Simplify Out[2]= 88 1, 0, 0 < , 8 0, 1, 0 < , 8 0, 0, 1 << In[3]:= Det @ H D Simplify Out[3]= 1 Finding , j , and when | h 33 | 1 Consider a general transition matrix H with entries h ij . In[4]:= H = Table @ Symbol @ "h" <> ToString @ i D <> ToString @ j DD , 8 i, 3 < , 8 j, 3 <D Out[4]= 88 h11, h12, h13 < , 8 h21, h22, h23 < , 8 h31, h32, h33 << Necessary conditions Suppose that h 33 1. Since each column of H corresponds to a unit vector, it follows that h 33 < 1. Then, there exists a unique H 0, L such that cos H L = h 33 and sin H L 0. Since each column of H corresponds to a unit vector In[5]:= H @@ 3, 1 DD 2 + H @@ 3, 2 DD 2 == 1- H @@ 3, 3 DD 2 . h33 fi Cos @ D Simplify Out[5]= h31 2 + h32 2 Sin @ D 2 It follows that h 31 , h 32 sin H L . Given this value of it follows that there exists a unique...
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This note was uploaded on 11/17/2011 for the course TAM 412 taught by Professor Weaver during the Spring '08 term at University of Illinois, Urbana Champaign.

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worksheet4.20100126.4b5f93ea568e15.47128035 - Worksheet on...

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