Data types

Vectors

In CelestLab most functions are vectorized. Arguments of functions are generally vectors of size:

• (1,N) for a set of scalars
• (3,N) for a set of 3D-vectors (for example: position, velocity, ...)
• (6,N) for a set of orbital elements (See orbital elements)
• (3,3,N) for a set of frame transformation matrices

Quaternions

The quaternion q associated with the rotation of axis u and angle alpha and defined relative to frame R is given by:

with (ux,uy,uz): coordinates of axis u in frame R.

Note that a quaternion in CelestLab is not an array, but a mlist. When a quaternion is printed on the screen, the components appear in the order: q0, q1, q2, q3.

CelestLab contains operators for the management of quaternions:

• Creation: see CL_rot_defQuat
• Operations with scalars: q*alpha (same as alpha*q), q/alpha, alpha/q, -q
• Operations with row vectors: q.*alpha (same as alpha.*q), q./alpha, alpha./q
• Logical operators on quaternions: q1==q2
• Operations with 2 quaternions: q1*q2 , q1/q2, q1+q2, q1-q2
• Other operations:

  size: size(q)

  norm: norm(q)

  inverse: inv(q)

  conjugate: conj(q) = ~q = q'

  extraction of real and imaginary part: real(q) = q.r, imag(q) = q.i

  concatenation: [q1,q2,...,qn]

  extraction: q2 = q1([i,j,k,...])

  insertion: q2([i,j,k,...]) = q1

  suppression: q2([i,j,k,...]) = []

Hypermatrices

Scilab's hypermatrix type (mlist 'hm') is partly overloaded in CelestLab. Hypermatrices should be of dimension 3.

A hypermatrix is considered as a set of matrices: A = [A1, A2,..., An], where Ak = A(:,:,k).

A matrix (when multiplied by a hypermatrix) is considered as a set of column vectors: B = [B1, B2,..., Bn], where Bk = B(:,k).

The overloaded operations are:

• Multiplication of a hypermatrix by a hypermatrix:

  

• Multiplication of a hypermatrix by a matrix:

  

  

• Transposition of a hypermatrix:

  

• Other operations: B = acos(A), defined by: B(:) = acos(A(:)).