venv/lib/python3.11/site-packages/sympy/physics/optics/utils.py

"""
**Contains**

* refraction_angle
* fresnel_coefficients
* deviation
* brewster_angle
* critical_angle
* lens_makers_formula
* mirror_formula
* lens_formula
* hyperfocal_distance
* transverse_magnification
"""

__all__ = ['refraction_angle',
           'deviation',
           'fresnel_coefficients',
           'brewster_angle',
           'critical_angle',
           'lens_makers_formula',
           'mirror_formula',
           'lens_formula',
           'hyperfocal_distance',
           'transverse_magnification'
           ]

from sympy.core.numbers import (Float, I, oo, pi, zoo)
from sympy.core.singleton import S
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.functions.elementary.miscellaneous import sqrt
from sympy.functions.elementary.trigonometric import (acos, asin, atan2, cos, sin, tan)
from sympy.matrices.dense import Matrix
from sympy.polys.polytools import cancel
from sympy.series.limits import Limit
from sympy.geometry.line import Ray3D
from sympy.geometry.util import intersection
from sympy.geometry.plane import Plane
from sympy.utilities.iterables import is_sequence
from .medium import Medium


def refractive_index_of_medium(medium):
    """
    Helper function that returns refractive index, given a medium
    """
    if isinstance(medium, Medium):
        n = medium.refractive_index
    else:
        n = sympify(medium)
    return n


def refraction_angle(incident, medium1, medium2, normal=None, plane=None):
    """
    This function calculates transmitted vector after refraction at planar
    surface. ``medium1`` and ``medium2`` can be ``Medium`` or any sympifiable object.
    If ``incident`` is a number then treated as angle of incidence (in radians)
    in which case refraction angle is returned.

    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, sequence or a number
        Incident vector or angle of incidence
    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.

    Returns
    =======

    Returns an angle of refraction or a refracted ray depending on inputs.

    Examples
    ========

    >>> from sympy.physics.optics import refraction_angle
    >>> from sympy.geometry import Point3D, Ray3D, Plane
    >>> from sympy.matrices import Matrix
    >>> from sympy import symbols, pi
    >>> 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)))
    >>> round(refraction_angle(pi/6, 1.2, 1.5), 5)
    0.41152
    """

    n1 = refractive_index_of_medium(medium1)
    n2 = refractive_index_of_medium(medium2)

    # check if an incidence angle was supplied instead of a ray
    try:
        angle_of_incidence = float(incident)
    except TypeError:
        angle_of_incidence = None

    try:
        critical_angle_ = critical_angle(medium1, medium2)
    except (ValueError, TypeError):
        critical_angle_ = None

    if angle_of_incidence is not None:
        if normal is not None or plane is not None:
            raise ValueError('Normal/plane not allowed if incident is an angle')

        if not 0.0 <= angle_of_incidence < pi*0.5:
            raise ValueError('Angle of incidence not in range [0:pi/2)')

        if critical_angle_ and angle_of_incidence > critical_angle_:
            raise ValueError('Ray undergoes total internal reflection')
        return asin(n1*sin(angle_of_incidence)/n2)

    # Treat the incident as ray below
    # A flag to check whether to return Ray3D or not
    return_ray = False

    if plane is not None and normal is not None:
        raise ValueError("Either plane or normal is acceptable.")

    if not isinstance(incident, Matrix):
        if is_sequence(incident):
            _incident = Matrix(incident)
        elif isinstance(incident, Ray3D):
            _incident = Matrix(incident.direction_ratio)
        else:
            raise TypeError(
                "incident should be a Matrix, Ray3D, or sequence")
    else:
        _incident = incident

    # If plane is provided, get direction ratios of the normal
    # to the plane from the plane else go with `normal` param.
    if plane is not None:
        if not isinstance(plane, Plane):
            raise TypeError("plane should be an instance of geometry.plane.Plane")
        # If we have the plane, we can get the intersection
        # point of incident ray and the plane and thus return
        # an instance of Ray3D.
        if isinstance(incident, Ray3D):
            return_ray = True
            intersection_pt = plane.intersection(incident)[0]
        _normal = Matrix(plane.normal_vector)
    else:
        if not isinstance(normal, Matrix):
            if is_sequence(normal):
                _normal = Matrix(normal)
            elif isinstance(normal, Ray3D):
                _normal = Matrix(normal.direction_ratio)
                if isinstance(incident, Ray3D):
                    intersection_pt = intersection(incident, normal)
                    if len(intersection_pt) == 0:
                        raise ValueError(
                            "Normal isn't concurrent with the incident ray.")
                    else:
                        return_ray = True
                        intersection_pt = intersection_pt[0]
            else:
                raise TypeError(
                    "Normal should be a Matrix, Ray3D, or sequence")
        else:
            _normal = normal

