[c@`sdZddlmZmZmZmZddlmZddlZ ddl m Z m Z m Z mZddlmZmZddlmZd d lmZd d lmZd d lmZd d lmZmZd dlmZdddddddddddddddddd d!d"d#d$d%d&d'd(d)gZd e j Z!e j"e j#e j$j%Z&d*e fd+YZ'd,e'fd-YZ(ed.d/d0d1e'fd2YZ)d3e fd4YZ*d5e fd6YZ+d7e fd8YZ,d9e fd:YZ-d;e fd<YZ.d=e fd>YZ/d?e fd@YZ0dAe fdBYZ1dCe fdDYZ2dEe fdFYZ3dGe fdHYZ4dIe fdJYZ5dKe fdLYZ6dMe fdNYZ7dOe fdPYZ8dQe fdRYZ9dSe fdTYZ:dUe fdVYZ;dWe fdXYZ<dYe fdZYZ=d[e fd\YZ>d]e fd^YZ?d_e fd`YZ@dae fdbYZAdce fddYZBdee fdfYZCdge fdhYZDdS(iuMathematical models.i(tabsolute_importtunicode_literalstdivisiontprint_function(t OrderedDictNi(tFittable1DModeltFittable2DModeltModeltModelDefinitionError(t ParametertInputParameterError(tellipse_extenti(tmap(tgaussian_sigma_to_fwhm(tunits(tQuantityt UnitsError(t deprecatedu AiryDisk2DuMoffat1DuMoffat2DuBox1DuBox2DuConst1DuConst2Du Ellipse2DuDisk2DuBaseGaussian1Du Gaussian1DuGaussianAbsorption1Du Gaussian2DuLinear1Du Lorentz1Du MexicanHat1Du MexicanHat2DuRedshiftScaleFactoruScaleuSersic1DuSersic2DuShiftuSine1Du Trapezoid1DuTrapezoidDisk2DuRing2DuVoigt1DtBaseGaussian1DcB`sbeZdZeddZeddZedddedfZddZ e dZ RS( uBase class for 1D Gaussian models. Do not use this directly. See Also -------- Gaussian1D, GaussianAbsorption1D tdefaultiitboundsg@cC`s(|j}||j}||||fS(u Tuple defining the default ``bounding_box`` limits, ``(x_low, x_high)`` Parameters ---------- factor : float The multiple of `stddev` used to define the limits. The default is 5.5, corresponding to a relative error < 1e-7. Examples -------- >>> from astropy.modeling.models import Gaussian1D >>> model = Gaussian1D(mean=0, stddev=2) >>> model.bounding_box (-11.0, 11.0) This range can be set directly (see: `Model.bounding_box `) or by using a different factor, like: >>> model.bounding_box = model.bounding_box(factor=2) >>> model.bounding_box (-4.0, 4.0) (tmeantstddev(tselftfactortx0tdx((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyt bounding_box/s  cC`s |jtS(u$Gaussian full width at half maximum.(RR (R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pytfwhmOsN( t__name__t __module__t__doc__R t amplitudeRt FLOAT_EPSILONtNoneRRtpropertyR(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR s  t Gaussian1DcB`sDeZdZedZedZedZdZRS(u One dimensional Gaussian model. Parameters ---------- amplitude : float Amplitude of the Gaussian. mean : float Mean of the Gaussian. stddev : float Standard deviation of the Gaussian. Notes ----- Model formula: .. math:: f(x) = A e^{- \frac{\left(x - x_{0}\right)^{2}}{2 \sigma^{2}}} Examples -------- >>> from astropy.modeling import models >>> def tie_center(model): ... mean = 50 * model.stddev ... return mean >>> tied_parameters = {'mean': tie_center} Specify that 'mean' is a tied parameter in one of two ways: >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3, ... tied=tied_parameters) or >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3) >>> g1.mean.tied False >>> g1.mean.tied = tie_center >>> g1.mean.tied Fixed parameters: >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3, ... fixed={'stddev': True}) >>> g1.stddev.fixed True or >>> g1 = models.Gaussian1D(amplitude=10, mean=5, stddev=.3) >>> g1.stddev.fixed False >>> g1.stddev.fixed = True >>> g1.stddev.fixed True .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Gaussian1D plt.figure() s1 = Gaussian1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() See Also -------- Gaussian2D, Box1D, Moffat1D, Lorentz1D cC`s%|tjd||d|dS(u, Gaussian1D model function. gi(tnptexp(txR RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pytevaluatescC`shtjd|d||d}|||||d}||||d|d}|||gS(u8 Gaussian1D model function derivatives. gii(R%R&(R'R RRt d_amplitudetd_meantd_stddev((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyt fit_derivs#cC`s+|jjdkrdSi|jjd6SdS(Nux(RtunitR"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyt input_unitsscC`s1td|dfd|dfd|dfgS(Numeanuxustddevu amplitudeuy(R(Rt inputs_unitt outputs_unit((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyt_parameter_units_for_data_unitss ( RRRt staticmethodR(R,R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR$Us P u2.0t alternativeu&Gaussian1D and subtract it off Const1DtGaussianAbsorption1DcB`s5eZdZdZedZedZRS(ua One dimensional Gaussian absorption line model. Parameters ---------- amplitude : float Amplitude of the gaussian absorption. mean : float Mean of the gaussian. stddev : float Standard deviation of the gaussian. Notes ----- Model formula: .. math:: f(x) = 1 - A e^{- \frac{\left(x - x_{0}\right)^{2}}{2 \sigma^{2}}} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt import warnings from astropy.modeling.models import GaussianAbsorption1D from astropy.utils.exceptions import AstropyDeprecationWarning plt.figure() with warnings.catch_warnings(): warnings.simplefilter('ignore', AstropyDeprecationWarning) s1 = GaussianAbsorption1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -3, 2]) plt.show() See Also -------- Gaussian1D cO`stt|j||dS(N(tsuperR4t__init__(Rtargstkwargs((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR6scC`sdtj||||S(u6 GaussianAbsorption1D model function. g?