This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.View Full 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:= f = 8 h11 1t + h12 2t + h13 3t , h21 1t + h22 2t + h23 3t , h31 1t + h32 2t + h33 3t < Out= 8 h11 1t + h12 2t + h13 3t , h21 1t + h22 2t + h23 3t , h31 1t + h32 2t + h33 3t < The inverse is then In:= Solve @8 f @@ 1 DD x, f @@ 2 DD y, f @@ 3 DD z < , 8 1t , 2t , 3t <D Simplify Out= :: 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.
- Spring '08