synospec.util.frame module

Provides a set of coordinate frames.

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Copyright © 2020, Kyle B. Westfall

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class synospec.util.frame.SemiMajorAxisCoo(xc=None, yc=None, rot=None, pa=None, ell=None)

Bases: object

Calculate the semi-major axis coordinates given a set of input parameters following \({\mathbf x} = {\mathbf A}^{-1}\ {\mathbf b}\), where

\[ \begin{align}\begin{aligned}\begin{split}{\mathbf A} = \left[ \begin{array}{rrrrrr} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ \cos\psi & \sin\psi & -1 & 0 & 0 & 0 \\ -\sin\psi & \cos\psi & 0 & -1 & 0 & 0 \\ 0 & 0 & \sin\phi_0 & \cos\phi_0 & -1 & 0 \\ 0 & 0 & -\cos\phi_0 & \sin\phi_0 & 0 & \varepsilon-1 \end{array} \right]\end{split}\\\begin{split}{\mathbf b} = \left[ \begin{array}{r} x_f \\ y_f \\ -x_0 \\ -y_0 \\ 0 \\ 0 \end{array} \right]\end{split}\end{aligned}\end{align} \]

such that

\[\begin{split}{\mathbf x} = \left[ \begin{array}{r} x_f \\ y_f \\ x_s \\ y_s \\ x_a \\ y_a \end{array} \right]\end{split}\]

and:

• \(\psi\) is the Cartesian rotation of the focal-plane relative to the sky-plane (+x toward East; +y toward North),

• \(\phi_0\) is the on-sky position angle of the major axis of the ellipse, defined as the angle from North through East

• \(\varepsilon=1-b/a\) is the ellipticity based on the the semi-minor to semi-major axis ratio (\(b/a\)).

• \((x_f,y_f)\) is the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

• \((x_s,y_s)\) is the on-sky position of \((x_f,y_f)\) relative to the center of the ellipse, and

• \((x_a,y_a)\) is the Cartesian position of \((x_f,y_f)\) in units of the semi-major axis.

This form is used such that \({\mathbf A}\) need only be defined once per class instance.

The class also allows for inverse calculations, i.e., calculating the focal-plane positions provide the semi-major axis coordinates. In this case,

\[ \begin{align}\begin{aligned}\begin{split}{\mathbf C} = \left[ \begin{array}{rrrr} \cos\psi & \sin\psi & -1 & 0 \\ -\sin\psi & \cos\psi & 0 & -1 \\ 0 & 0 & \sin\phi_0 & \cos\phi_0 \\ 0 & 0 & -\cos\phi_0 & \sin\phi_0 \end{array} \right]\end{split}\\\begin{split}{\mathbf d} = \left[ \begin{array}{r} -x_0 \\ -y_0 \\ x_a \\ y_a (1-\varepsilon) \end{array} \right]\end{split}\end{aligned}\end{align} \]

such that

\[\begin{split}{\mathbf f} = \left[ \begin{array}{r} x_f \\ y_f \\ x_s \\ y_s \end{array} \right]\end{split}\]

and \({\mathbf f} = {\mathbf C}^{-1}\ {\mathbf d}\).

Parameters

• xc (float) – Same as \(x_0\), defined above

• yc (float) – Same as \(y_0\), defined above

• rot (float) – Same as \(\psi\), defined above

• pa (float) – Same as \(\phi_0\), defined above

• ell (float) – Same as \(\varepsilon\), defined above

xc,yc

a reference on-sky position relative to the center of the ellipse (galaxy center); same as \((x_0,y_0)\) defined above

Type

float,float

rot

Cartesian rotation of the focal-plane relative to the sky-plane (+x toward East; +y toward North); same as \(\psi\) defined above

Type

float

pa

On-sky position angle of the major axis of the ellipse, defined as the angle from North through East and is the same as \(\phi_0\) defined above

Type

float

ell

Ellipticity define as \(\varepsilon=1-b/a\), based on the semi-minor to semi-major axis ratio (\(b/a\)) of the ellipse.

Type

float

A

The coordinate transformation matrix

Type

numpy.ndarray

Alu

The lu array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf A}\)

Type

numpy.ndarray

Apiv

The piv array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf A}\)

Type

numpy.ndarray

B

The vector \({\mathbf b}\), as defined above, used to calculate \({\mathbf x} = {\mathbf A}^{-1}\ {\mathbf b}\)

Type

numpy.ndarray

C

The coordinate transformation matrix use for the inverse operations

Type

numpy.ndarray

Clu

The lu array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf C}\)

Type

numpy.ndarray

Cpiv

The piv array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf C}\)

Type

numpy.ndarray

D

The vector \({\mathbf d}\), as defined above, used to calculate \({\mathbf f} = {\mathbf C}^{-1}\ {\mathbf d}\)

Type

numpy.ndarray

_calculate_cartesian(r, theta)

Invert the calculation of the semi-major-axis polar coordinates to calculate the semi-major-axis Cartesian coordinates \((x_a,y_a)\) using

\[\begin{split}x_a &= \pm R / \sqrt{1 + \tan^2\theta}\\ y_a &= -x_a\ \tan\theta\end{split}\]

where \(x_a\) is negative when \(\pi/2 \leq \theta < 3\pi/2\).

Parameters

• r (array-like) – The semi-major-axis polar coordinates \((R,\theta)\).

• theta (array-like) – The semi-major-axis polar coordinates \((R,\theta)\).

Returns

The semi-major-axis Cartesian coordinates: \(x_a, y_a\).