(R$R((R'R RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`s4ddl}tt|jtj||||S(uB GaussianAbsorption1D model function derivatives. iN(toperatortlistR tnegR$R,(R'R RRR9((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,s (RRRR6R2R(R,(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR4s/ t Gaussian2DcB`seZdZeddZeddZeddZeddZeddZeddZ ej ej ej dddddZ e dZe dZdd Zed Zed Ze d Zd ZRS(ul Two dimensional Gaussian model. Parameters ---------- amplitude : float Amplitude of the Gaussian. x_mean : float Mean of the Gaussian in x. y_mean : float Mean of the Gaussian in y. x_stddev : float or None Standard deviation of the Gaussian in x before rotating by theta. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (1). y_stddev : float or None Standard deviation of the Gaussian in y before rotating by theta. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (1). theta : float, optional Rotation angle in radians. The rotation angle increases counterclockwise. Must be None if a covariance matrix (``cov_matrix``) is provided. If no ``cov_matrix`` is given, ``None`` means the default value (0). cov_matrix : ndarray, optional A 2x2 covariance matrix. If specified, overrides the ``x_stddev``, ``y_stddev``, and ``theta`` defaults. Notes ----- Model formula: .. math:: f(x, y) = A e^{-a\left(x - x_{0}\right)^{2} -b\left(x - x_{0}\right) \left(y - y_{0}\right) -c\left(y - y_{0}\right)^{2}} Using the following definitions: .. math:: a = \left(\frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right) b = \left(\frac{\sin{\left (2 \theta \right )}}{2 \sigma_{x}^{2}} - \frac{\sin{\left (2 \theta \right )}}{2 \sigma_{y}^{2}}\right) c = \left(\frac{\sin^{2}{\left (\theta \right )}}{2 \sigma_{x}^{2}} + \frac{\cos^{2}{\left (\theta \right )}}{2 \sigma_{y}^{2}}\right) If using a ``cov_matrix``, the model is of the form: .. math:: f(x, y) = A e^{-0.5 \left(\vec{x} - \vec{x}_{0}\right)^{T} \Sigma^{-1} \left(\vec{x} - \vec{x}_{0}\right)} where :math:`\vec{x} = [x, y]`, :math:`\vec{x}_{0} = [x_{0}, y_{0}]`, and :math:`\Sigma` is the covariance matrix: .. math:: \Sigma = \left(\begin{array}{ccc} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{array}\right) :math:`\rho` is the correlation between ``x`` and ``y``, which should be between -1 and +1. Positive correlation corresponds to a ``theta`` in the range 0 to 90 degrees. Negative correlation corresponds to a ``theta`` in the range of 0 to -90 degrees. See [1]_ for more details about the 2D Gaussian function. See Also -------- Gaussian1D, Box2D, Moffat2D References ---------- .. [1] https://en.wikipedia.org/wiki/Gaussian_function Riigc K`s|dkri|dkr*|jjj}n|dkrH|jjj}n|dkr&|jjj}q&n|dk s|dk s|dk rtdntj|}|j dkrt dntj j |\} } tj | \}}| dddf} tj| d| d}|jdi|djdtdf|djdtdftt|jd |d |d |d |d |d||dS(Nu3Cannot specify both cov_matrix and x/y_stddev/thetaiuCovariance matrix must be 2x2iiuboundsux_stddevuy_stddevR tx_meanty_meantx_stddevty_stddevttheta(ii(R"t __class__R?RR@RAR R%tarraytshapet ValueErrortlinalgteigtsqrttarctan2t setdefaultR!R5R<R6( RR R=R>R?R@RAt cov_matrixR8teig_valsteig_vecsty_vec((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR6cs,    $cC`s |jtS(u)Gaussian full width at half maximum in X.(R?R (R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pytx_fwhmscC`s |jtS(u)Gaussian full width at half maximum in Y.(R@R (R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyty_fwhmsg@cC`sp||j}||j}|jj}t|||\}}|j||j|f|j||j|ffS(u Tuple defining the default ``bounding_box`` limits in each dimension, ``((y_low, y_high), (x_low, x_high))`` The default offset from the mean is 5.5-sigma, corresponding to a relative error < 1e-7. The limits are adjusted for rotation. Parameters ---------- factor : float, optional The multiple of `x_stddev` and `y_stddev` used to define the limits. The default is 5.5. Examples -------- >>> from astropy.modeling.models import Gaussian2D >>> model = Gaussian2D(x_mean=0, y_mean=0, x_stddev=1, y_stddev=2) >>> model.bounding_box ((-11.0, 11.0), (-5.5, 5.5)) This range can be set directly (see: `Model.bounding_box `) or by using a different factor like: >>> model.bounding_box = model.bounding_box(factor=2) >>> model.bounding_box ((-4.0, 4.0), (-2.0, 2.0)) (R?R@RAtvalueR R>R=(RRtatbRARtdy((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs    cC`stj|d}tj|d} tjd|} |d} |d} ||} ||}d|| | | }d| | | | }d| | || }|tj|| d|| |||d S(u!Two dimensional Gaussian functionig@g?(R%tcostsinR&(R'tyR R=R>R?R@RAtcost2tsint2tsin2ttxstd2tystd2txdifftydiffRRRStc((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s     c)C`sYtj|}tj|} tj|d} tj|d} tjd|} tjd|} |d}|d}|d}|d}||}||}|d}|d}d| || |}d| || |}d| || |}|tj||||||| }| |d|d|}| |}| |}| || |}| |}| |}| } | |}!| |}"||}#|d||||}$|||d||}%|||||||!| }&|||||||"| }'||||||| | }(|#|$|%|&|'|(gS(uGTwo dimensional Gaussian function derivative with respect to parametersig@ig?g?