ó ~9­\c @ sjdZddlmZddddddd d d d g Zdd lmZmZmZmZm Z m Z m Z m Z m Z mZmZmZmZmZddlmZddlmZmZddlmZddlmZddlmZddd„Zd„Z ddd„Z!d„Z"d„Z#d„Z$dddd„Z%dddd„Z&d„Z'd„Z(dS(sÎ **Contains** * refraction_angle * fresnel_coefficients * deviation * brewster_angle * critical_angle * lens_makers_formula * mirror_formula * lens_formula * hyperfocal_distance * transverse_magnification iÿÿÿÿ(tdivisiontrefraction_anglet deviationtfresnel_coefficientstbrewster_angletcritical_angletlens_makers_formulatmirror_formulat lens_formulathyperfocal_distancettransverse_magnification(tSymboltsympifytsqrttMatrixtacostootLimittatan2tasintcostsinttantItcancel(t is_sequence(tRay3DtPoint3D(t intersection(tPlanei(tMediumcC sït}|d k r-|d k r-tdƒ‚nt|tƒs‡t|ƒrWt|ƒ}qt|tƒrxt|jƒ}qtdƒ‚n|}|d k rôt|t ƒs·tdƒ‚nt|tƒrât }|j |ƒd}nt|j ƒ}n²t|tƒs t|ƒrt|ƒ}q¦t|tƒr‘t|jƒ}t|tƒrt ||ƒ}t |ƒdkr{tdƒ‚qŽt }|d}qq¦tdƒ‚n|}d \} } t|tƒrÍ|j} n t|ƒ} t|tƒrô|j} n t|ƒ} | | } ttg|D]} | d^qƒƒ} ttg|D]} | d^q@ƒƒ}|| }||}|j|ƒ }d| dd|d}|jr§dS| || |t|ƒ|}|| }|sÛ|St|d |ƒSd S( s1 This function calculates transmitted vector after refraction at planar surface. `medium1` and `medium2` can be `Medium` or any sympifiable object. If `incident` is an object of `Ray3D`, `normal` also has to be an instance of `Ray3D` in order to get the output as a `Ray3D`. Please note that if plane of separation is not provided and normal is an instance of `Ray3D`, normal will be assumed to be intersecting incident ray at the plane of separation. This will not be the case when `normal` is a `Matrix` or any other sequence. If `incident` is an instance of `Ray3D` and `plane` has not been provided and `normal` is not `Ray3D`, output will be a `Matrix`. Parameters ========== incident : Matrix, Ray3D, or sequence Incident vector medium1 : sympy.physics.optics.medium.Medium or sympifiable Medium 1 or its refractive index medium2 : sympy.physics.optics.medium.Medium or sympifiable Medium 2 or its refractive index normal : Matrix, Ray3D, or sequence Normal vector plane : Plane Plane of separation of the two media. Examples ======== >>> from sympy.physics.optics import refraction_angle >>> from sympy.geometry import Point3D, Ray3D, Plane >>> from sympy.matrices import Matrix >>> from sympy import symbols >>> n = Matrix([0, 0, 1]) >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) >>> refraction_angle(r1, 1, 1, n) Matrix([ [ 1], [ 1], [-1]]) >>> refraction_angle(r1, 1, 1, plane=P) Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1)) With different index of refraction of the two media >>> n1, n2 = symbols('n1, n2') >>> refraction_angle(r1, n1, n2, n) Matrix([ [ n1/n2], [ n1/n2], [-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]]) >>> refraction_angle(r1, n1, n2, plane=P) Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1))) s%Either plane or normal is acceptable.s/incident should be a Matrix, Ray3D, or sequences3plane should be an instance of geometry.plane.Planeis.Normal isn't concurrent with the incident ray.s-Normal should be a Matrix, Ray3D, or sequenceiitdirection_ratioN(NN(tFalsetNonet ValueErrort isinstanceRRRRt TypeErrorRtTrueRt normal_vectortlenRtrefractive_indexR R tsumtdott is_negative(tincidenttmedium1tmedium2tnormaltplanet return_rayt _incidenttintersection_ptt_normaltn1tn2tetatit mag_incidentt mag_normaltc1tcs2tdrs((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR'sl;            ))     c C s8t|tƒr|j}n t|ƒ}t|tƒrB|j}n t|ƒ}t|t|ƒ|ƒ}yt||ƒ}Wntk r”d}nX|dks­||krXt||ƒ t||ƒ}t ||ƒt ||ƒ}dt|ƒt |ƒt||ƒ} dt|ƒt |ƒt||ƒt ||ƒ} ||| | gS||} t t |ƒt t t|ƒd| dƒt |ƒt t t|ƒd| dƒƒ}t | dt |ƒt t t|ƒd| dƒ| dt |ƒt t t|ƒd| dƒƒ}||gSdS(sB This function uses Fresnel equations to calculate reflection and transmission coefficients. Those are obtained for both polarisations when the electric field vector is in the plane of incidence (labelled 'p') and when the electric