SpECTRE  v2022.05.05
domain::CoordinateMaps::Frustum Class Reference

A reorientable map from the cube to a frustum. More...

#include <Frustum.hpp>

Public Member Functions

 Frustum (const std::array< std::array< double, 2 >, 4 > &face_vertices, double lower_bound, double upper_bound, OrientationMap< 3 > orientation_of_frustum, bool with_equiangular_map=false, double projective_scale_factor=1.0, bool auto_projective_scale_factor=false, double sphericity=0.0)
 Frustum (Frustum &&)=default
 Frustum (const Frustum &)=default
Frustumoperator= (const Frustum &)=default
Frustumoperator= (Frustum &&)=default
template<typename T >
std::array< tt::remove_cvref_wrap_t< T >, 3 > operator() (const std::array< T, 3 > &source_coords) const
std::optional< std::array< double, 3 > > inverse (const std::array< double, 3 > &target_coords) const
 Returns std::nullopt if \(z\) is at or beyond the \(z\)-coordinate of the apex of the pyramid, tetrahedron, or triangular prism that is formed by extending the Frustum (for a \(z\)-oriented Frustum). The inverse function is only callable with doubles because the inverse might fail if called for a point out of range, and it is unclear what should happen if the inverse were to succeed for some points in a DataVector but fail for other points.
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFramejacobian (const std::array< T, 3 > &source_coords) const
template<typename T >
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrameinv_jacobian (const std::array< T, 3 > &source_coords) const
void pup (PUP::er &p)
bool is_identity () const

Static Public Attributes

static constexpr size_t dim = 3


bool operator== (const Frustum &lhs, const Frustum &rhs)

Detailed Description

A reorientable map from the cube to a frustum.

A frustum with rectangular bases.

Description and Specifiable Parameters

A map from the logical cube to the volume determined by interpolating between two parallel rectangles each perpendicular to the \(z\)-axis. A Frustum map \(\vec{x}(\xi,\eta,\zeta)\) is determined by specifying two heights \(z_1 = \) lower_bound and \(z_2 = \) upper_bound for the positions of the rectangular bases of the frustum along the \(z-\)axis, and the sizes of the rectangles are determined by the eight values passed to face_vertices:

\begin{align*} &\textrm{lower x upper base} : x^{+\zeta}_{-\xi,-\eta} = x(-1,-1,1)\\ &\textrm{lower x lower base} : x^{-\zeta}_{-\xi,-\eta} = x(-1,-1,-1)\\ &\textrm{upper x upper base} : x^{+\zeta}_{+\xi,+\eta} = x(1,1,1)\\ &\textrm{upper x lower base} : x^{-\zeta}_{+\xi,+\eta} = x(1,1,-1)\\ &\textrm{lower y upper base} : y^{+\zeta}_{-\xi,-\eta} = y(-1,-1,1)\\ &\textrm{lower y lower base} : y^{-\zeta}_{-\xi,-\eta} = y(-1,-1,-1)\\ &\textrm{upper y upper base} : y^{+\zeta}_{+\xi,+\eta} = y(1,1,1)\\ &\textrm{upper y lower base} : y^{-\zeta}_{+\xi,+\eta} = y(1,1,-1)\end{align*}

As an example, consider a frustum along the z-axis, with the lower base starting at (x,y) = (-2.0,3.0) and extending to (2.0,5.0), and with the upper base extending from (0.0,1.0) to (1.0,3.0). The corresponding value for face_vertices is {{{{-2.0,3.0}}, {{2.0,5.0}}, {{0.0,1.0}}, {{1.0,3.0}}}}.

In the case where the two rectangles are geometrically similar, the volume formed is a geometric frustum. However, this coordinate map generalizes to rectangles which need not be similar. The user may reorient the frustum by passing an OrientationMap to the constructor. If with_equiangular_map is true, then this coordinate map applies a tangent function mapping to the logical \(\xi\) and \(\eta\) coordinates. We then refer to the generalized logical coordinates as \(\Xi\) and \(\mathrm{H}\). If projective_scale_factor is set to a quantity other than unity, then this coordinate map applies a rational function mapping to the logical \(\zeta\) coordinate. This generalized logical coordinate is referred to as \(\mathrm{Z}\). If auto_projective_scale_factor is true, the user-specified projective_scale_factor is ignored and an appropriate value for projective_scale_factor is computed based on the values passed to face_vertices. See the page on redistributing gridpoints to see more detailed information on equiangular variables and projective scaling.

