1999-2000 Lecture Notes on Astrometry

Rotation (Euclidean) Distance-Invariant

Finite Rotation: Matrix representation

Orthogonality

22 xX

xX R

TT

T

RRIRR

RRR

1

2T2

or -

xxxx

1999-2000 Lecture Notes on Astrometry

Infinitesimal Rotational Displacement Antisymmetric　Matrix

Vector Product

xθx

0

0

0

xy

xz

yz

z

y

x

θ

1999-2000 Lecture Notes on Astrometry

Finite Rotation

Expressions: Matrix, Spinol, Quarternion

Rotation = Matrix Operation Rot. Matrix = Set ofBasis Vectors (= Triad) ZYX eeeR

X

YZ

1999-2000 Lecture Notes on Astrometry

Euler’s Theorem Any Finite Rotation = 3 Basic

Rotation

Euler angles: 3 Angles of Basic Rotations

)()()(,, ijkijk RRRRR

,,,, 1kjiijk RR

1999-2000 Lecture Notes on Astrometry

Basic Rotation Rotation around z-axis by angle

)()(3 zRR X

Y

xy P

1999-2000 Lecture Notes on Astrometry

Basic Rotation (contd.) Rotation around j-axis by angle

Inverse Rotation

)(jR

jj RR 1

1999-2000 Lecture Notes on Astrometry

Basic Rotation Matrix

Example: Equatorial – Ecliptic Obliquity of Ecliptic

100

0cossin

0sincos

)(3

R

1R

1999-2000 Lecture Notes on Astrometry

Basic Rotation Matrix (contd.) Small Angle Approximation

j

jjj

jj e

e

IR

IIR 33

000

00

00

1999-2000 Lecture Notes on Astrometry

Angular Velocity

ωe

e

jj

j

jjj

jjj

dt

d

dt

d

R

IRR

j

jj

dt

deω

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

313313 ,, RRRR

1999-2000 Lecture Notes on Astrometry

Euler Angles (3-1-3)

CCSSS

SCCCCSSSCCCS

SSCCSSCSCSCC

,,313R

coscossinsinsin

sincoscoscoscossinsinsincoscoscossin

sinsincoscossinsincossincossincoscos

1999-2000 Lecture Notes on Astrometry

Euler Angles

X

Z

YN

P

1999-2000 Lecture Notes on Astrometry

Demerit of 3-1-3 Sequence

0,,313 IR

Degeneration in case of small angles

Solution: 3-2-1-like Sequences

1999-2000 Lecture Notes on Astrometry

3-2-3 Sequence y-convention: precession

Conic Rotation Rotation around a fixed direction

cos

sinsin

cossin

n

,,323R

AAA z ,,323 RP

I+ sin 1 cos n n n

1999-2000 Lecture Notes on Astrometry

Other Sequences 1-3-1: Nutation

2-1-3: Polar Motion + Sidereal Rotation

1-2-3: Aerodynamics, Attitude Control Best Recommended

AA ,,131RN

pp xy ,,312RWS

1999-2000 Lecture Notes on Astrometry

Small Angle Rotation

I

RRRR

CCSCS

SCCSSCCSSSCS

SSCSCCSSSCCC

)()()(,, 123123

1999-2000 Lecture Notes on Astrometry

Rotational Velocity

xvxvV

xX

RdtdR

R

R