field vector is perpendicular to the plane of incidence (labelled 's'). There are four real coefficients unless the incident ray reflects in total internal in which case there are two complex ones. Angle of incidence is the angle between the incident ray and the surface normal. ``medium1`` and ``medium2`` can be ``Medium`` or any sympifiable object. Parameters ========== angle_of_incidence : sympifiable medium1 : Medium or sympifiable Medium 1 or its refractive index medium2 : Medium or sympifiable Medium 2 or its refractive index Returns a list with four real Fresnel coefficients: [reflection p (TM), reflection s (TE), transmission p (TM), transmission s (TE)] If the ray is undergoes total internal reflection then returns a list of two complex Fresnel coefficients: [reflection p (TM), reflection s (TE)] Examples ======== >>> from sympy.physics.optics import fresnel_coefficients >>> fresnel_coefficients(0.3, 1, 2) [0.317843553417859, -0.348645229818821, 0.658921776708929, 0.651354770181179] >>> fresnel_coefficients(0.6, 2, 1) [-0.235625382192159 - 0.971843958291041*I, 0.816477005968898 - 0.577377951366403*I] References ========== https://en.wikipedia.org/wiki/Fresnel_equations iN(R#RR(R RRRR"R!RRRRR ( tangle_of_incidenceR-R.R5R6tangle_of_refractiont(angle_of_total_internal_reflection_onsettR_stR_ptT_stT_ptn((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR³s>/         ($    ' 'c C søt|||d|d|ƒ}|dkrôt|tƒrKt|jƒ}nt|tƒs¥t|ƒrut|ƒ}q«t|tƒr–t|jƒ}q«tdƒ‚n|}|dkrt|tƒst|ƒrát|ƒ}qt|tƒrt|jƒ}qtdƒ‚q)|}nt|jƒ}t t g|D]}|d^q6ƒƒ} t t g|D]}|d^q_ƒƒ} t t g|D]}|d^qˆƒƒ} || }|| }|| }t |j |ƒƒ}t |j |ƒƒ} || SdS(s1 This function calculates the angle of deviation of a ray due to refraction at planar surface. Parameters ========== incident : Matrix, Ray3D, or sequence Incident vector medium1 : sympy.physics.optics.medium.Medium or sympifiable Medium 1 or its refractive index medium2 : sympy.physics.optics.medium.Medium or sympifiable Medium 2 or its refractive index normal : Matrix, Ray3D, or sequence Normal vector plane : Plane Plane of separation of the two media. Examples ======== >>> from sympy.physics.optics import deviation >>> from sympy.geometry import Point3D, Ray3D, Plane >>> from sympy.matrices import Matrix >>> from sympy import symbols >>> n1, n2 = symbols('n1, n2') >>> n = Matrix([0, 0, 1]) >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) >>> deviation(r1, 1, 1, n) 0 >>> deviation(r1, n1, n2, plane=P) -acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3) R/R0is/incident should be a Matrix, Ray3D, or sequences-normal should be a Matrix, Ray3D, or sequenceiN( RR#RRRRR$R!R&R R)RR*( R,R-R.R/R0t refractedR2R4R8R9R:t mag_refractedtr((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR sF$        )))   cC sgd\}}t|tƒr'|j}n t|ƒ}t|tƒrN|j}n t|ƒ}t||ƒS(s­ This function calculates the Brewster's angle of incidence to Medium 2 from Medium 1 in radians. Parameters ========== medium 1 : Medium or sympifiable Refractive index of Medium 1 medium 2 : Medium or sympifiable Refractive index of Medium 1 Examples ======== >>> from sympy.physics.optics import brewster_angle >>> brewster_angle(1, 1.33) 0.926093295503462 N(NN(R!R#RR(R R(R-R.R5R6((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR\s     cC s‡d\}}t|tƒr'|j}n t|ƒ}t|tƒrN|j}n t|ƒ}||krutdƒ‚nt||ƒSdS(sÑ This function calculates the critical angle of incidence (marking the onset of total internal) to Medium 2 from Medium 1 in radians. Parameters ========== medium 1 : Medium or sympifiable Refractive index of Medium 1 medium 2 : Medium or sympifiable Refractive index of Medium 1 Examples ======== >>> from sympy.physics.optics import critical_angle >>> critical_angle(1.33, 1) 0.850908514477849 s0Total internal reflection impossible for n1 < n2N(NN(R!R#RR(R R"R(R-R.R5R6((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyRs      