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// coords.hh -- coordinate systems and conversions
// $Id$
//
// local_coords - conversions between various local coordinate systems
// global_coords - conversions between global and local coordinates
//
//
// prerequisites:
// "jt/util++.hh"
// fp.hh
//
//******************************************************************************
//
// We use the following terminology for coordinates:
// {a,b,c} == any one of (a,b,c) == a | b | c
// ((a,b,c)) == any two of (a,b,c) == (a,b) | (a,c) | (b,c)
// (r,(a,b,c)) == r and any two of (a,b,c) == (r,a,b) | (r,a,c) | (r,b,c)
//
//
// We use the following coordinates and coordinate systems:
// global_(x,y,z) global Cartesian x,y,z coordinates
// (these are the coordinates Cactus gives us)
// (x,y,z) local Cartesian x,y,z coordinates relative to a
// local origin point (stored in this object);
// This origin should be chosen to be somewhere
// at least vaguely in the "middle" of our
// apparent horizon
// (r,theta,phi) polar spherical coordinates about the local
// origin point
// (r,(mu,nu,phi)) patch coordinates about the local origin point
// mu arctan(y/z) = rotation about local x axis
// nu arctan(x/z) = rotation about local y axis
// phi arctan(y/x) = rotation about local z axis
// = usual polar spherical phi
// (rho,sigma) generic patch angular coordinates, these will
// be whichever two of (mu,nu,phi) are nonsingular
// throughout the patch
// tau the third member of (mu,nu,phi) (the one which
// is neither rho nor sigma)
//
//*****************************************************************************
//
// The various local_coords functions provide conversions between
// different local coordinate systems.
namespace local_coords
{
//
// Bugs:
// These coordinate-conversion functions aren't optimally efficient:
// they do lots of could-be-optimized away arithmetic and trig calls.
// But in practice this doesn't matter, since we don't call these
// functions inside inner loops. (We cache their values at grid
// points.)
//
// ((mu,nu,phi)) --> the 3rd
fp phi_of_mu_nu(fp mu, fp nu );
fp nu_of_mu_phi(fp mu, fp phi);
fp mu_of_nu_phi(fp nu, fp phi);
// (r,(mu,nu,phi)) <--> (x,y,z)
// ... on the (local) unit sphere, i.e. r == 1.0
void xyz_of_mu_nu (fp mu, fp nu , fp &x, fp &y, fp &z);
void xyz_of_mu_phi(fp mu, fp phi, fp &x, fp &y, fp &z);
void xyz_of_nu_phi(fp nu, fp phi, fp &x, fp &y, fp &z);
void mu_nu_of_xyz(fp x, fp y, fp z, fp &mu, fp &nu );
void mu_phi_of_xyz(fp x, fp y, fp z, fp &mu, fp &phi);
void nu_phi_of_xyz(fp x, fp y, fp z, fp &nu, fp &phi);
// ... with an arbitrary r coordinate
void xyz_of_r_mu_nu (fp r, fp mu, fp nu , fp &x, fp &y, fp &z);
void xyz_of_r_mu_phi(fp r, fp mu, fp phi, fp &x, fp &y, fp &z);
void xyz_of_r_nu_phi(fp r, fp nu, fp phi, fp &x, fp &y, fp &z);
void r_mu_nu_of_xyz(fp x, fp y, fp z, fp &r, fp &mu, fp &nu );
void r_mu_phi_of_xyz(fp x, fp y, fp z, fp &r, fp &mu, fp &phi);
void r_nu_phi_of_xyz(fp x, fp y, fp z, fp &r, fp &nu, fp &phi);
// usual polar spherical (r,theta,phi) <--> (x,y,z)
// ... on the (local) unit sphere, i.e. r == 1.0
void xyz_of_theta_phi(fp theta, fp phi, fp &x, fp &y, fp &z);
void theta_phi_of_xyz(fp x, fp y, fp z, fp &theta, fp &phi);
// ... with an arbitrary r coordinate
void xyz_of_r_theta_phi(fp r, fp theta, fp phi, fp &x, fp &y, fp &z);
void r_theta_phi_of_xyz(fp x, fp y, fp z, fp &r, fp &theta, fp &phi);
// ((mu,nu,phi)) <--> usual polar spherical (theta,phi)
// ... note phi is the same coordinate in both systems
void theta_phi_of_mu_nu (fp mu, fp nu , fp &ps_theta, fp &ps_phi);
void theta_phi_of_mu_phi(fp mu, fp phi, fp &ps_theta, fp &ps_phi);
void theta_phi_of_nu_phi(fp nu, fp phi, fp &ps_theta, fp &ps_phi);
void mu_nu_of_theta_phi(fp ps_theta, fp ps_phi, fp &mu, fp &nu );
void mu_phi_of_theta_phi(fp ps_theta, fp ps_phi, fp &mu, fp &phi);
void nu_phi_of_theta_phi(fp ps_theta, fp ps_phi, fp &nu, fp &phi);
};
//******************************************************************************
//
// This class stores the origin point of our local coordinates, and
// provides conversions between local and global coordinates.
//
class global_coords
{
public:
// global (x,y,z) --> local (x,y,z)
fp local_x_of_global_x(fp global_x) const
{ return global_x - origin_x_; }
fp local_y_of_global_y(fp global_y) const
{ return global_y - origin_y_; }
fp local_z_of_global_z(fp global_z) const
{ return global_z - origin_z_; }
// local (x,y,z) --> global (x,y,z)
fp global_x_of_local_x(fp local_x) const
{ return origin_x_ + local_x; }
fp global_y_of_local_y(fp local_y) const
{ return origin_y_ + local_y; }
fp global_z_of_local_z(fp local_z) const
{ return origin_z_ + local_z; }
// get global (x,y,z) coordinates of local origin point
fp origin_x() const { return origin_x_; }
fp origin_y() const { return origin_y_; }
fp origin_z() const { return origin_z_; }
// ***** constructor, destructor *****
// constructor: specify global (x,y,z) coordinates of local origin point
global_coords(fp origin_x_in, fp origin_y_in, fp origin_z_in)
: origin_x_(origin_x_in),
origin_y_(origin_y_in),
origin_z_(origin_z_in)
{ }
// destructor: compiler-generated no-op is ok
// ***** copy constructor, assignment operator *****
private:
// we forbid copying and passing by value
// by declaring the copy constructor and assignment operator
// private, but never defining them
global_coords(const global_coords &rhs);
global_coords& operator=(const global_coords &rhs);
// ***** member variables *****
private:
// global (x,y,z) coordinates of local origin point
fp origin_x_, origin_y_, origin_z_;
};
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