mangadap.util.geometry module¶
Provides a set of utility functions dealing with computational geometry.
Copyright © 2019, SDSS-IV/MaNGA Pipeline Group
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class
mangadap.util.geometry.
SemiMajorAxisCoo
(xc=None, yc=None, rot=None, pa=None, ell=None)[source]¶ 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
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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
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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
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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
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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
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A
¶ The coordinate transformation matrix
- Type
numpy.ndarray
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Alu
¶ The lu array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf A}\)
- Type
numpy.ndarray
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Apiv
¶ The piv array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf A}\)
- Type
numpy.ndarray
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B
¶ The vector \({\mathbf b}\), as defined above, used to calculate \({\mathbf x} = {\mathbf A}^{-1}\ {\mathbf b}\)
- Type
numpy.ndarray
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C
¶ The coordinate transformation matrix use for the inverse operations
- Type
numpy.ndarray
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Clu
¶ The lu array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf C}\)
- Type
numpy.ndarray
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Cpiv
¶ The piv array returned by scipy.linalg.lu_factor, which is used to calculate the LU decomposition of \({\mathbf C}\)
- Type
numpy.ndarray
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D
¶ The vector \({\mathbf d}\), as defined above, used to calculate \({\mathbf f} = {\mathbf C}^{-1}\ {\mathbf d}\)
- Type
numpy.ndarray
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_calculate_cartesian
(r, theta)[source]¶ 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
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_calculate_polar
(x, y)[source]¶ 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
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_defined
()[source]¶ Determine if the object is defined such that its methods can be used to convert between coordinate systems.
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_setA
()[source]¶ Set the transformation matrix and calculate its LU decomposition for forward operations.
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_setB
(x, y)[source]¶ Set the on-sky coordinate vector for forward operations.
- Parameters
x (float) – Single values for use in calculating the semi-major-axis coordinates.
y (float) – Single values for use in calculating the semi-major-axis coordinates.
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_setC
()[source]¶ Set the transformation matrix and calculate its LU decomposition for inverse operations.
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_setD
(x, y)[source]¶ Set the semi-major-axis coordinate vector for inverse operations.
- Parameters
x (float) – Single values for use in calculating the on-sky focal plane coordinates.
y (float) – Single values for use in calculating the on-sky focal plane coordinates.
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cartesian
(x, y)[source]¶ 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
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cartesian_invert
(x, y)[source]¶ 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
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coo
(x, y)[source]¶ 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 ofcartesian()
andpolar()
, 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
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polar
(x, y)[source]¶ 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
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polar_invert
(r, theta)[source]¶ 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
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solve
(x, y)[source]¶ 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.
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solve_inverse
(x, y)[source]¶ 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.
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mangadap.util.geometry.
point_inside_polygon
(polygon, point)[source]¶ Determine if a point is inside a polygon using the winding number.
- Parameters
polygon (numpy.ndarray) – An Nx2 array containing the x,y coordinates of a polygon. The points should be ordered either counter-clockwise or clockwise.
point (numpy.ndarray) – A 2-element array defining the x,y position of the point to use as a reference for the winding number.
- Returns
True if the point is inside the polygon.
- Return type
bool
Warning
If the point is on the polygon (or very close to it w.r.t. the machine precision), the returned value is False.
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mangadap.util.geometry.
polygon_area
(x, y)[source]¶ Compute the area of a polygon using the Shoelace formula.
Inspired by this discussion.
- Parameters
x (numpy.ndarray) – Vector with the Cartesian x-coordinates of the polygon vertices.
y (numpy.ndarray) – Vector with the Cartesian y-coordinates of the polygon vertices.
- Returns
Polygon area
- Return type
float
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mangadap.util.geometry.
polygon_winding_number
(polygon, point)[source]¶ Determine the winding number of a 2D polygon about a point. The code does not check if the polygon is simple (no interesecting line segments). Algorithm taken from Numerical Recipies Section 21.4.
- Parameters
polygon (numpy.ndarray) – An Nx2 array containing the x,y coordinates of a polygon. The points should be ordered either counter-clockwise or clockwise.
point (numpy.ndarray) – A 2-element array defining the x,y position of the point to use as a reference for the winding number.
- Returns
Winding number of polygon w.r.t. point
- Return type
int
- Raises
ValueError – Raised if polygon is not 2D, if polygon does not have two columns, or if point is not a 2-element array.