Auxiliary code

Helpers functions necessary to the core code and plotting functions.

frame_transformations.py

File frame_transformations.py Contains functions that for frame transformations and rotations

Authors:mdelvallevaro, LucaZampieri (2018)

Note

In this file, when there is a reference, unless explicitly stated otherwise, it refers to Lindegren main article: “The astronometric core solution for the Gaia mission - overview of models, algorithms, and software implementation” by L. Lindegren, U. Lammer, D. Hobbs, W. O’Mullane, U. Bastian, and J.Hernandez The reference is usually made in the following way: Ref. Paper [LUDW2011] eq. [1]

frame_transformations.adp_to_cartesian(alpha, delta, parallax)

Convert coordinates from (alpha, delta, parallax) format into the (x, y, z) format.

Parameters:

• azimuth – [rad]
• altitude – [rad]
• parallax – [mas]

Returns:

[parsec](x, y, z) array in parsecs.

frame_transformations.compute_ljk(epsilon)

Ref. [Lind2001] (Lindegren, SAG-LL-35, Eq.1)

Calculates ecliptic triad vectors with respect to BCRS-frame.

Parameters:epsilon – obliquity of the equator. Returns:

np.array, np.array, np.array of the ecliptic triad:

• l: is a unit vector toward (alpha, delta) = (0,0)`)
• n: is a unit vector towards delta = 90°
• m = n x l

frame_transformations.compute_pqr(alpha, delta)

Ref. Paper [LUDW2011] eq. [5]

Can be used also with numpy arrays

Computes the p, q, r parameters

Parameters:

• alpha – [rad] astronomic parameter alpha
• delta – [rad] astronomic parameter alpha

Returns:

p, q, r

frame_transformations.lmn_to_xyz(attitude, vector)

Ref. Paper [LUDW2011] eq. [9] Goes from the non-rotating (lmn) CoMRS frame to the rotating (xyz) SRS frame

Note

The attitude Quaternion q(t) gives the rotation from (lmn) to (xyz) (lmn) being the CoMRS (C), and (xyz) the SRS (S). The relation between the two frames is given by: {S’v,0} = q^-1 {C’v,0} q for an any vector v

Parameters:

• attitude – Quaternion object
• vector – array of 3D

Returns:

the coordinates in XYZ-frame of the input vector.

frame_transformations.polar_to_direction(alpha, delta)

Convert polar angles to unit direction vector

Parameters:

• alpha – [rad]
• delta – [rad]

Returns:

3D np.array unit vector

frame_transformations.quat_to_vector(quat)

Parameters:quat – [quaternion] Quaternion to transform into vector Returns:3D array made with x,y,z components of the quaternion

frame_transformations.rotate_by_quaternion(quaternion, vector)

Ref. Paper [LUDW2011] eq. [9] rotate vector by quaternion

frame_transformations.transform_twoPi_into_halfPi(deltas)

Tranforms an angle in range [0-2*pi] to range [-pi/2, pi/2] by substracting 2pi to any angle greater than pi.

Warning

The input angles have to be defined between [0,pi/2] and [3pi/2, 2pi]

Parameters:delta – input angles Returns:modified angles

frame_transformations.vector_to_alpha_delta(vector, two_pi=False)

Ref. Paper [LUDW2011] eq. [96] Convert carthesian coordinates of a vector into its corresponding polar coordinates (0 - 2*pi)

Parameters:

• vector – [array] X,Y,Z coordinates in CoMRS frame (non-rotating)
• two_pi – [bool] if True return delta in [0,2pi] instead of [-pi/2,pi/2]

Returns:

[rad][rad] alpha, delta

frame_transformations.vector_to_quat(vector)

Transform vector to quaternion by setting x,y,z components of the quaternion with x,y,z components of the vector.

Parameters:vector – vector to transform to quaternion Returns:quaternion created from vector

frame_transformations.xyz_to_lmn(attitude, vector)

Ref. Paper [LUDW2011] eq. [9] Go from the rotating (xyz) SRS frame to the non-rotating (lmn) CoMRS frame

Note

The attitude Quaternion q(t) gives the rotation from (lmn) to (xyz) (lmn) being the CoMRS (C), and (xyz) the SRS (S). The relation between the two frames is given by: {C’v,0} = q {S’v,0} q^-1 for an any vector v

Parameters:

• attitude – Quaternion object
• vector – array of 3D

Returns:

the coordinates in LMN-frame of the input vector.

frame_transformations.zero_to_two_pi_to_minus_pi_pi(angle, unit='radians')

Tranforms an angle in range [0-2*pi] to range [-pi, pi] by substracting 2pi to any angle greater than pi.

Info: Can be used with numpy arrays

Parameters:

• angle – [rad] angle or array of angles in [0-2*pi] format
• unit – [str] specify if the input data is in radians or degrees

Returns:

angle in the [-pi, pi] format

helpers.py

File helpers.py

Helper functions for the analytic scanner

Contains: (at least)

• compute_intersection
• compute_angle

Author:LucaZampieri (2018)

helpers.check_symmetry(a, tol=1e-12)

Check the symmetry of array a. True if symmetric up to tolerance

helpers.compute_angle(v1, v2)

Computes the angle between two Vectors :param vi: vector between which you want to compute the angle for each i=1:2 :returns: [float] [deg] angle between the vectors

helpers.compute_intersection(x1, y1, x2, y2, x3, y3, x4, y4, segment=True)

Return intersection of two lines (or segments) if it exists, raise an error otherwise. :param xi: x-coordinate of segment i for i=1:4 :param yi: y-coordinate of segment i for i=1:4 :param segment: [bool]

Returns:

• (x, y) tuple with x and y coordinartes of the intersection point
• [list] error_msg list

helpers.get_rotation_matrix(v1, v2)

Get the rotation matrix necessary to go from v1 to v2 :param vi: 3D vector as np.array To rotate vector v1 into v2 then do r@v1

helpers.get_sparse_diagonal_matrix_from_half_band(half_band)

it assumes we have the upper hal of the band, i.e. we some zeros at the bottom of band for column index greater than 0

helpers.normalize(v, tol=1e-10)

return normalized version of v

helpers.plot_sparsity_pattern(A, tick_frequency)

Parameters:A – np array containing the matrix

helpers.sec(x)

Stable if x close to 0

helpers.symmetrize_triangular_matrix(a)

Symmetrize an already triangular matrix (lower or upper) :param a: upper ot lower triangular matrix :returns: corresponding symmetric matrix

analytic_plots.py