(R%RURVR&()R'RWR R=R>R?R@RAtcosttsintRXRYtcos2tRZR[R\txstd3tystd3R]R^txdiff2tydiff2RRRSR_tgt da_dthetat da_dx_stddevt da_dy_stddevt db_dthetat db_dx_stddevt db_dy_stddevt dc_dthetat dc_dx_stddevt dc_dy_stddevtdg_dAt dg_dx_meant dg_dy_meant dg_dx_stddevt dg_dy_stddevt dg_dtheta((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,sT                  cC`sJ|jjdkr(|jjdkr(dSi|jjd6|jjd6SdS(Nuxuy(R=R-R"R>(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.s$c C`sz|d|dkr#tdntd|dfd|dfd|dfd|dfdtjfd |d fgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_meanuy_meanux_stddevuy_stddevuthetau amplitudeuz(RRtutrad(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s    N(RRRR R R=R>R?R@RARR"R6R#RORPRR2R(R,R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR< s"M  ( &/tShiftcB`seZdZd Zd ZeddZeZeZ eZ e dZ e dZ edZedZedZRS( uv Shift a coordinate. Parameters ---------- offset : float Offset to add to a coordinate. uxRicC`s+|jjdkrdSi|jjd6SdS(Nux(toffsetR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.'scC`s|j}|jd9_|S(u,One dimensional inverse Shift model functioni(tcopyRz(Rtinv((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pytinverse.s cC`sZt|tjr'|j}|j}nt|tjrN|j}|||S||SdS(u$One dimensional Shift model functionN(t isinstanceRwRR-RQ(R'Rzt return_unit((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(5s    cC`s|S(u=Evaluate the implicit term (x) of one dimensional Shift model((R'((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pytsum_of_implicit_termsAscG`stj|}|gS(u@One dimensional Shift model derivative with respect to parameter(R%t ones_like(R'tparamstd_offset((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,Fs(ux(ux(RRRtinputstoutputsR RztTruetlineartinput_units_stricttinput_units_allow_dimensionlessR#R.R}R2R(RR,(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRys tScalecB`s}eZdZdZd ZeddZeZeZ eZ eZ e dZ e dZedZedZRS( u Multiply a model by a factor. Parameters ---------- factor : float Factor by which to scale a coordinate. uxRicC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.cscC`s |j}d|j|_|S(u,One dimensional inverse Scale model functioni(R{R(RR|((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR}js cC`sTt|tjr'|j}|j}nt|tjrH|j||S||SdS(u$One dimensional Scale model functionN(R~RwRR-RQ(R'RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(qs   cG`s |}|gS(u@One dimensional Scale model derivative with respect to parameter((R'Rtd_factor((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,|s(ux(ux(RRRRRR RRRtfittableRRR#R.R}R2R(R,(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRNs tRedshiftScaleFactorcB`sPeZdZeddddZedZedZedZ RS(u One dimensional redshift scale factor model. Parameters ---------- z : float Redshift value. Notes ----- Model formula: .. math:: f(x) = x (1 + z) t descriptionuredshiftRicC`s d||S(u2One dimensional RedshiftScaleFactor model functioni((R'tz((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`s |}|gS(u4One dimensional RedshiftScaleFactor model derivative((R'Rtd_z((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,scC`s(|j}dd|jd|_|S(u!Inverse RedshiftScaleFactor modelg?(R{R(RR|((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR}s ( RRRR RR2R(R,R#R}(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs tSersic1DcB`sheZdZeddZeddZeddZdZe dZ e dZ dZ RS(u One dimensional Sersic surface brightness profile. Parameters ---------- amplitude : float Surface brightness at r_eff. r_eff : float Effective (half-light) radius n : float Sersic Index. See Also -------- Gaussian1D, Moffat1D, Lorentz1D Notes ----- Model formula: .. math:: I(r)=I_e\exp\left\{-b_n\left[\left(\frac{r}{r_{e}}\right)^{(1/n)}-1\right]\right\} The constant :math:`b_n` is defined such that :math:`r_e` contains half the total luminosity, and can be solved for numerically. .. math:: \Gamma(2n) = 2\gamma (b_n,2n) Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Sersic1D import matplotlib.pyplot as plt plt.figure() plt.subplot(111, xscale='log', yscale='log') s1 = Sersic1D(amplitude=1, r_eff=5) r=np.arange(0, 100, .01) for n in range(1, 10): s1.n = n plt.plot(r, s1(r), color=str(float(n) / 15)) plt.axis([1e-1, 30, 1e-2, 1e3]) plt.xlabel('log Radius') plt.ylabel('log Surface Brightness') plt.text(.25, 1.5, 'n=1') plt.text(.25, 300, 'n=10') plt.xticks([]) plt.yticks([]) plt.show() References ---------- .. [1] http://ned.ipac.caltech.edu/level5/March05/Graham/Graham2.html RiicC`s|jdkrOyddlm}||_WqOtk rKtdqOXn|tj|jd|d ||d|dS(u(One dimensional Sersic profile function.i(t gammaincinvu%Sersic1D model requires scipy > 0.11.ig?iN(t _gammaincinvR"t scipy.specialRREt ImportErrorR%R&(tclstrR tr_efftnR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s   cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`s$td|dfd|dfgS(Nur_effuxu amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1sN(RRRR R RRR"Rt classmethodR(R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs>tSine1DcB`sqeZdZeddZeddZeddZedZedZ e dZ dZ RS(uL One dimensional Sine model. Parameters ---------- amplitude : float Oscillation amplitude frequency : float Oscillation frequency phase : float Oscillation phase See Also -------- Const1D, Linear1D Notes ----- Model formula: .. math:: f(x) = A \sin(2 \pi f x + 2 \pi p) Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Sine1D plt.figure() s1 = Sine1D(amplitude=1, frequency=.25) r=np.arange(0, 