ParabolicSag#

class optika.sags.ParabolicSag(focal_length=<Quantity inf mm>, *, transformation=None, parameters_slope_error=None, parameters_roughness=None, parameters_microroughness=None)[source]#

Bases: AbstractConicSag

A parabolic sag profile.

Attributes

`conic`	The conic constant of this conic section.
`focal_length`	The focal length of this parabola.
`parameters_microroughness`	The microroughness parameters for this sag profile.
`parameters_roughness`	The roughness parameters for this sag profile.
`parameters_slope_error`	The slope error parameters for this sag profile.
`radius`	The effective radius of this conic section.
`shape`	The array shape of this object.
`transformation`	The transformation between the surface coordinate system and the sag coordinate system.

Methods

`__init__`([focal_length, transformation, ...])
`intercept`(rays)	Compute the intercept of a ray with this parabolic surface.
`normal`(position)	The vector perpendicular to the surface at the given position.
`propagate_rays`(rays)	For the given input rays, calculate new rays based off of their interation with this object.
`to_string`([prefix])	Public-facing version of the `__repr__` method that allows for defining a prefix string, which can be used to calculate how much whitespace to add to the beginning of each line of the result.

Inheritance Diagram

Parameters:

focal_length (Quantity | AbstractScalar)
transformation (None | AbstractTransformation)
parameters_slope_error (None | SlopeErrorParameters)
parameters_roughness (None | RoughnessParameters)
parameters_microroughness (None | RoughnessParameters)

intercept(rays)[source]#

Compute the intercept of a ray with this parabolic surface.

Parameters:: rays (AbstractRayVectorArray) – The rays to find the intercept for.
Return type:: RayVectorArray

Notes

The equation for a paraboloid is

\[z(x, y) = \frac{x^2 + y^2}{4 f},\]

where \(x\), \(y\), and \(z\) are points on the paraboloid and \(f\) is the focal length of the paraboloid. The equation for a line is

\[\mathbf{x} = \mathbf{o} + d \mathbf{u},\]

where \(\mathbf{o}\) is the starting point of the line, \(\mathbf{u}\) is a unit vector pointing in the direction of the line, and \(d\) is the distance from the origin to the intercept with the paraboloid. Combining these equations gives

\[o_z + d u_z = \frac{(o_x + d u_x)^2 + (o_y + d u_y)^2}{4 f},\]

which can then be solved for \(d\) using the quadratic equation,

\[d = \frac{-o_x u_x - o_y u_y + 2 f u_z - \text{sgn}(f u_z) \sqrt{-o_y^2 u_x^2 - o_x^2 u_y^2 + 2 o_y u_y (o_x u_x - 2 f u_z) + 4 f (o_z (u_x^2 + u_y^2) - o_x u_x u_z + f u_z^2} }{u_x^2 + u_y^2}.\]

If the line is parallel to the \(z\) axis, then the above equation is singular and we need to solve the corresponding linear equation to find

\[d = \frac{o_x^2 + o_y^2 - 4 f o_z}{4 f u_z}.\]

normal(position)[source]#

The vector perpendicular to the surface at the given position.

Parameters:: position (AbstractCartesian3dVectorArray) – The location on the surface to evaluate the normal vector
Return type:: AbstractCartesian3dVectorArray

propagate_rays(rays)#

For the given input rays, calculate new rays based off of their interation with this object.

Parameters:: rays (AbstractRayVectorArray) – a set of input rays that will interact with this object
Return type:: AbstractRayVectorArray

to_string(prefix=None)#

Public-facing version of the __repr__ method that allows for defining a prefix string, which can be used to calculate how much whitespace to add to the beginning of each line of the result.

Parameters:: prefix (None | str) – an optional string, the length of which is used to calculate how much whitespace to add to the result.
Return type:: str

property conic: int#: The conic constant of this conic section.

focal_length: Quantity | AbstractScalar = <Quantity inf mm>#: The focal length of this parabola.

parameters_microroughness: None | RoughnessParameters = None#: The microroughness parameters for this sag profile.

parameters_roughness: None | RoughnessParameters = None#: The roughness parameters for this sag profile.

parameters_slope_error: None | SlopeErrorParameters = None#: The slope error parameters for this sag profile.

property radius: Quantity | AbstractScalar#: The effective radius of this conic section.

property shape: dict[str, int]#: The array shape of this object.

transformation: None | AbstractTransformation = None#: The transformation between the surface coordinate system and the sag coordinate system.