# Introduction to Astrometry 位置天文学入門

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```					    Rotation
   (Euclidean) Distance-Invariant
X   2
 x 
2

   Finite Rotation: Matrix representation
X  Rx
   Orthogonality
Rx      2
 x R Rx  x
T       T                   2

R R  I or R  R
T                         -1           T
1999-2000            Lecture Notes on Astrometry
Infinitesimal Rotational
Displacement
   Antisymmetric Matrix
 0      z           y 
                           
   z     0              x 
       x                
 y                       0              x 
 
   Vector Product                           θ   y 
x  θ  x                                
 z
1999-2000        Lecture Notes on Astrometry
Finite Rotation
 Expressions: Matrix, Spinol, Quarternion
 Rotation = Matrix Operation

 Rot. Matrix = Set of
Z
Y

R  e X eY e Z 
X

1999-2000      Lecture Notes on Astrometry
Euler’s Theorem
   Any Finite Rotation = 3 Basic Rotation
R  R ijk  ,  ,    R k ( )R j (  )R i ( )

R ,  , 
ijk
1
 R kji   , , 
   Euler angles: 3 Angles of Basic
Rotations

1999-2000         Lecture Notes on Astrometry
Basic Rotation
   Rotation around z-axis by angle 
Y
y                        P
x

R 3 ( )  R z ( )
                   X

1999-2000              Lecture Notes on Astrometry
Basic Rotation (contd.)
   Rotation around j-axis by angle 
R j ( )
   Inverse Rotation

R  
j
1
 R j  

1999-2000              Lecture Notes on Astrometry
Basic Rotation Matrix
 cos                    sin       0
                                     
R 3 ( )    sin                  cos       0
 0                                   
                                0   1
   Example: Equatorial – Ecliptic R1  
   Obliquity of Ecliptic 
1999-2000        Lecture Notes on Astrometry
Basic Rotation Matrix (contd.)
   Small Angle Approximation
 0  0
          
R 3    I     0 0   I   e 3 
 0 0 0
          
          
  R j  j   I    j e j  
          
j                j        
1999-2000          Lecture Notes on Astrometry
Angular Velocity
d j
ω                ej
j   dt
          
 R   R j  j   I    j e j  
          
j                  j        
dR      d j 
  
          e j   ω  

dt     j  dt        
1999-2000          Lecture Notes on Astrometry
Euler Rotation
   3x2x2 = 12 different combinations
   3-1-3 Sequence (= x-convention)
    Most popular (Euler angles)
    Used to describe rotational dynamics

R 313  , ,    R 3  R1  R 3  

1999-2000             Lecture Notes on Astrometry
Euler Angles (3-1-3)

 C C  S C S            C S  S C C     S S 
                                                        
R 313  , ,      S C  C C S          S S  C C C    C S 
       S S                             S C    C 
                                                        
 cos cos  sin  cos sin     cos sin   sin  cos cos sin  sin  
                                                                          
   sin  cos  cos cos sin   sin  sin   cos cos cos cos sin  
           sin  sin                      sin  cos           cos 
                                                                          

1999-2000              Lecture Notes on Astrometry
Z
Euler Angles

P


X                                   
                                 Y
N

1999-2000       Lecture Notes on Astrometry
Demerit of 3-1-3 Sequence
   Degeneration in case of small angles

  
     
R 313  , ,    I   0  
  
     
   Solution: 3-2-1-like Sequences
1999-2000        Lecture Notes on Astrometry
3-2-3 Sequence
   y-convention: precession
P  R 323   A , A , z A 
   Conic Rotation                        sin  cos  
Rotation around
             

n   sin  sin  
a fixed direction                   cos 
R 323  ,  ,                                  

 I+ sin  n   1  cos   n  n 
1999-2000            Lecture Notes on Astrometry
Other Sequences
   1-3-1: Nutation
N  R131  A ,  , A   
   2-1-3: Polar Motion + Sidereal Rotation
WS  R 312 , y p , x p 
   1-2-3: Aerodynamics, Attitude Control
   Best Recommended
1999-2000         Lecture Notes on Astrometry
Small Angle Rotation
R123  ,  ,    R 3 ( )R 2 (  )R1 ( )
 C C        C S  S  S C                C S  C  S S 
                                                                  
   S C      S S  S  C C               S S  C  C S 
 S                  C  S                          C  C       
                                                                 
 
 
 I    
 
 
1999-2000             Lecture Notes on Astrometry
Rotational Velocity
X  Rx 
dR
V  Rv     x  Rv  x
dt

1999-2000   Lecture Notes on Astrometry

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