10, .01) for amplitude in range(1,4): s1.amplitude = amplitude plt.plot(r, s1(r), color=str(0.25 * amplitude), lw=2) plt.axis([0, 10, -5, 5]) plt.show() RiicC`s>t|||}t|tr-|j}n|tj|S(u#One dimensional Sine model function(tTWOPIR~RRQR%RV(R'R t frequencytphasetargument((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(=s cC`s~tjt||t|}t||tjt||t|}t|tjt||t|}|||gS(u%One dimensional Sine model derivative(R%RVRRU(R'R RRR)t d_frequencytd_phase((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,Js    cC`s/|jjdkrdSid|jjd6SdS(Ng?ux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.UscC`s(td|ddfd|dfgS(Nu frequencyuxiu amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1\s( RRRR R RRR2R(R,R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR s,  tLinear1DcB`sweZdZeddZeddZeZedZ edZ e dZ e dZ dZRS( u( One dimensional Line model. Parameters ---------- slope : float Slope of the straight line intercept : float Intercept of the straight line See Also -------- Const1D Notes ----- Model formula: .. math:: f(x) = a x + b RiicC`s |||S(u#One dimensional Line model function((R'tslopet intercept((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(|scC`s|}tj|}||gS(u@One dimensional Line model derivative with respect to parameters(R%R(R'RRtd_slopet d_intercept((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,scC`s4|jd}|j |j}|jd|d|S(NiRR(RRRB(Rt new_slopet new_intercept((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR}s cC`sG|jjdkr(|jjdkr(dSi|jj|jjd6SdS(Nux(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.s$cC`s,td|dfd|d|dfgS(Nu interceptuyuslopeux(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s(RRRR RRRRR2R(R,R#R}R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRastPlanar2DcB`s_eZdZeddZeddZeddZeZe dZ e dZ RS(u Two dimensional Plane model. Parameters ---------- slope_x : float Slope of the straight line in X slope_y : float Slope of the straight line in Y intercept : float Z-intercept of the straight line See Also -------- Linear1D, Polynomial2D Notes ----- Model formula: .. math:: f(x, y) = a x + b y + c RiicC`s|||||S(u$Two dimensional Plane model function((R'RWtslope_xtslope_yR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`s(|}|}tj|}|||gS(uATwo dimensional Plane model derivative with respect to parameters(R%R(R'RWRRRt d_slope_xt d_slope_yR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,s( RRRR RRRRRR2R(R,(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRst Lorentz1DcB`s}eZdZeddZeddZeddZedZedZ ddZ e dZ d Z RS( u_ One dimensional Lorentzian model. Parameters ---------- amplitude : float Peak value x_0 : float Position of the peak fwhm : float Full width at half maximum See Also -------- Gaussian1D, Box1D, MexicanHat1D Notes ----- Model formula: .. math:: f(x) = \frac{A \gamma^{2}}{\gamma^{2} + \left(x - x_{0}\right)^{2}} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Lorentz1D plt.figure() s1 = Lorentz1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() RiicC`s(||dd||d|ddS(u)One dimensional Lorentzian model functiong@i((R'R tx_0R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`ss|d|d||d}||d|d||d||d}d|||d|}|||gS(uFOne dimensional Lorentzian model derivative with respect to parametersii((R'R RRR)td_x_0td_fwhm((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,s icC`s(|j}||j}||||fS(urTuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of FWHM used to define the limits. Default is chosen to include most (99%) of the area under the curve, while still showing the central feature of interest. (RR(RRRR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs  cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.!scC`s1td|dfd|dfd|dfgS(Nux_0uxufwhmu amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1(s (RRRR R RRR2R(R,RR#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs- tVoigt1DcB`seZdZeddZeddZeddejZedej dZ ej ddddgddd d gd d d d gd dddggZ e dZe dZedZdZRS(u! One dimensional model for the Voigt profile. Parameters ---------- x_0 : float Position of the peak amplitude_L : float The Lorentzian amplitude fwhm_L : float The Lorentzian full width at half maximum fwhm_G : float The Gaussian full width at half maximum See Also -------- Gaussian1D, Lorentz1D Notes ----- Algorithm for the computation taken from McLean, A. B., Mitchell, C. E. J. & Swanston, D. M. Implementation of an efficient analytical approximation to the Voigt function for photoemission lineshape analysis. Journal of Electron Spectroscopy and Related Phenomena 69, 125-132 (1994) Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Voigt1D import matplotlib.pyplot as plt plt.figure() x = np.arange(0, 10, 0.01) v1 = Voigt1D(x_0=5, amplitude_L=10, fwhm_L=0.5, fwhm_G=0.9) plt.plot(x, v1(x)) plt.show() Riiigq= ףpgQIg??g~:?g?g~:ؿgX9vӿg.1?g/$?g~k g/$g~k ?cC`s|j\}}}} tjtjd} ||d| |} tj| dtjf} || |} tj| dtjf} tj|| || | || |d| |ddd} ||tjtj| || S(Ni.taxisi(t_abcdR%RHtlogt atleast_1dtnewaxistsumtpi(RR'Rt amplitude_Ltfwhm_Ltfwhm_GtAtBtCtDtsqrt_ln2tXtYtV((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(dsAc