Return type

numpy.ndarray

_calculate_polar(x, y)

Calculate the polar coordinates (radius and azimuth) provided the Cartesian semi-major-axis coordinates \((x_a,y_a)\) using

\[\begin{split}R &= \sqrt{x_a^2 + y_a^2} \\ \theta &= \tan^{-1}\left(\frac{-y_a}{x_a}\right)\end{split}\]

Parameters

• x (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

• y (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

Returns

The semi-major-axis polar coordinates: \(R, \theta\).

Return type

numpy.ndarray

_defined()

Determine if the object is defined such that its methods can be used to convert between coordinate systems.

_get_B(x, y)

Set the on-sky coordinate vector for forward operations.

Parameters

• x (numpy.ndarray) – On-sky Cartesian coordinate.

• y (numpy.ndarray) – On-sky Cartesian coordinate.

Returns

Array prepared for the matrix solution.

Return type

numpy.ndarray

_get_D(x, y)

Set the semi-major-axis coordinate vector for inverse operations.

Parameters

• x (numpy.ndarray) – Semi-major axis Cartesian coordinate.

• y (numpy.ndarray) – Semi-major axis Cartesian coordinate.

Returns

Array prepared for the matrix solution.

Return type

numpy.ndarray

_setA()

Set the transformation matrix and calculate its LU decomposition for forward operations.

_setC()

Set the transformation matrix and calculate its LU decomposition for inverse operations.

cartesian(x, y)

Calculate \({\mathbf x}\) using solve() for the provided \((x_f,y_f)\) and return the semi-major-axis Cartesian and coordinates, \((x_a,y_a)\).

Parameters

• x (array-like) – The coordinate \((x_f,y_f)\), which is the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

• y (array-like) – The coordinate \((x_f,y_f)\), which is the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

Returns

Two arrays with the semi-major-axis Cartesian coordinates, \(x_a, y_a\).

Return type

numpy.ndarray

cartesian_invert(x, y)

Calculate \({\mathbf f}\) using solve() for the provided \((x_a,y_a)\) and return focal-plane cartesian coordinates \((x_f,y_f)\).

Parameters

• x (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

• y (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

Returns

The focal-plane Cartesian coordinates \((x_f,y_f)\).

Return type

numpy.ndarray

coo(x, y)

Calculate \({\mathbf x}\) using solve() for the provided \((x_f,y_f)\) and return the semi-major-axis Cartesian and polar coordinates, \((x_a,y_a)\) and \((R,\theta)\). This combines the functionality of cartesian() and polar(), and so is more efficient than using these both separately.

Parameters

• x (array-like) – The coordinates \((x_f,y_f)\), which are the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

• y (array-like) – The coordinates \((x_f,y_f)\), which are the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

Returns

Four arrays with the semi-major-axis Cartesian and polar coordinates: \(x_a, y_a, R, \theta\).

Return type

numpy.ndarray

polar(x, y)

Calculate \({\mathbf x}\) using solve() for the provided \((x_f,y_f)\) and return the semi-major-axis polar coordinates, \((R,\theta)\), where

\[\begin{split}R &= \sqrt{x_a^2 + y_a^2} \\ \theta &= \tan^{-1}\left(\frac{-y_a}{x_a}\right)\end{split}\]

Parameters

• x (array-like) – The coordinate \((x_f,y_f)\), which is the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

• y (array-like) – The coordinate \((x_f,y_f)\), which is the sky-right, focal-plane position relative to a reference on-sky position \((x_0,y_0)\) relative to the center of the ellipse (galaxy center),

Returns

Two arrays with the semi-major-axis polar coordinates: \(R, \theta\).

Return type

numpy.ndarray

polar_invert(r, theta)

Calculate \({\mathbf f}\) using solve() for the provided \((R,\theta)\) and return focal-plane cartesian coordinates \((x_f,y_f)\).

Parameters

• r (array-like) – The semi-major-axis polar coordinates \((R,\theta)\).

• theta (array-like) – The semi-major-axis polar coordinates \((R,\theta)\).

Returns

Two arrays with the focal-plane Cartesian coordinates \((x_f,y_f)\).

Return type

numpy.ndarray

solve(x, y)

Use scipy.linalg.lu_solve to solve \({\mathbf x} = {\mathbf A}^{-1}\ {\mathbf b}\).

Parameters

• x (array-like) – The coordinates \((x_f,y_f)\), which are the sky-right, focal-plane Cartesian coordinates relative to a reference on-sky position \((x_0,y_0)\), which is relative to the center of the ellipse (galaxy center).

• y (array-like) – The coordinates \((x_f,y_f)\), which are the sky-right, focal-plane Cartesian coordinates relative to a reference on-sky position \((x_0,y_0)\), which is relative to the center of the ellipse (galaxy center).

Returns

The \({\mathbf x}\) vectors (separated by rows) as defined by the solution to \({\mathbf A}^{-1}\ {\mathbf b}\)

Return type

numpy.ndarray

Raises

ValueError – Raised if object was not properly defined or if the X and Y arrays do not have the same size.

solve_inverse(x, y)

Use scipy.linalg.lu_solve to solve \({\mathbf f} = {\mathbf C}^{-1}\ {\mathbf d}\).

Parameters

• x (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

• y (array-like) – The semi-major-axis Cartesian coordinates \((x_a,y_a)\).

Returns

The \({\mathbf f}\) vector as defined by the solution to \({\mathbf C}^{-1}\ {\mathbf d}\)

Return type

numpy.ndarray

Raises

ValueError – Raised if object was not properly defined or if the X and Y arrays do not have the same size.