Coordinate Map and Jacobian

In terms of the face_vertices variables, we define the following auxiliary variables:

\begin{align*}\Sigma^x &= \frac{1}{4}(x^{+\zeta}_{-\xi,-\eta} + x^{+\zeta}_{+\xi,+\eta} + x^{-\zeta}_{-\xi,-\eta} + x^{-\zeta}_{+\xi,+\eta}) \\ \Delta^x_{\zeta} &= \frac{1}{4}(x^{+\zeta}_{-\xi,-\eta} + x^{+\zeta}_{+\xi,+\eta} - x^{-\zeta}_{-\xi,-\eta} - x^{-\zeta}_{+\xi,+\eta}) \\ \Delta^x_{\xi} &= \frac{1}{4}(x^{+\zeta}_{+\xi,+\eta} - x^{+\zeta}_{-\xi,-\eta} + x^{-\zeta}_{+\xi,+\eta} - x^{-\zeta}_{-\xi,-\eta}) \\ \Delta^x_{\xi\zeta} &= \frac{1}{4}(x^{+\zeta}_{+\xi,+\eta} - x^{+\zeta}_{-\xi,-\eta} - x^{-\zeta}_{+\xi,+\eta} + x^{-\zeta}_{-\xi,-\eta}) \\ \Sigma^y &= \frac{1}{4}(y^{+\zeta}_{-\xi,-\eta} + y^{+\zeta}_{+\xi,+\eta} + y^{-\zeta}_{-\xi,-\eta} + y^{-\zeta}_{+\xi,+\eta}) \\ \Delta^y_{\zeta} &= \frac{1}{4}(y^{+\zeta}_{-\xi,-\eta} + y^{+\zeta}_{+\xi,+\eta} - y^{-\zeta}_{-\xi,-\eta} - y^{-\zeta}_{+\xi,+\eta}) \\ \Delta^y_{\eta} &= \frac{1}{4}(y^{+\zeta}_{+\xi,+\eta} - y^{+\zeta}_{-\xi,-\eta} + y^{-\zeta}_{+\xi,+\eta} - y^{-\zeta}_{-\xi,-\eta}) \\ \Delta^y_{\eta\zeta} &= \frac{1}{4}(y^{+\zeta}_{+\xi,+\eta} - y^{+\zeta}_{-\xi,-\eta} - y^{-\zeta}_{+\xi,+\eta} + y^{-\zeta}_{-\xi,-\eta}) \\ \Sigma^z &= \frac{z_1 + z_2}{2}\\ \Delta^z &= \frac{z_2 - z_1}{2} \end{align*}

The full map is then given by:

\[\vec{x}(\xi,\eta,\zeta) = \begin{bmatrix} \Sigma^x + \Delta^x_{\xi}\Xi + (\Delta^x_{\zeta} + \Delta^x_{\xi\zeta}\Xi)\mathrm{Z}\\ \Sigma^y + \Delta^y_{\eta}\mathrm{H} + (\Delta^y_{\zeta} + \Delta^y_{\eta\zeta}\mathrm{H})\mathrm{Z}\\ \Sigma^z + \Delta^z\mathrm{Z}\\ \end{bmatrix}\]

With Jacobian:

\[J = \begin{bmatrix} (\Delta^x_{\xi} + \Delta^x_{\xi\zeta}\mathrm{Z})\Xi' & 0 & (\Delta^x_{\zeta}+ \Delta^x_{\xi\zeta}\Xi)\mathrm{Z}' \\ 0 & (\Delta^y_{\eta} + \Delta^y_{\eta\zeta}\mathrm{Z})\mathrm{H}' & (\Delta^y_{\zeta} + \Delta^y_{\eta\zeta}\mathrm{H})\mathrm{Z}'\\ 0 & 0 & \Delta^z\mathrm{Z}' \\ \end{bmatrix} \]

Suggested values for the projective scale factor

This section assumes familiarity with projective scaling as discussed on the page on redistributing gridpoints.

When constructing a Frustum map, it is not immediately obvious what value of \(w_{\delta}\) to use in the projective map; here we present a choice of \(w_{\delta}\) that will produce minimal grid distortion. We cover two cases: the first is the special case in which the two rectangular Frustum bases are related by a simple scale factor; the second is the general case.

Trapezoids from squares. (Davide Cervone)

As seen in Cervone's [Cubes and Hypercubes Rotating] (http://www.math.union.edu/~dpvc/math/4D/rotation/welcome.html), there is a special case in which the inverse projection of a trapezoid is not another trapezoid, but a rectangle where the bases are congruent. Most often one will want to use the special value of \(w_{\delta}\) where this occurs. This value of \(w_{\delta}\) corresponds to the projective transformation mapping a rectangular prism in an ambient 4D space with congruent faces at \(w=1\) and \(w=w_{\delta}\) to the frustum in the plane \(w=1\):

\[w_{\delta} = \frac{L_1}{L_2}\]

where \(\frac{L_1}{L_2}\) is the ratio between any pair of corresponding side lengths of the \(z_1\) and \(z_2\)-bases of the frustum, respectively.