C`s|j\}}}} tjtjd} ||d| |} tj| ddtjf} || |} tj| ddtjf} ||tjtj| |} || || | |}| |d| |d}tj||dd}tj| |d| ||tj|dd}tj||d| ||tj|dd}| |d| || ||| |||| || || |d||||||g}|S(NiRi( RR%RHRRRRRtsquare(RR'RRRRRRRRRRRtconstanttalphatbetaRtdVdxtdVdytdyda((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,rs""""66 2cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`s>td|dfd|dfd|dfd|dfgS(Nux_0uxufwhm_Lufwhm_Gu amplitude_Luy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s  (RRRR RRR%RRRRRCRRR(R,R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.s)tConst1DcB`sYeZdZeddZeZedZedZ e dZ dZ RS(u One dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const2D Notes ----- Model formula: .. math:: f(x) = A Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Const1D plt.figure() s1 = Const1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() RicC`s|jdkr:tj|dt}|j|jn|tj|dt}t|tr{t|d|j dtS|SdS(u'One dimensional Constant model functionitsubokR-R{N( tsizeR%t empty_liketFalsetfilltitemRR~RR-(R'R ((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`stj|}|gS(uDOne dimensional Constant model derivative with respect to parameters(R%R(R'R R)((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,scC`sdS(N(R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`std|dfgS(Nu amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s( RRRR R RRR2R(R,R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs'tConst2DcB`sJeZdZeddZeZedZe dZ dZ RS(u Two dimensional Constant model. Parameters ---------- amplitude : float Value of the constant function See Also -------- Const1D Notes ----- Model formula: .. math:: f(x, y) = A RicC`s|jdkr:tj|dt}|j|jn|tj|dt}t|tr{t|d|j dtS|SdS(u'Two dimensional Constant model functioniRR-R{N( RR%RRRRRR~RR-(R'RWR ((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(scC`sdS(N(R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR. scC`std|dfgS(Nu amplitudeuz(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s( RRRR R RRR2R(R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs t Ellipse2DcB`seZdZeddZeddZeddZeddZeddZeddZ e dZ e dZ e dZdZRS(uA A 2D Ellipse model. Parameters ---------- amplitude : float Value of the ellipse. x_0 : float x position of the center of the disk. y_0 : float y position of the center of the disk. a : float The length of the semimajor axis. b : float The length of the semiminor axis. theta : float The rotation angle in radians of the semimajor axis. The rotation angle increases counterclockwise from the positive x axis. See Also -------- Disk2D, Box2D Notes ----- Model formula: .. math:: f(x, y) = \left \{ \begin{array}{ll} \mathrm{amplitude} & : \left[\frac{(x - x_0) \cos \theta + (y - y_0) \sin \theta}{a}\right]^2 + \left[\frac{-(x - x_0) \sin \theta + (y - y_0) \cos \theta}{b}\right]^2 \leq 1 \\ 0 & : \mathrm{otherwise} \end{array} \right. Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Ellipse2D from astropy.coordinates import Angle import matplotlib.pyplot as plt import matplotlib.patches as mpatches x0, y0 = 25, 25 a, b = 20, 10 theta = Angle(30, 'deg') e = Ellipse2D(amplitude=100., x_0=x0, y_0=y0, a=a, b=b, theta=theta.radian) y, x = np.mgrid[0:50, 0:50] fig, ax = plt.subplots(1, 1) ax.imshow(e(x, y), origin='lower', interpolation='none', cmap='Greys_r') e2 = mpatches.Ellipse((x0, y0), 2*a, 2*b, theta.degree, edgecolor='red', facecolor='none') ax.add_patch(e2) plt.show() RiicC`s||}||} tj|} tj|} || | | } ||  | | } | |d| |ddk}tj|g|g}t|trt|d|jdtS|SdS(u'Two dimensional Ellipse model function.ig?R-R{N(R%RURVtselectR~RR-R(R'RWR Rty_0RRRSRAtxxtyyR`Rat numerator1t numerator2t in_ellipsetresult((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(bs   cC`sh|j}|j}|jj}t|||\}}|j||j|f|j||j|ffS(uu Tuple defining the default ``bounding_box`` limits. ``((y_low, y_high), (x_low, x_high))`` (RRRSRARQR RR(RRRRSRARRT((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRts    cC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.sc C`sz|d|dkr#tdntd|dfd|dfd|dfd|dfdtjfd |d fgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0uaubuthetau amplitudeuz(RRRwRx(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s    (RRRR R RRRRRSRAR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRsDtDisk2DcB`seZdZeddZeddZeddZeddZedZ e dZ e dZ dZ RS(uf Two dimensional radial symmetric Disk model. Parameters ---------- amplitude : float Value of the disk function x_0 : float x position center of the disk y_0 : float y position center of the disk R_0 : float Radius of the disk See Also -------- Box2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(r) = \left \{ \begin{array}{ll} A & : r \leq R_0 \\ 0 & : r > R_0 \end{array} \right. RiicC`sl||d||d}tj||dkg|g}t|trdt|d|jdtS|SdS(u#Two dimensional Disk model functioniR-R{N(R%RR~RR-R(R'RWR RRtR_0trrR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s "cC`s>|j|j|j|jf|j|j|j|jffS(uu Tuple defining the default ``bounding_box`` limits. ``((y_low, y_high), (x_low, x_high))`` (RRR(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRscC`sJ|jjdkr(|jjdkr(dSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.s$cC`sa|d|dkr#tdntd|dfd|dfd|dfd|dfgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0uR_0u