    eta = n1/n2  # Relative index of refraction
    # Calculating magnitude of the vectors
    mag_incident = sqrt(sum(i**2 for i in _incident))
    mag_normal = sqrt(sum(i**2 for i in _normal))
    # Converting vectors to unit vectors by dividing
    # them with their magnitudes
    _incident /= mag_incident
    _normal /= mag_normal
    c1 = -_incident.dot(_normal)  # cos(angle_of_incidence)
    cs2 = 1 - eta**2*(1 - c1**2)  # cos(angle_of_refraction)**2
    if cs2.is_negative:  # This is the case of total internal reflection(TIR).
        return S.Zero
    drs = eta*_incident + (eta*c1 - sqrt(cs2))*_normal
    # Multiplying unit vector by its magnitude
    drs = drs*mag_incident
    if not return_ray:
        return drs
    else:
        return Ray3D(intersection_pt, direction_ratio=drs)


def fresnel_coefficients(angle_of_incidence, medium1, medium2):
    """
    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
    =======

    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
    ==========

    .. [1] https://en.wikipedia.org/wiki/Fresnel_equations
    """
    if not 0 <= 2*angle_of_incidence < pi:
        raise ValueError('Angle of incidence not in range [0:pi/2)')

    n1 = refractive_index_of_medium(medium1)
    n2 = refractive_index_of_medium(medium2)

    angle_of_refraction = asin(n1*sin(angle_of_incidence)/n2)
    try:
        angle_of_total_internal_reflection_onset = critical_angle(n1, n2)
    except ValueError:
        angle_of_total_internal_reflection_onset = None

    if angle_of_total_internal_reflection_onset is None or\
    angle_of_total_internal_reflection_onset > angle_of_incidence:
        R_s = -sin(angle_of_incidence - angle_of_refraction)\
                /sin(angle_of_incidence + angle_of_refraction)
        R_p = tan(angle_of_incidence - angle_of_refraction)\
                /tan(angle_of_incidence + angle_of_refraction)
        T_s = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
                /sin(angle_of_incidence + angle_of_refraction)
        T_p = 2*sin(angle_of_refraction)*cos(angle_of_incidence)\
                /(sin(angle_of_incidence + angle_of_refraction)\
                *cos(angle_of_incidence - angle_of_refraction))
        return [R_p, R_s, T_p, T_s]
    else:
        n = n2/n1
        R_s = cancel((cos(angle_of_incidence)-\
                I*sqrt(sin(angle_of_incidence)**2 - n**2))\
                /(cos(angle_of_incidence)+\
                I*sqrt(sin(angle_of_incidence)**2 - n**2)))
        R_p = cancel((n**2*cos(angle_of_incidence)-\
                I*sqrt(sin(angle_of_incidence)**2 - n**2))\
                /(n**2*cos(angle_of_incidence)+\
                I*sqrt(sin(angle_of_incidence)**2 - n**2)))
        return [R_p, R_s]


def deviation(incident, medium1, medium2, normal=None, plane=None):
    """
    This function calculates the angle of deviation of a ray
    due to refraction at planar surface.

    Parameters
    ==========

    incident : Matrix, Ray3D, sequence or float
        Incident vector or angle of incidence
    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.

    Returns angular deviation between incident and refracted rays

    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)
    >>> round(deviation(0.1, 1.2, 1.5), 5)
    -0.02005
    """
    refracted = refraction_angle(incident,
                                 medium1,
                                 medium2,
                                 normal=normal,
                                 plane=plane)
    try:
        angle_of_incidence = Float(incident)
    except TypeError:
        angle_of_incidence = None

    if angle_of_incidence is not None:
        return float(refracted) - angle_of_incidence

    if refracted != 0:
        if isinstance(refracted, Ray3D):
            refracted = Matrix(refracted.direction_ratio)

        if not isinstance(incident, Matrix):
            if is_sequence(incident):
                _incident = Matrix(incident)
            elif isinstance(incident, Ray3D):
                _incident = Matrix(incident.direction_ratio)
            else:
                raise TypeError(
                    "incident should be a Matrix, Ray3D, or sequence")
        else:
            _incident = incident

        if plane is None:
            if not isinstance(normal, Matrix):
                if is_sequence(normal):
                    _normal = Matrix(normal)
                elif isinstance(normal, Ray3D):
                    _normal = Matrix(normal.direction_ratio)
                else:
                    raise TypeError(
                        "normal should be a Matrix, Ray3D, or sequence")
            else:
                _normal = normal
        else:
            _normal = Matrix(plane.normal_vector)

        mag_incident = sqrt(sum(i**2 for i in _incident))
        mag_normal = sqrt(sum(i**2 for i in _normal))
        mag_refracted = sqrt(sum(i**2 for i in refracted))
        _incident /= mag_incident
        _normal /= mag_normal
        refracted /= mag_refracted
        i = acos(_incident.dot(_normal))
        r = acos(refracted.dot(_normal))
        return i - r


def brewster_angle(medium1, medium2):
    """
    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

    """

    n1 = refractive_index_of_medium(medium1)
    n2 = refractive_index_of_medium(medium2)

    return atan2(n2, n1)

def critical_angle(medium1, medium2):
    """
    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

    """

    n1 = refractive_index_of_medium(medium1)
    n2 = refractive_index_of_medium(medium2)

    if n2 > n1:
        raise ValueError('Total internal reflection impossible for n1 < n2')
    else:
        return asin(n2/n1)



def lens_makers_formula(n_lens, n_surr, r1, r2, d=0):
    """
    This function calculates focal length of a 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.
    d : sympifiable, optional
        Thickness of lens, default value is 0.