cC s†t|tƒr|j}n t|ƒ}t|tƒrB|j}n t|ƒ}t|ƒ}t|ƒ}d|||d|d|S(sD This function calculates focal length of a thin lens. It follows cartesian sign convention. Parameters ========== n_lens : Medium or sympifiable Index of refraction of lens. n_surr : Medium or sympifiable Index of reflection of surrounding. r1 : sympifiable Radius of curvature of first surface. r2 : sympifiable Radius of curvature of second surface. Examples ======== >>> from sympy.physics.optics import lens_makers_formula >>> lens_makers_formula(1.33, 1, 10, -10) 15.1515151515151 i(R#RR(R (tn_lenstn_surrtr1tr2((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR§s      cC sÔ|r!|r!|r!tdƒ‚nt|ƒ}t|ƒ}t|ƒ}|tkr`tdƒ}n|tkr{tdƒ}n|tkr–tdƒ}n|dkrT|tkrè|tkrètt|||||tƒ|tƒjƒS|tkrt|||||tƒjƒS|tkrDt|||||tƒjƒS||||S|dkr|tkr¦|tkr¦tt|||||tƒ|tƒjƒS|tkrÔt|||||tƒjƒS|tkrt|||||tƒjƒS||||S|dkrÐ|tkrd|tkrdtt|||||tƒ|tƒjƒS|tkr’t|||||tƒjƒS|tkrÀt|||||tƒjƒS||||SdS(sÍ This function provides one of the three parameters when two of them are supplied. This is valid only for paraxial rays. Parameters ========== focal_length : sympifiable Focal length of the mirror. u : sympifiable Distance of object from the pole on the principal axis. v : sympifiable Distance of the image from the pole on the principal axis. Examples ======== >>> from sympy.physics.optics import mirror_formula >>> from sympy.abc import f, u, v >>> mirror_formula(focal_length=f, u=u) f*u/(-f + u) >>> mirror_formula(focal_length=f, v=v) f*v/(-f + v) >>> mirror_formula(u=u, v=v) u*v/(u + v) s"Please provide only two parameterstutvtfN(R"R RR R!Rtdoit(t focal_lengthRMRNt_ut_vt_f((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyRÏsF       . " " . " " . " "cC sÔ|r!|r!|r!tdƒ‚nt|ƒ}t|ƒ}t|ƒ}|tkr`tdƒ}n|tkr{tdƒ}n|tkr–tdƒ}n|dkrT|tkrè|tkrètt|||||tƒ|tƒjƒS|tkrt|||||tƒjƒS|tkrDt|||||tƒjƒS||||S|dkr|tkr¦|tkr¦tt|||||tƒ|tƒjƒS|tkrÔt|||||tƒjƒS|tkrt|||||tƒjƒS||||S|dkrÐ|tkrd|tkrdtt|||||tƒ|tƒjƒS|tkr’t|||||tƒjƒS|tkrÀt|||||tƒjƒS||||SdS(s× This function provides one of the three parameters when two of them are supplied. This is valid only for paraxial rays. Parameters ========== focal_length : sympifiable Focal length of the mirror. u : sympifiable Distance of object from the optical center on the principal axis. v : sympifiable Distance of the image from the optical center on the principal axis. Examples ======== >>> from sympy.physics.optics import lens_formula >>> from sympy.abc import f, u, v >>> lens_formula(focal_length=f, u=u) f*u/(f + u) >>> lens_formula(focal_length=f, v=v) f*v/(f - v) >>> lens_formula(u=u, v=v) u*v/(u - v) s"Please provide only two parametersRMRNRON(R"R RR R!RRP(RQRMRNRRRSRT((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyRsF       . " " . " " . " "cC s8t|ƒ}t|ƒ}t|ƒ}d|||dS(sš Parameters ========== f: sympifiable Focal length of a given lens N: sympifiable F-number of a given lens c: sympifiable Circle of Confusion (CoC) of a given image format Example ======= >>> from sympy.physics.optics import hyperfocal_distance >>> from sympy.abc import f, N, c >>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2) 9.47 ii(R (ROtNtc((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR Xs   cC s!t|ƒ}t|ƒ}|| S(su Calculates the transverse magnification, which is the ratio of the image size to the object size. Parameters ========== so: sympifiable Lens-object distance si: sympifiable Lens-image distance Example ======= >>> from sympy.physics.optics import transverse_magnification >>> transverse_magnification(30, 15) -2 (R (tsitso((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyR ts  N()t__doc__t __future__Rt__all__tsympyR R R RRRRRRRRRRRtsympy.core.compatibilityRtsympy.geometry.lineRRtsympy.geometry.utilRtsympy.geometry.planeRtmediumRR!RRRRRRRRR R (((s9lib/python2.7/site-packages/sympy/physics/optics/utils.pyts6 ^Œ XQ # ( (ED