Coverage for local/lib/python2.7/site-packages/sage/geometry/hyperbolic_space/hyperbolic_coercion.py : 100%

# -*- coding: utf-8 -*- 

""" 

Coercion Maps Between Hyperbolic Plane Models 

This module implements the coercion maps between different hyperbolic 

plane models. 

AUTHORS: 

- Travis Scrimshaw (2014): initial version 

""" 

#*********************************************************************** 

# Copyright (C) 2014 Travis Scrimshaw <tscrim at ucdavis.edu> 

# 

# Distributed under the terms of the GNU General Public License (GPL) 

# as published by the Free Software Foundation; either version 2 of 

# the License, or (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#*********************************************************************** 

from sage.categories.morphism import Morphism 

from sage.symbolic.all import I 

from sage.matrix.constructor import matrix 

from sage.modules.free_module_element import vector 

from sage.rings.integer import Integer 

from sage.rings.infinity import infinity 

from sage.functions.other import real, imag, sqrt 

from sage.misc.lazy_import import lazy_import 

lazy_import('sage.misc.misc', 'attrcall') 

class HyperbolicModelCoercion(Morphism): 

""" 

Abstract base class for morphisms between the hyperbolic models. 

""" 

def _repr_type(self): 

""" 

Return the type of morphism. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = UHP.coerce_map_from(PD) 

sage: phi._repr_type() 

'Coercion Isometry' 

""" 

return "Coercion Isometry" 

def _call_(self, x): 

""" 

Return the image of ``x`` under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = UHP.coerce_map_from(PD) 

sage: phi(PD.get_point(0.5+0.5*I)) 

Point in UHP 2.00000000000000 + 1.00000000000000*I 

sage: psi = HM.coerce_map_from(UHP) 

sage: psi(UHP.get_point(I)) 

Point in HM (0, 0, 1) 

It is an error to try to convert a boundary point to a model 

that doesn't support boundary points:: 

sage: psi(UHP.get_point(infinity)) 

Traceback (most recent call last): 

... 

NotImplementedError: boundary points are not implemented for the Hyperboloid Model 

It is an error to try to convert a boundary point to a model 

that doesn't support boundary points:: 

sage: psi(UHP(infinity)) 

Traceback (most recent call last): 

... 

NotImplementedError: boundary points are not implemented for the Hyperboloid Model 

""" 

C = self.codomain() 

if not C.is_bounded() and self.domain().is_bounded() and x.is_boundary(): 

msg = u"boundary points are not implemented for the {}" 

raise NotImplementedError(msg.format(C.name())) 

y = self.image_coordinates(x.coordinates()) 

if self.domain().is_bounded(): 

bdry = x.is_boundary() 

else: 

bdry = C.boundary_point_in_model(y) 

return C.element_class(C, y, bdry, check=False, **x.graphics_options()) 

def convert_geodesic(self, x): 

""" 

Convert the geodesic ``x`` of the domain into a geodesic of 

the codomain. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = UHP.coerce_map_from(PD) 

sage: phi.convert_geodesic(PD.get_geodesic(0.5+0.5*I, -I)) 

Geodesic in UHP from 2.00000000000000 + 1.00000000000000*I to 0 

""" 

return self.codomain().get_geodesic(self(x.start()), self(x.end()), 

**x.graphics_options()) 

def convert_isometry(self, x): 

""" 

Convert the hyperbolic isometry ``x`` of the domain into an 

isometry of the codomain. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(UHP) 

sage: I2 = UHP.get_isometry(identity_matrix(2)) 

sage: phi.convert_isometry(I2) 

Isometry in HM 

[1 0 0] 

[0 1 0] 

[0 0 1] 

""" 

C = self.codomain() 

return C._Isometry(C, self.image_isometry_matrix(x._matrix), check=False) 

def __invert__(self): 

""" 

Return the inverse coercion of ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = UHP.coerce_map_from(PD) 

sage: ~phi 

Coercion Isometry morphism: 

From: Hyperbolic plane in the Upper Half Plane Model 

To: Hyperbolic plane in the Poincare Disk Model 

""" 

return self.domain().coerce_map_from(self.codomain()) 

############ 

# From UHP # 

############ 

class CoercionUHPtoPD(HyperbolicModelCoercion): 

""" 

Coercion from the UHP to PD model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(UHP) 

sage: phi.image_coordinates(I) 

0 

""" 

if x == infinity: 

return I 

return (x - I) / (Integer(1) - I*x) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(UHP) 

sage: phi.image_isometry_matrix(identity_matrix(2)) 

[1 0] 

[0 1] 

""" 

if x.det() < 0: 

