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Rotation

Rotation. (Euclidean) Distance-Invariant Finite Rotation: Matrix representation Orthogonality. Infinitesimal Rotational Displacement. Antisymmetric Matrix Vector Product. Finite Rotation. Expressions: Matrix, Spinol, Quarternion Rotation = Matrix Operation Rot. Matrix = Set of

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Rotation

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  1. Rotation • (Euclidean) Distance-Invariant • Finite Rotation: Matrix representation • Orthogonality Lecture Notes on Astrometry

  2. Infinitesimal Rotational Displacement • AntisymmetricMatrix • Vector Product Lecture Notes on Astrometry

  3. Finite Rotation • Expressions: Matrix, Spinol, Quarternion • Rotation = Matrix Operation • Rot. Matrix = Set of Basis Vectors (= Triad) Z Y X Lecture Notes on Astrometry

  4. Euler’s Theorem • Any Finite Rotation = 3 Basic Rotation • Euler angles: 3 Angles of Basic Rotations Lecture Notes on Astrometry

  5. Y y P x q X Basic Rotation • Rotation around z-axis by angle q Lecture Notes on Astrometry

  6. Basic Rotation (contd.) • Rotation around j-axis by angle q • Inverse Rotation Lecture Notes on Astrometry

  7. Basic Rotation Matrix • Example: Equatorial – Ecliptic • Obliquity of Ecliptic Lecture Notes on Astrometry

  8. Basic Rotation Matrix (contd.) • Small Angle Approximation Lecture Notes on Astrometry

  9. Angular Velocity Lecture Notes on Astrometry

  10. Euler Rotation • 3x2x2 = 12 different combinations • 3-1-3 Sequence (= x-convention) • Most popular (Euler angles) • Used to describe rotational dynamics Lecture Notes on Astrometry

  11. Euler Angles (3-1-3) Lecture Notes on Astrometry

  12. Euler Angles Z P f q X y Y N Lecture Notes on Astrometry

  13. Demerit of 3-1-3 Sequence • Degeneration in case of small angles • Solution: 3-2-1-like Sequences Lecture Notes on Astrometry

  14. 3-2-3 Sequence • y-convention: precession • Conic Rotation • Rotation around a fixed direction Lecture Notes on Astrometry

  15. Other Sequences • 1-3-1: Nutation • 2-1-3: Polar Motion + Sidereal Rotation • 1-2-3: Aerodynamics, Attitude Control • Best Recommended Lecture Notes on Astrometry

  16. Small Angle Rotation Lecture Notes on Astrometry

  17. Rotational Velocity Lecture Notes on Astrometry

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