amplitudeuz(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s   (RRRR R RRRR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs  tRing2DcB`seZdZeddZeddZeddZeddZeddZej ej ej ej ej d dZ e dZ edZedZdZRS( u7 Two dimensional radial symmetric Ring model. Parameters ---------- amplitude : float Value of the disk function x_0 : float x position center of the disk y_0 : float y position center of the disk r_in : float Inner radius of the ring width : float Width of the ring. r_out : float Outer Radius of the ring. Can be specified instead of width. See Also -------- Disk2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(r) = \left \{ \begin{array}{ll} A & : r_{in} \leq r \leq r_{out} \\ 0 & : \text{else} \end{array} \right. Where :math:`r_{out} = r_{in} + r_{width}`. Riic K`s|dk r:||jjkr-tdn||}n|dkrU|jj}ntt|jd|d|d|d|d||dS(Nu6Cannot specify both width and outer radius separately.R RRtr_intwidth(R"RRR R5RR6(RR RRRRtr_outR8((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR6s    c C`s||d||d}tj||dk|||dk}tj|g|g} t|trt| d|jdtS| SdS(u$Two dimensional Ring model function.iR-R{N(R%t logical_andRR~RR-R( R'RWR RRRRRtr_rangeR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR((s *cC`sB|j|j}|j||j|f|j||j|ffS(un Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` (RRRR(Rtdr((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR5scC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.BscC`sn|d|dkr#tdntd|dfd|dfd|dfd|dfd|d fgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0ur_inuwidthu amplitudeuz(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1Js   N(RRRR R RRRRRR"R6R2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs%    tDelta1DcB`seZdZdZRS(u%One dimensional Dirac delta function.cC`stddS(NuNot implemented(R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR6Zs(RRRR6(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRWstDelta2DcB`seZdZdZRS(u%Two dimensional Dirac delta function.cC`stddS(NuNot implemented(R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR6as(RRRR6(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR^stBox1DcB`sqeZdZeddZeddZeddZedZe dZ e dZ dZ RS(u One dimensional Box model. Parameters ---------- amplitude : float Amplitude A x_0 : float Position of the center of the box function width : float Width of the box See Also -------- Box2D, TrapezoidDisk2D Notes ----- Model formula: .. math:: f(x) = \left \{ \begin{array}{ll} A & : x_0 - w/2 \leq x \leq x_0 + w/2 \\ 0 & : \text{else} \end{array} \right. Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Box1D plt.figure() s1 = Box1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() RiicC`sytj|||dk|||dk}tj|g|gd}t|trqt|d|jdtS|SdS(u"One dimensional Box model functiong@iR-R{N(R%RRR~RR-R(R'R RRtinsideR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s .cC`s%|jd}|j||j|fS(uc Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` i(RR(RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`s1td|dfd|dfd|dfgS(Nux_0uxuwidthu amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s ( RRRR R RRR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRes3  tBox2DcB`seZdZeddZeddZeddZeddZeddZe dZ e dZ e dZ dZRS(u Two dimensional Box model. Parameters ---------- amplitude : float Amplitude A x_0 : float x position of the center of the box function x_width : float Width in x direction of the box y_0 : float y position of the center of the box function y_width : float Width in y direction of the box See Also -------- Box1D, Gaussian2D, Moffat2D Notes ----- Model formula: .. math:: f(x, y) = \left \{ \begin{array}{ll} A : & x_0 - w_x/2 \leq x \leq x_0 + w_x/2 \text{ and} \\ & y_0 - w_y/2 \leq y \leq y_0 + w_y/2 \\ 0 : & \text{else} \end{array} \right. Riic C`stj|||dk|||dk}tj|||dk|||dk}tjtj||g|gd} t|trt| d|jdtS| SdS(u"Two dimensional Box model functiong@iR-R{N(R%RRR~RR-R( R'RWR RRtx_widthty_widthtx_rangety_rangeR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s'cC`sL|jd}|jd}|j||j|f|j||j|ffS(un Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` i(RRRR(RRRT((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs  cC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR. scC`sKtd|dfd|dfd|dfd|dfd|dfgS( Nux_0uxuy_0uyux_widthuy_widthu amplitudeuz(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s    (RRRR R RRRRR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs#t Trapezoid1DcB`seZdZeddZeddZeddZeddZedZ e dZ e dZ dZ RS(ue One dimensional Trapezoid model. Parameters ---------- amplitude : float Amplitude of the trapezoid x_0 : float Center position of the trapezoid width : float Width of the constant part of the trapezoid. slope : float Slope of the tails of the trapezoid See Also -------- Box1D, Gaussian1D, Moffat1D Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Trapezoid1D plt.figure() s1 = Trapezoid1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() RiicC`s||d}||d}|||}|||}tj||k||k} tj||k||k} tj||k||k} |||} |} |||}tj| | | g| | |g}t|trt|d|jdtS|SdS(u(One dimensional Trapezoid model functiong@R-R{N(R%RRR~RR-R(R'R