# x = I * x 

return matrix([[1,-I],[-I,1]]) * x * matrix([[1,I],[I,1]]).conjugate()/Integer(2) 

return matrix([[1,-I],[-I,1]]) * x * matrix([[1,I],[I,1]])/Integer(2) 

class CoercionUHPtoKM(HyperbolicModelCoercion): 

""" 

Coercion from the UHP to KM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(UHP) 

sage: phi.image_coordinates(3 + I) 

(6/11, 9/11) 

""" 

if x == infinity: 

return (0, 1) 

return ((2*real(x))/(real(x)**2 + imag(x)**2 + 1), 

(real(x)**2 + imag(x)**2 - 1)/(real(x)**2 + imag(x)**2 + 1)) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(UHP) 

sage: phi.image_isometry_matrix(identity_matrix(2)) 

[1 0 0] 

[0 1 0] 

[0 0 1] 

""" 

return SL2R_to_SO21(x) 

class CoercionUHPtoHM(HyperbolicModelCoercion): 

""" 

Coercion from the UHP to HM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(UHP) 

sage: phi.image_coordinates(3 + I) 

(3, 9/2, 11/2) 

""" 

return vector((real(x)/imag(x), 

(real(x)**2 + imag(x)**2 - 1)/(2*imag(x)), 

(real(x)**2 + imag(x)**2 + 1)/(2*imag(x)))) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: UHP = HyperbolicPlane().UHP() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(UHP) 

sage: phi.image_isometry_matrix(identity_matrix(2)) 

[1 0 0] 

[0 1 0] 

[0 0 1] 

""" 

return SL2R_to_SO21(x) 

########### 

# From PD # 

########### 

class CoercionPDtoUHP(HyperbolicModelCoercion): 

""" 

Coercion from the PD to UHP model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: PD = HyperbolicPlane().PD() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(PD) 

sage: phi.image_coordinates(0.5+0.5*I) 

2.00000000000000 + 1.00000000000000*I 

sage: phi.image_coordinates(0) 

I 

sage: phi.image_coordinates(I) 

+Infinity 

sage: phi.image_coordinates(-I) 

0 

""" 

if x == I: 

return infinity 

return (x + I)/(Integer(1) + I*x) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES: 

We check that orientation-reversing isometries behave as they 

should:: 

sage: PD = HyperbolicPlane().PD() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(PD) 

sage: phi.image_isometry_matrix(matrix([[0,I],[I,0]])) 

[-1 0] 

[ 0 -1] 

""" 

from sage.geometry.hyperbolic_space.hyperbolic_isometry import HyperbolicIsometryPD 

if not HyperbolicIsometryPD._orientation_preserving(x): 

return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]).conjugate() / Integer(2) 

return matrix([[1,I],[I,1]]) * x * matrix([[1,-I],[-I,1]]) / Integer(2) 

class CoercionPDtoKM(HyperbolicModelCoercion): 

""" 

Coercion from the PD to KM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: PD = HyperbolicPlane().PD() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(PD) 

sage: phi.image_coordinates(0.5+0.5*I) 

(0.666666666666667, 0.666666666666667) 

""" 

return (2*real(x)/(Integer(1) + real(x)**2 + imag(x)**2), 

2*imag(x)/(Integer(1) + real(x)**2 + imag(x)**2)) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: PD = HyperbolicPlane().PD() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(PD) 

sage: phi.image_isometry_matrix(matrix([[0,I],[I,0]])) 

[-1 0 0] 

[ 0 1 0] 

[ 0 0 -1] 

""" 

return SL2R_to_SO21(matrix(2, [1, I, I, 1]) * x * 

matrix(2, [1, -I, -I, 1]) / Integer(2)) 

class CoercionPDtoHM(HyperbolicModelCoercion): 

""" 

Coercion from the PD to HM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: PD = HyperbolicPlane().PD() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(PD) 

sage: phi.image_coordinates(0.5+0.5*I) 

(2.00000000000000, 2.00000000000000, 3.00000000000000) 

""" 

return vector((2*real(x)/(1 - real(x)**2 - imag(x)**2), 

2*imag(x)/(1 - real(x)**2 - imag(x)**2), 

(real(x)**2 + imag(x)**2 + 1) / 

(1 - real(x)**2 - imag(x)**2))) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: PD = HyperbolicPlane().PD() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(PD) 

sage: phi.image_isometry_matrix(matrix([[0,I],[I,0]])) 

[-1 0 0] 

[ 0 1 0] 

[ 0 0 -1] 

""" 

return SL2R_to_SO21(matrix(2, [1, I, I, 1]) * x * 

matrix(2, [1, -I, -I, 1]) / Integer(2)) 

########### 

# From KM # 

########### 

class CoercionKMtoUHP(HyperbolicModelCoercion): 

""" 

Coercion from the KM to UHP model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(KM) 

sage: phi.image_coordinates((0, 0)) 

