Science/Memorandum Series

The Memorandum Series is a compilation of handouts that I have prepared mainly as reminder (hence the name “Memorandum”) for various topics I deal with. They are also intended as teaching materials; as such, many of them contain supplemental files, e.g. software implementations of the algorithms and theory presented therein.

If you have any feedback, comments, error reports, ideas and enhancement requests regarding the Memorandum Series, do not hesitate to use my bug tracker.

M001: Keplerian Orbit Elements to Cartesian State Vectors

This Memorandum contains an algorithm for converting a traditional set of Keplerian Orbit Elements (semi-major axis, eccentricity, argument of periapsis, longitude of ascending node, inclination, and mean anomaly) to cartesian state vectors (position and velocity vector).

Keywords:
Kepler, Orbit, Elements, Cartesian, State, Vector, Rotation, Matrix

---

Downloads:

• Memorandum
• Supplemental Files (if any)

M002: Cartesian State Vectors to Keplerian Orbit Elements

This Memorandum contains an algorithm for converting cartesian state vectors (position and velocity vector) to a traditional set of Keplerian Orbit Elements (semi-major axis, eccentricity, argument of periapsis, longitude of ascending node, inclination, and mean anomaly).

Keywords:
Kepler, Orbit, Elements, Cartesian, State, Vector

---

Downloads:

• Memorandum
• Supplemental Files (if any)

M003: Trigonometric Functions

A formulary of trigonometric functions, providing the following sections:

• Trigonometric functions and their definitions
• Shorthands
• Important values of the basic trigonometric functions
• Pythagorean and related trigonometric identities
• Symmetry, shifts and periodicity
• Addition theorems, sum and difference identities of two functions
• Products of two functions
• Power reduction formula
• Multiple-angle formulæ
• Half-angle formulæ
• Inverse trigonometric functions
  • Definition of the inverse trigonometric functions
  • Relationships between trigonometric functions and inverse trigonometric functions
  • Relationships among the inverse trigonometric functions
• Power series expansions, infinite product formulæ
• Relationship to the complex exponential function

Keywords:
trigonometric, functions, sine, cosine, tangent, secant, cosecant, versin, vercosin, coversin, covercosin, haversin, havercosin, hacoversin, hacovercosin, exsec, excsc, crd, sinc, cis, inverse, complex, exponential, power, series, expansion, euler, identity, pythagorean

---

Downloads:

• Memorandum
• Supplemental Files (if any)

M005: Astrodynamical Constants

A listing of commonly used constants in astrodynamics as well as in space science and engineering with their latest values according to recent publications. This Memorandum is going to be updated with new constants from time to time.

---

Downloads:

• Memorandum
• Supplemental Files (if any)

M006: Quaternions and Spatial Rotation

A compact introduction into quaternions and their application for spatial rotations, providing the following contents:

• brief introduction into quaternions, their components, notations, and properties
• quaternion algebra formulary summarizing basic properties (quaternion equality, hypercomplex conjugate, norm, and inverse), basic operations (quaternion addition and subtraction, multiplication, division, normalization, cross and dot product), the exponential and logarithmic functions (quaternion exponential function, natural logarithm, logarithmic functions, power and root functions) as well as trigonometric and hyperbolic functions of quaternions
• application of quaternions in the context of spatial rotations, quaternion rotation operators, rotations and transformations
• conversion algorithms:
  • Euler Angles to Quaternion (for all 12 possible rotation sequences)
  • Quaternion to Euler Angles (for all 12 possible rotation sequences)
  • Direction Cosine Matrix (DCM) to Quaternion
  • Quaternion to Direction Cosine Matrix (DCM)
  • Euler Angles to Rotation Matrix (for all 12 possible rotation sequences)
• time derivative of a rotation quaternion
• quaternion interpolation algorithms
  • linear interpolation (LERP)
  • spherical linear interpolation (SLERP)

The appendix of this Memorandum also contains the interface and exception documentation for the MATLAB Quaternion Class (supplement to this Memorandum).

Keywords:
quaternions, spatial rotations, components, notations, properties, quaternion algebra formulary, quaternion equality, quaternion hypercomplex conjugate, quaternion norm, quaternion inverse, quaternion addition, quaternion subtraction, quaternion multiplication, quaternion division, quaternion normalization, quaternion cross product, quaternion dot product, quaternion exponential function, quaternion natural logarithm, quaternion logarithmic functions, quaternion power functions, quaternion root functions, quaternion trigonometric functions, quaternion hyperbolic functions, quaternion rotation operators, transformations, conversion algorithms, Euler angles to quaternion, quaternion to Euler angles, direction cosine matrix (DCM) to quaternion, quaternion to direction cosine matrix (DCM), euler angles to rotation matrix, quaternion time derivative, quaternion interpolation algorithms, linear interpolation (LERP), spherical linear interpolation (SLERP)

---

Downloads:

• Memorandum
• Supplemental Files (if any)