RRRtx2tx3tx1tx4trange_atrange_btrange_ctval_atval_btval_cR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(Ks$cC`s3|jd|j|j}|j||j|fS(uc Tuple defining the default ``bounding_box`` limits. ``(x_low, x_high))`` i(RR RR(RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRdscC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.pscC`sFtd|dfd|dfd|d|dfd|dfgS(Nux_0uxuwidthuslopeuyu amplitude(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1ws (RRRR R RRRR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs( tTrapezoidDisk2DcB`seZdZeddZeddZeddZeddZeddZe dZ e dZ e dZ dZRS(u Two dimensional circular Trapezoid model. Parameters ---------- amplitude : float Amplitude of the trapezoid x_0 : float x position of the center of the trapezoid y_0 : float y position of the center of the trapezoid R_0 : float Radius of the constant part of the trapezoid. slope : float Slope of the tails of the trapezoid in x direction. See Also -------- Disk2D, Box2D Riic C`stj||d||d}||k}tj||k||||k} |} ||||} tj|| g| | g} t|trt| d|jdtS| SdS(u-Two dimensional Trapezoid Disk model functioniR-R{N(R%RHRRR~RR-R( R'RWR RRRRRtrange_1trange_2tval_1tval_2R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s# &cC`sI|j|j|j}|j||j|f|j||j|ffS(un Tuple defining the default ``bounding_box``. ``((y_low, y_high), (x_low, x_high))`` (RR RRR(RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRscC`sJ|jjdkr(|jjdkr(dSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.s$cC`sv|d|dkr#tdntd|dfd|dfd|dfd|d|dfd |dfgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0uR_0uslopeuzu amplitude(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s  (RRRR R RRRRR2R(R#RR.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR~s t MexicanHat1DcB`sneZdZeddZeddZeddZedZddZ e dZ dZ RS( u One dimensional Mexican Hat model. Parameters ---------- amplitude : float Amplitude x_0 : float Position of the peak sigma : float Width of the Mexican hat See Also -------- MexicanHat2D, Box1D, Gaussian1D, Trapezoid1D Notes ----- Model formula: .. math:: f(x) = {A \left(1 - \frac{\left(x - x_{0}\right)^{2}}{\sigma^{2}}\right) e^{- \frac{\left(x - x_{0}\right)^{2}}{2 \sigma^{2}}}} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import MexicanHat1D plt.figure() s1 = MexicanHat1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -2, 4]) plt.show() RiicC`s8||dd|d}|dd|tj| S(u*One dimensional Mexican Hat model functionii(R%R&(R'R Rtsigmatxx_ww((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(sg$@cC`s(|j}||j}||||fS(uTuple defining the default ``bounding_box`` limits, ``(x_low, x_high)``. Parameters ---------- factor : float The multiple of sigma used to define the limits. (RR(RRRR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs  cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`s1td|dfd|dfd|dfgS(Nux_0uxusigmau amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s ( RRRR R RRR2R(RR#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs/ t MexicanHat2DcB`sqeZdZeddZeddZeddZeddZedZ e dZ dZ RS(uY Two dimensional symmetric Mexican Hat model. Parameters ---------- amplitude : float Amplitude x_0 : float x position of the peak y_0 : float y position of the peak sigma : float Width of the Mexican hat See Also -------- MexicanHat1D, Gaussian2D Notes ----- Model formula: .. math:: f(x, y) = A \left(1 - \frac{\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2}}{\sigma^{2}}\right) e^{\frac{- \left(x - x_{0}\right)^{2} - \left(y - y_{0}\right)^{2}}{2 \sigma^{2}}} RiicC`s@||d||dd|d}|d|tj| S(u*Two dimensional Mexican Hat model functionii(R%R&(R'RWR RRRtrr_ww((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(Hs&cC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.OscC`sa|d|dkr#tdntd|dfd|dfd|dfd|dfgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0usigmau amplitudeuz(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1Ws   ( RRRR R RRRR2R(R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR$st AiryDisk2DcB`s}eZdZeddZeddZeddZeddZdZ dZ e dZ e dZdZRS(u Two dimensional Airy disk model. Parameters ---------- amplitude : float Amplitude of the Airy function. x_0 : float x position of the maximum of the Airy function. y_0 : float y position of the maximum of the Airy function. radius : float The radius of the Airy disk (radius of the first zero). See Also -------- Box2D, TrapezoidDisk2D, Gaussian2D Notes ----- Model formula: .. math:: f(r) = A \left[\frac{2 J_1(\frac{\pi r}{R/R_z})}{\frac{\pi r}{R/R_z}}\right]^2 Where :math:`J_1` is the first order Bessel function of the first kind, :math:`r` is radial distance from the maximum of the Airy function (:math:`r = \sqrt{(x - x_0)^2 + (y - y_0)^2}`), :math:`R` is the input ``radius`` parameter, and :math:`R_z = 1.2196698912665045`). For an optical system, the radius of the first zero represents the limiting angular resolution and is approximately 1.22 * lambda / D, where lambda is the wavelength of the light and D is the diameter of the aperture. See [1]_ for more details about the Airy disk. References ---------- .. [1] https://en.wikipedia.org/wiki/Airy_disk Riic C`sJ|jdkrry@ddlm}m}|dddtj|_||_Wqrtk rnt dqrXntj ||d||d||j} t | t r| j tj} ntj| j} tj| | dk} d|j| | d| | dk 0.11.ig@R{N(t_rzR"RRRR%Rt_j1RERRHR~Rtto_valueRwtdimensionless_unscaledtonesRDR( RR'RWR RRtradiusRRRRtrt((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(s"  .