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LSST Data Management Base Package
|
Functions | |
| covar_to_ellipse (sigma_x_sq, sigma_y_sq, cov_xy, degrees=False) | |
| gauss2dint (xdivsigma) | |
| lsst.gauss2d.utils.covar_to_ellipse | ( | sigma_x_sq, | |
| sigma_y_sq, | |||
| cov_xy, | |||
| degrees = False ) |
Convert covariance matrix terms to ellipse major axis, axis ratio and
position angle representation.
Parameters
----------
sigma_x_sq, sigma_y_sq : `float` or array-like
x- and y-axis squared standard deviations of a 2-dimensional normal
distribution (diagonal terms of its covariance matrix).
Must be scalar or identical length array-likes.
cov_xy : `float` or array-like
x-y covariance of a of a 2-dimensional normal distribution
(off-diagonal term of its covariance matrix).
Must be scalar or identical length array-likes.
degrees : `bool`
Whether to return the position angle in degrees instead of radians.
Returns
-------
r_major, axrat, angle : `float` or array-like
Converted major-axis radius, axis ratio and position angle
(counter-clockwise from the +x axis) of the ellipse defined by
each set of input covariance matrix terms.
Notes
-----
The eigenvalues from the determinant of a covariance matrix are:
|a-m b|
|b c-m|
det = (a-m)(c-m) - b^2 = ac - (a+c)m + m^2 - b^2 = m^2 - (a+c)m + (ac-b^2)
Solving:
m = ((a+c) +/- sqrt((a+c)^2 - 4(ac-b^2)))/2
...or equivalently:
m = ((a+c) +/- sqrt((a-c)^2 + 4b^2))/2
Unfortunately, the latter simplification is not as well-behaved
in floating point math, leading to square roots of negative numbers when
one of a or c is very close to zero.
The values from this function should match those from
`Ellipse.make_ellipse_major` to within rounding error, except in the
special case of sigma_x == sigma_y == 0, which returns a NaN axis ratio
here by default. This function mainly intended to be more convenient
(and possibly faster) for array-like inputs.
Definition at line 25 of file utils.py.
| lsst.gauss2d.utils.gauss2dint | ( | xdivsigma | ) |
Return the fraction of the total surface integral of a 2D Gaussian
contained with a given multiple of its dispersion.
Parameters
----------
xdivsigma : `float`
The multiple of the dispersion to integrate to.
Returns
-------
frac : `float`
The fraction of the surface integral contained within an ellipse of
size `xdivsigma`.
Notes
-----
This solution can be computed as follows:
https://www.wolframalpha.com/input/?i=
Integrate+2*pi*x*exp(-x%5E2%2F(2*s%5E2))%2F(s*sqrt(2*pi))+dx+from+0+to+r
Definition at line 81 of file utils.py.