    Examples
    ========

    >>> from sympy.physics.optics import lens_makers_formula
    >>> from sympy import S
    >>> lens_makers_formula(1.33, 1, 10, -10)
    15.1515151515151
    >>> lens_makers_formula(1.2, 1, 10, S.Infinity)
    50.0000000000000
    >>> lens_makers_formula(1.33, 1, 10, -10, d=1)
    15.3418463277618

    """

    if isinstance(n_lens, Medium):
        n_lens = n_lens.refractive_index
    else:
        n_lens = sympify(n_lens)
    if isinstance(n_surr, Medium):
        n_surr = n_surr.refractive_index
    else:
        n_surr = sympify(n_surr)
    d = sympify(d)

    focal_length = 1/((n_lens - n_surr) / n_surr*(1/r1 - 1/r2 + (((n_lens - n_surr) * d) / (n_lens * r1 * r2))))

    if focal_length == zoo:
        return S.Infinity
    return focal_length


def mirror_formula(focal_length=None, u=None, v=None):
    """
    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)

    """
    if focal_length and u and v:
        raise ValueError("Please provide only two parameters")

    focal_length = sympify(focal_length)
    u = sympify(u)
    v = sympify(v)
    if u is oo:
        _u = Symbol('u')
    if v is oo:
        _v = Symbol('v')
    if focal_length is oo:
        _f = Symbol('f')
    if focal_length is None:
        if u is oo and v is oo:
            return Limit(Limit(_v*_u/(_v + _u), _u, oo), _v, oo).doit()
        if u is oo:
            return Limit(v*_u/(v + _u), _u, oo).doit()
        if v is oo:
            return Limit(_v*u/(_v + u), _v, oo).doit()
        return v*u/(v + u)
    if u is None:
        if v is oo and focal_length is oo:
            return Limit(Limit(_v*_f/(_v - _f), _v, oo), _f, oo).doit()
        if v is oo:
            return Limit(_v*focal_length/(_v - focal_length), _v, oo).doit()
        if focal_length is oo:
            return Limit(v*_f/(v - _f), _f, oo).doit()
        return v*focal_length/(v - focal_length)
    if v is None:
        if u is oo and focal_length is oo:
            return Limit(Limit(_u*_f/(_u - _f), _u, oo), _f, oo).doit()
        if u is oo:
            return Limit(_u*focal_length/(_u - focal_length), _u, oo).doit()
        if focal_length is oo:
            return Limit(u*_f/(u - _f), _f, oo).doit()
        return u*focal_length/(u - focal_length)


def lens_formula(focal_length=None, u=None, v=None):
    """
    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)

    """
    if focal_length and u and v:
        raise ValueError("Please provide only two parameters")

    focal_length = sympify(focal_length)
    u = sympify(u)
    v = sympify(v)
    if u is oo:
        _u = Symbol('u')
    if v is oo:
        _v = Symbol('v')
    if focal_length is oo:
        _f = Symbol('f')
    if focal_length is None:
        if u is oo and v is oo:
            return Limit(Limit(_v*_u/(_u - _v), _u, oo), _v, oo).doit()
        if u is oo:
            return Limit(v*_u/(_u - v), _u, oo).doit()
        if v is oo:
            return Limit(_v*u/(u - _v), _v, oo).doit()
        return v*u/(u - v)
    if u is None:
        if v is oo and focal_length is oo:
            return Limit(Limit(_v*_f/(_f - _v), _v, oo), _f, oo).doit()
        if v is oo:
            return Limit(_v*focal_length/(focal_length - _v), _v, oo).doit()
        if focal_length is oo:
            return Limit(v*_f/(_f - v), _f, oo).doit()
        return v*focal_length/(focal_length - v)
    if v is None:
        if u is oo and focal_length is oo:
            return Limit(Limit(_u*_f/(_u + _f), _u, oo), _f, oo).doit()
        if u is oo:
            return Limit(_u*focal_length/(_u + focal_length), _u, oo).doit()
        if focal_length is oo:
            return Limit(u*_f/(u + _f), _f, oo).doit()
        return u*focal_length/(u + focal_length)

def hyperfocal_distance(f, N, c):
    """

    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
    >>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2)
    9.47
    """

    f = sympify(f)
    N = sympify(N)
    c = sympify(c)

    return (1/(N * c))*(f**2)

def transverse_magnification(si, so):
    """

    Calculates the transverse magnification upon reflection in a mirror,
    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

    """

    si = sympify(si)
    so = sympify(so)

    return (-(si/so))