I 

sage: phi.image_coordinates((0, 1)) 

+Infinity 

""" 

if tuple(x) == (0, 1): 

return infinity 

return (-x[0]/(x[1] - 1) 

+ I*(-(sqrt(-x[0]**2 - x[1]**2 + 1) - x[0]**2 - x[1]**2 + 1) 

/ ((x[1] - 1)*sqrt(-x[0]**2 - x[1]**2 + 1) + x[1] - 1))) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(KM) 

sage: m = matrix([[5/3,0,4/3], [0,1,0], [4/3,0,5/3]]) 

sage: phi.image_isometry_matrix(m) 

[2*sqrt(1/3) sqrt(1/3)] 

[ sqrt(1/3) 2*sqrt(1/3)] 

""" 

return SO21_to_SL2R(x) 

class CoercionKMtoPD(HyperbolicModelCoercion): 

""" 

Coercion from the KM to PD model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(KM) 

sage: phi.image_coordinates((0, 0)) 

0 

""" 

return (x[0]/(1 + (1 - x[0]**2 - x[1]**2).sqrt()) 

+ I*x[1]/(1 + (1 - x[0]**2 - x[1]**2).sqrt())) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(KM) 

sage: m = matrix([[5/3,0,4/3], [0,1,0], [4/3,0,5/3]]) 

sage: phi.image_isometry_matrix(m) 

[2*sqrt(1/3) sqrt(1/3)] 

[ sqrt(1/3) 2*sqrt(1/3)] 

""" 

return (matrix(2,[1,-I,-I,1]) * SO21_to_SL2R(x) * 

matrix(2,[1,I,I,1])/Integer(2)) 

class CoercionKMtoHM(HyperbolicModelCoercion): 

""" 

Coercion from the KM to HM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(KM) 

sage: phi.image_coordinates((0, 0)) 

(0, 0, 1) 

""" 

return (vector((2*x[0], 2*x[1], 1 + x[0]**2 + x[1]**2)) 

/ (1 - x[0]**2 - x[1]**2)) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: KM = HyperbolicPlane().KM() 

sage: HM = HyperbolicPlane().HM() 

sage: phi = HM.coerce_map_from(KM) 

sage: m = matrix([[5/3,0,4/3], [0,1,0], [4/3,0,5/3]]) 

sage: phi.image_isometry_matrix(m) 

[5/3 0 4/3] 

[ 0 1 0] 

[4/3 0 5/3] 

""" 

return x 

########### 

# From HM # 

########### 

class CoercionHMtoUHP(HyperbolicModelCoercion): 

""" 

Coercion from the HM to UHP model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(HM) 

sage: phi.image_coordinates( vector((0,0,1)) ) 

I 

""" 

return -((x[0]*x[2] + x[0]) + I*(x[2] + 1)) / ((x[1] - 1)*x[2] 

- x[0]**2 - x[1]**2 + x[1] - 1) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: UHP = HyperbolicPlane().UHP() 

sage: phi = UHP.coerce_map_from(HM) 

sage: phi.image_isometry_matrix(identity_matrix(3)) 

[1 0] 

[0 1] 

""" 

return SO21_to_SL2R(x) 

class CoercionHMtoPD(HyperbolicModelCoercion): 

""" 

Coercion from the HM to PD model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(HM) 

sage: phi.image_coordinates( vector((0,0,1)) ) 

0 

""" 

return x[0]/(1 + x[2]) + I*(x[1]/(1 + x[2])) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: PD = HyperbolicPlane().PD() 

sage: phi = PD.coerce_map_from(HM) 

sage: phi.image_isometry_matrix(identity_matrix(3)) 

[1 0] 

[0 1] 

""" 

return (matrix(2,[1,-I,-I,1]) * SO21_to_SL2R(x) * 

matrix(2,[1,I,I,1])/Integer(2)) 

class CoercionHMtoKM(HyperbolicModelCoercion): 

""" 

Coercion from the HM to KM model. 

""" 

def image_coordinates(self, x): 

""" 

Return the image of the coordinates of the hyperbolic point ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(HM) 

sage: phi.image_coordinates( vector((0,0,1)) ) 

(0, 0) 

""" 

return (x[0]/(1 + x[2]), x[1]/(1 + x[2])) 

def image_isometry_matrix(self, x): 

""" 

Return the image of the matrix of the hyperbolic isometry ``x`` 

under ``self``. 

EXAMPLES:: 

sage: HM = HyperbolicPlane().HM() 

sage: KM = HyperbolicPlane().KM() 

sage: phi = KM.coerce_map_from(HM) 

sage: phi.image_isometry_matrix(identity_matrix(3)) 

[1 0 0] 

[0 1 0] 

[0 0 1] 

""" 

return x 

##################################################################### 

## Helper functions 

def SL2R_to_SO21(A): 

r""" 

Given a matrix in `SL(2, \RR)` return its irreducible representation in 

`O(2,1)`. 

