worksheet3.20100125.4b5e7a90905183.84074720

worksheet3.20100125.4b5e7a90905183.84074720 - Worksheet on...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Worksheet on transition matrices Exploring proper orthogonal matrices A global chart Given a triplet of perpendicular lines labeled X , Y , and Z and intersecting at a point O , we saw how a global chart for the configuration of a particle located at P can be constructed using the signed distances from O to the orthogonal projections of P onto X , Y , and Z , respectively. We write for the corresponding one-to-one map and 1 = x , 2 = y , and 3 = z for the correspond- ing coordinate functions. A linear transformation Define the function f Now consider three new quantities, say 1 , 2 , and 3 and write = I 1 , 2 , 3 M T . Define the function f : R 3 fi R 3 such that f H L = I f 1 I 1 , 2 , 3 M , f 2 I 1 , 2 , 3 M , f 3 I 1 , 2 , 3 MM T , where f 1 I 1 , 2 , 3 M = h 11 1 + h 12 2 + h 13 3 , f 2 I 1 , 2 , 3 M = h 21 1 + h 22 2 + h 23 3 , and f 3 I 1 , 2 , 3 M = h 31 1 + h 32 2 + h 33 3 . Consider the equation = f H L . For every triplet of values of 1 , 2 , and 3 there is a unique triplet of values of 1 , 2 , and 3 . Since the relationship is linear in in 1 , 2 , and 3 it can generally be inverted. Here, In[1]:= f = 8 h11 1t + h12 2t + h13 3t , h21 1t + h22 2t + h23 3t , h31 1t + h32 2t + h33 3t < Out[1]= 8 h11 1t + h12 2t + h13 3t , h21 1t + h22 2t + h23 3t , h31 1t + h32 2t + h33 3t < The inverse is then In[2]:= Solve @8 f @@ 1 DD x, f @@ 2 DD y, f @@ 3 DD z < , 8 1t , 2t , 3t <D Simplify Out[2]= :: 1t fi h23 h32 x- h22 h33 x- h13 h32 y + h12 h33 y + h13 h22 z- h12 h23 z h13 h22 h31- h12 h23 h31- h13 h21 h32 + h11 h23 h32...
View Full Document

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.

Page1 / 4

worksheet3.20100125.4b5e7a90905183.84074720 - Worksheet on...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online