For the general case one will want to use the value:

\[w_{\delta} = \frac{\sqrt{(x^{-\zeta}_{+\xi,+\eta} - x^{-\zeta}_{-\xi,+\eta}) (y^{-\zeta}_{+\xi,+\eta} - y^{-\zeta}_{+\xi,-\eta})}} {\sqrt{(x^{+\zeta}_{+\xi,+\eta} - x^{+\zeta}_{-\xi,+\eta}) (y^{+\zeta}_{+\xi,+\eta} - y^{+\zeta}_{+\xi,-\eta})}}\]

This is the value for \(w_{\delta}\) used by this CoordinateMap when auto_projective_scale_factor is true.

Bulged Frustum Coordinate Map and Jacobian

Each of the frustum faces in the frustum map given above are flat, but the upper +z face of the frustum can be bulged out by setting a non-zero value for the sphericity, where a value of 0.0 corresponds to the usual flat- face, and a value of 1.0 corresponds to a value of fully spherical. Using OrientationMaps allows the user to create a set of frustums that fully cover a spherical surface. The radius of the sphere is determined by the corner of the frustum that is furthest from the origin.

The full map is given by:

\[\vec{x}(\xi,\eta,\zeta) = \begin{bmatrix} \Sigma^x + \Delta^x_{\xi}\Xi + (\Delta^x_{\zeta} + \Delta^x_{\xi\zeta}\Xi)\mathrm{Z}\\ \Sigma^y + \Delta^y_{\eta}\mathrm{H} + (\Delta^y_{\zeta} + \Delta^y_{\eta\zeta}\mathrm{H})\mathrm{Z}\\ \Sigma^z + \Delta^z\mathrm{Z}\\ \end{bmatrix} + s \frac{1 + \mathrm{Z}}{2}\left(\frac{R}{|\vec{\sigma}_{\mathrm{+z}}|}-1\right) \vec{\sigma}_{\mathrm{+z}} \]

where \(R\) is the radius, \(s\) is the sphericity, and \(\vec{\sigma}_{\mathrm{+z}}\) is given by:

\[ \vec{\sigma}_{\mathrm{+z}} = \begin{bmatrix} \Sigma^x + \Delta^x_{\xi}\Xi + \Delta^x_{\zeta} + \Delta^x_{\xi\zeta}\Xi\\ \Sigma^y + \Delta^y_{\eta}\mathrm{H} + \Delta^y_{\zeta} + \Delta^y_{\eta\zeta}\mathrm{H}\\ \Sigma^z + \Delta^z\\ \end{bmatrix}. \]

The Jacobian is:

\[J = \begin{bmatrix} (\Delta^x_{\xi} + \Delta^x_{\xi\zeta}\mathrm{Z})\Xi' & 0 & (\Delta^x_{\zeta}+ \Delta^x_{\xi\zeta}\Xi)\mathrm{Z}' \\ 0 & (\Delta^y_{\eta} + \Delta^y_{\eta\zeta}\mathrm{Z})\mathrm{H}' & (\Delta^y_{\zeta} + \Delta^y_{\eta\zeta}\mathrm{H})\mathrm{Z}'\\ 0 & 0 & \Delta^z\mathrm{Z}' \\ \end{bmatrix} + \frac{s}{2} \left\{ \left(\frac{R}{|\vec{\sigma}_{\mathrm{+z}}|}-1\right) \vec{\sigma}_{\mathrm{+z}}\hat{\zeta}^{\intercal}\mathrm{Z}'+ (1+\mathrm{Z})\left(\frac{R}{|\vec{\sigma}_{\mathrm{+z}}|}\left( \mathbb{1}-\frac{\vec{\sigma}_{\mathrm{+z}} \vec{\sigma}_{\mathrm{+z}}^{\intercal}} {|\vec{\sigma}_{\mathrm{+z}}|^2}\right)-\mathbb{1}\right) J_{\sigma}\right\}, \]

where \(\hat{\zeta}\) is the row vector \([0, 0, 1]\), and \(J_{\sigma}\) is the Jacobian of the upper +z surface, given by:

\[ J_{\sigma} = \begin{bmatrix} (\Delta^x_{\xi} + \Delta^x_{\xi\zeta})\Xi' & 0 & 0 \\ 0 & (\Delta^y_{\eta} + \Delta^y_{\eta\zeta})\mathrm{H}' & 0 \\ 0 & 0 & 0 \\ \end{bmatrix}. \]

The documentation for this class was generated from the following file: