# Utilities¶

Contains

• refraction_angle
• deviation
• lens_makers_formula
• mirror_formula
• lens_formula
• hyperfocal_distance
sympy.physics.optics.utils.refraction_angle(incident, medium1, medium2, normal=None, plane=None)[source]

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)))

sympy.physics.optics.utils.deviation(incident, medium1, medium2, normal=None, plane=None)[source]

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)

sympy.physics.optics.utils.lens_makers_formula(n_lens, n_surr, r1, r2)[source]

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

sympy.physics.optics.utils.mirror_formula(focal_length=None, u=None, v=None)[source]

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)

sympy.physics.optics.utils.lens_formula(focal_length=None, u=None, v=None)[source]

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)

sympy.physics.optics.utils.hyperfocal_distance(f, N, c)[source]
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