% cC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.scC`sa|d|dkr#tdntd|dfd|dfd|dfd|dfgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0uradiusu amplitudeuz(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1s   N(RRRR R RRR R"RR RR(R#R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRcs)tMoffat1DcB`seZdZeddZeddZeddZeddZedZ e dZ e dZ edZ dZRS( u One dimensional Moffat model. Parameters ---------- amplitude : float Amplitude of the model. x_0 : float x position of the maximum of the Moffat model. gamma : float Core width of the Moffat model. alpha : float Power index of the Moffat model. See Also -------- Gaussian1D, Box1D Notes ----- Model formula: .. math:: f(x) = A \left(1 + \frac{\left(x - x_{0}\right)^{2}}{\gamma^{2}}\right)^{- \alpha} Examples -------- .. plot:: :include-source: import numpy as np import matplotlib.pyplot as plt from astropy.modeling.models import Moffat1D plt.figure() s1 = Moffat1D() r = np.arange(-5, 5, .01) for factor in range(1, 4): s1.amplitude = factor s1.width = factor plt.plot(r, s1(r), color=str(0.25 * factor), lw=2) plt.axis([-5, 5, -1, 4]) plt.show() RiicC`s'd|jtjdd|jdS(u Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach `_. g@g?(tgammaR%RHR(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRscC`s|d|||d| S(u%One dimensional Moffat model functionii((R'R RRR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR( sc C`sd||d|d| }| ||d|d||d||}d|||||d|d||}| |tjd||d|d}||||gS(uBOne dimensional Moffat model derivative with respect to parametersiiii(R%R( R'R RRRtd_ARtd_gammatd_alpha((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR, s,cC`s+|jjdkrdSi|jjd6SdS(Nux(RR-R"(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR. scC`s1td|dfd|dfd|dfgS(Nux_0uxugammau amplitudeuy(R(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1! s (RRRR R RRRR#RR2R(R,R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRs0  tMoffat2DcB`seZdZeddZeddZeddZeddZeddZe dZ e dZ e dZ e dZdZRS( uk Two dimensional Moffat model. Parameters ---------- amplitude : float Amplitude of the model. x_0 : float x position of the maximum of the Moffat model. y_0 : float y position of the maximum of the Moffat model. gamma : float Core width of the Moffat model. alpha : float Power index of the Moffat model. See Also -------- Gaussian2D, Box2D Notes ----- Model formula: .. math:: f(x, y) = A \left(1 + \frac{\left(x - x_{0}\right)^{2} + \left(y - y_{0}\right)^{2}}{\gamma^{2}}\right)^{- \alpha} RiicC`s'd|jtjdd|jdS(u Moffat full width at half maximum. Derivation of the formula is available in `this notebook by Yoonsoo Bach `_. g@g?(RR%RHR(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyRL scC`s3||d||d|d}|d|| S(u%Two dimensional Moffat model functionii((R'RWR RRRRtrr_gg((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR(U s"c C`s||d||d|d}d|| }| ||d|d||dd|} | ||d|d||dd|} | |tjd|} d|||||d|} || | | | gS(uBTwo dimensional Moffat model derivative with respect to parametersiii(R%R( R'RWR RRRRRRRtd_y_0RR((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR,\ s""cC`s8|jjdkrdSi|jjd6|jjd6SdS(Nuxuy(RR-R"R(R((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR.j scC`sa|d|dkr#tdntd|dfd|dfd|dfd|dfgS( Nuxuyu(Units of 'x' and 'y' inputs should matchux_0uy_0ugammau amplitudeuz(RR(RR/R0((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR1r s   (RRRR R RRRRR#RR2R(R,R.R1(((sAlib/python2.7/site-packages/astropy/modeling/functional_models.pyR' s tSersic2DcB`seZdZeddZeddZeddZeddZeddZeddZ eddZ dZ e dZedZdZRS( u Two dimensional Sersic surface brightness profile. Parameters ---------- amplitude : float Surface brightness at r_eff. r_eff : float Effective (half-light) radius n : float Sersic Index. x_0 : float, optional x position of the center. y_0 : float, optional y position of the center. ellip : float, optional Ellipticity. theta : float, optional Rotation angle in radians, counterclockwise from the positive x-axis. See Also -------- Gaussian2D, Moffat2D Notes ----- Model formula: .. math:: I(x,y) = I(r) = I_e\exp\left\{-b_n\left[\left(\frac{r}{r_{e}}\right)^{(1/n)}-1\right]\right\} The constant :math:`b_n` is defined such that :math:`r_e` contains half the total luminosity, and can be solved for numerically. .. math:: \Gamma(2n) = 2\gamma (b_n,2n) Examples -------- .. plot:: :include-source: import numpy as np from astropy.modeling.models import Sersic2D import matplotlib.pyplot as plt x,y = np.meshgrid(np.arange(100), np.arange(100)) mod = Sersic2D(amplitude = 1, r_eff = 25, n=4, x_0=50, y_0=50, ellip=.5, theta=-1) img = mod(x, y) log_img = np.log10(img) plt.figure() plt.imshow(log_img, origin='lower', interpolation='nearest', vmin=-1, vmax=2) plt.xlabel('x') plt.ylabel('y') cbar = plt.colorbar() cbar.set_label('Log Brightness', rotation=270, labelpad=25) cbar.set_ticks([-1, 0, 1, 2], update_ticks=True) plt.show() References ---------- .. 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