Z-aligned circular involute. More...

#include <Involute.hh>

Type aliases
using	Intersections = Array< real_type, 3 >
	Safety cannot be trivially calculated.

using	StorageSpan = Span< real_type const, 6 >
	Safety cannot be trivially calculated.

using	Real2 = Array< real_type, 2 >
	Safety cannot be trivially calculated.

static CELER_CONSTEXPR_FUNCTION SurfaceType	surface_type ()
	Safety cannot be trivially calculated.

static CELER_CONSTEXPR_FUNCTION bool	simple_safety ()
	Safety cannot be trivially calculated.

	Involute (Real2 const &origin, real_type radius, real_type displacement, Chirality sign, real_type tmin, real_type tmax)
	Construct involute from parameters. More...

template<class R >
CELER_FUNCTION	Involute (Span< R, StorageSpan::extent >)
	Construct from raw data.

CELER_FUNCTION Real2 const &	origin () const
	X-Y center of the circular base of the involute.

CELER_FUNCTION real_type	r_b () const
	Involute circle's radius.

CELER_FUNCTION real_type	displacement_angle () const
	Displacement angle.

CELER_FUNCTION Chirality	sign () const
	Orientation of the involute curve.

CELER_FUNCTION real_type	tmin () const
	Get bounds of the involute.

CELER_FUNCTION real_type	tmax () const
	Safety cannot be trivially calculated.

CELER_FUNCTION StorageSpan	data () const
	Get a view to the data for type-deleted storage.

CELER_FUNCTION SignedSense	calc_sense (Real3 const &pos) const
	Determine the sense of the position relative to this surface. More...

CELER_FUNCTION Intersections	calc_intersections (Real3 const &pos, Real3 const &dir, SurfaceState on_surface) const
	Calculate all possible straight-line intersections with this surface.

CELER_FUNCTION Real3	calc_normal (Real3 const &pos) const
	Calculate outward normal at a position on the surface. More...

Detailed Description

Z-aligned circular involute.

An involute is a curve created by unwinding a chord from a shape. The involute implemented here is constructed from a circle and can be made to be clockwise (negative) or anti-clockwise (positive). This involute is the same type of that found in HFIR, and is necessary for building accurate models.

While the parameters define the curve itself have no restriction, except for the radius of involute being positive, the involute needs to be bounded for there to be finite solutions to the roots of the intersection points. Additionally this bound cannot exceed an interval of size of 2 to be able to determine if whether a particle is in, out, or on the involute surface. Lastly the involute geometry is fixed to be z-aligned.

\[ x = r_b (\cos(t+a) + t \sin(t+a)) + x_0 \]

\[ y = r_b (\sin(t+a) - t \cos(t+a)) + y_0 \]

where t is the normal angle of the tangent to the circle of involute with radius r_b from a starting angle of a ( \(r/h\) for a finite cone.

Constructor & Destructor Documentation

◆ Involute()

celeritas::Involute::Involute	(	Real2 const &	origin,
		real_type	radius,
		real_type	displacement,
		Chirality	sign,
		real_type	tmin,
		real_type	tmax
	)

explicit

Construct involute from parameters.

Parameters

origin	origin of the involute
radius	radius of the circle of involute
displacement	displacement angle of the involute
sign	chirality of involute
tmin	minimum tangent angle
tmax	maximum tangent angle

Member Function Documentation

◆ calc_normal()

CELER_FORCEINLINE_FUNCTION Real3 celeritas::Involute::calc_normal ( Real3 const & pos ) const

inline

Calculate outward normal at a position on the surface.

This is done by taking the derivative of the involute equation :

\[ n = {\sin(t+a) - \cos(t+a), 0} \]

◆ calc_sense()

CELER_FUNCTION SignedSense celeritas::Involute::calc_sense ( Real3 const & pos ) const

inline

Determine the sense of the position relative to this surface.

This is performed by first checking if the particle is outside the bounding circles given by tmin and tmax.

Then position is compared to a point on the involute at the same radius from the origin.

Next the point is determined to be inside if an involute that passes it, has an t that is within the bounds and an a greater than the involute being tested for. To do this, the tangent to the involute must be determined by calculating the intercepts of the circle of involute and circle centered at the midpoint of the origin and the point being tested for that extends to the origin. After which the angle of tangent point can be determined and used to calculate a via:

\[ a = angle - t_point \]

To calculate the intercept of two circles it is assumed that the two circles are x_prime axis. The coordinate along x_prime originating from the center of one of the circles is given by:

\[ x_prime = (d^2 - r^2 + R^2)/(2d) \]

Where d is the distance between the center of the two circles, r is the radius of the displaced circle, and R is the radius of the circle at the orgin. Then the point y_prime can be obatined from:

\[ y_prime = \pm sqrt(R^2 - x_prime^2) \]

Then to convert ( x_prime , y_prime ) to ( x, y ) the following rotation is applied:

\[ x = x_prime*cos(theta) - y_prime*sin(theta) y = y_prime*cos(theta) + x_prime*sin(theta) \]

Where theta is the angle between the x_prime axis and the x axis.