SpECTRE
v2023.01.13

Redistributes gridpoints within the unit sphere. More...
#include <SpecialMobius.hpp>
Public Member Functions  
SpecialMobius (double mu)  
SpecialMobius (SpecialMobius &&)=default  
SpecialMobius (const SpecialMobius &)=default  
SpecialMobius &  operator= (const SpecialMobius &)=default 
SpecialMobius &  operator= (SpecialMobius &&)=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 for target_coords outside the unit sphere. 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::NoFrame >  jacobian (const std::array< T, 3 > &source_coords) const 
template<typename T >  
tnsr::Ij< tt::remove_cvref_wrap_t< T >, 3, Frame::NoFrame >  inv_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 
Friends  
bool  operator== (const SpecialMobius &lhs, const SpecialMobius &rhs) 
Redistributes gridpoints within the unit sphere.
A special case of the conformal Mobius transformation that maps the unit ball to itself. This map depends on a single parameter, mu
\( = \mu\), which is the xcoordinate of the preimage of the origin under this map. This map has the fixed points \(x=1\) and \(x=1\). The map is singular for \(\mu=1\) but we have found that this map is accurate up to 12 decimal places for values of \(\mu\) up to 0.96.
We define the auxiliary variables
\[ r := \sqrt{x^2 + y^2 +z^2}\]
and
\[ \lambda := \frac{1}{1  2 x \mu + \mu^2 r^2}\]
The map corresponding to this transformation in cartesian coordinates is then given by:
\[\vec{x}'(x,y,z) = \lambda\begin{bmatrix} x(1+\mu^2)  \mu(1+r^2)\\ y(1\mu^2)\\ z(1\mu^2)\\ \end{bmatrix}\]
The inverse map is the same as the forward map with \(\mu\) replaced by \(\mu\).
This map is intended to be used only inside the unit sphere. A point inside the unit sphere maps to another point inside the unit sphere. The map can have undesirable behavior at certain points outside the unit sphere: The map is singular at \((x,y,z) = (1/\mu, 0, 0)\) (which is outside the unit sphere since \(\mu < 1\)). Moreover, a point on the \(x\)axis arbitrarily close to the singularity maps to an arbitrarily large value on the \(\pm x\)axis, where the sign depends on which side of the singularity the point is on.
A general Mobius transformation is a function on the complex plane, and takes the form \( f(z) = \frac{az+b}{cz+d}\), where \(z, a, b, c, d \in \mathbb{C}\), and \(adbc\neq 0\).
The special case used in this map is the function \( f(z) = \frac{z  \mu}{1  z\mu}\). This has the desired properties:
The threedimensional version of this map is obtained by rotating the disk in the plane about the xaxis.
This map is useful for performing transformations along the xaxis that preserve the unit disk. A concrete example of this is in the BBH domain, where two BBHs with a centerofmass at x= \(\mu\) can be shifted such that the new center of mass is now located at x=0. Additionally, the spherical shape of the outer wavezone is preserved and, as a mobius map, the spherical coordinate shapes of the black holes is also preserved.