Note that this is not the only homomorphism, but it is the only one 

that works in the context of the implemented 2D hyperbolic geometry 

models. 

EXAMPLES:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_coercion import SL2R_to_SO21 

sage: A = SL2R_to_SO21(identity_matrix(2)) 

sage: J = matrix([[1,0,0],[0,1,0],[0,0,-1]]) #Lorentzian Gram matrix 

sage: norm(A.transpose()*J*A - J) < 10**-4 

True 

""" 

a, b, c, d = (A/A.det().sqrt()).list() 

# Kill ~0 imaginary parts 

B = matrix(3, map(real, 

[a*d + b*c, a*c - b*d, a*c + b*d, a*b - c*d, 

Integer(1)/Integer(2)*a**2 - Integer(1)/Integer(2)*b**2 - 

Integer(1)/Integer(2)*c**2 + Integer(1)/Integer(2)*d**2, 

Integer(1)/Integer(2)*a**2 + Integer(1)/Integer(2)*b**2 - 

Integer(1)/Integer(2)*c**2 - Integer(1)/Integer(2)*d**2, 

a*b + c*d, Integer(1)/Integer(2)*a**2 - 

Integer(1)/Integer(2)*b**2 + Integer(1)/Integer(2)*c**2 - 

Integer(1)/Integer(2)*d**2, Integer(1)/Integer(2)*a**2 + 

Integer(1)/Integer(2)*b**2 + Integer(1)/Integer(2)*c**2 + 

Integer(1)/Integer(2)*d**2])) 

#B = B.apply_map(attrcall('real')) 

if A.det() > 0: 

return B 

else: 

# Orientation-reversing isometries swap the nappes of 

# the lightcone. This fixes that issue. 

return -B 

def SO21_to_SL2R(M): 

r""" 

A homomorphism from `SO(2, 1)` to `SL(2, \RR)`. 

Note that this is not the only homomorphism, but it is the only one 

that works in the context of the implemented 2D hyperbolic geometry 

models. 

EXAMPLES:: 

sage: from sage.geometry.hyperbolic_space.hyperbolic_coercion import SO21_to_SL2R 

sage: (SO21_to_SL2R(identity_matrix(3)) - identity_matrix(2)).norm() < 10**-4 

True 

""" 

#################################################################### 

# SL(2,R) is the double cover of SO (2,1)^+, so we need to choose # 

# a lift. I have formulas for the absolute values of each entry # 

# a,b ,c,d of the lift matrix(2,[a,b,c,d]), but we need to choose # 

# one entry to be positive. I choose d for no particular reason, # 

# unless d = 0, then we choose c > 0. The basic strategy for this # 

# function is to find the linear map induced by the SO(2,1) # 

# element on the Lie algebra sl(2, R). This corresponds to the # 

# Adjoint action by a matrix A or -A in SL(2,R). To find which # 

# matrix let X,Y,Z be a basis for sl(2,R) and look at the images # 

# of X,Y,Z as well as the second and third standard basis vectors # 

# for 2x2 matrices (these are traceless, so are in the Lie # 

# algebra). These corresponds to AXA^-1 etc and give formulas # 

# for the entries of A. # 

#################################################################### 

(m_1,m_2,m_3,m_4,m_5,m_6,m_7,m_8,m_9) = M.list() 

d = sqrt(Integer(1)/Integer(2)*m_5 - Integer(1)/Integer(2)*m_6 - 

Integer(1)/Integer(2)*m_8 + Integer(1)/Integer(2)*m_9) 

if M.det() > 0: # EPSILON? 

det_sign = 1 

elif M.det() < 0: # EPSILON? 

det_sign = -1 

if d > 0: # EPSILON? 

c = (-Integer(1)/Integer(2)*m_4 + Integer(1)/Integer(2)*m_7)/d 

b = (-Integer(1)/Integer(2)*m_2 + Integer(1)/Integer(2)*m_3)/d 

ad = det_sign*1 + b*c # ad - bc = pm 1 

a = ad/d 

else: # d is 0, so we make c > 0 

c = sqrt(-Integer(1)/Integer(2)*m_5 - Integer(1)/Integer(2)*m_6 + 

Integer(1)/Integer(2)*m_8 + Integer(1)/Integer(2)*m_9) 

d = (-Integer(1)/Integer(2)*m_4 + Integer(1)/Integer(2)*m_7)/c 

# d = 0, so ad - bc = -bc = pm 1. 

b = - (det_sign*1)/c 

a = (Integer(1)/Integer(2)*m_4 + Integer(1)/Integer(2)*m_7)/b 

A = matrix(2, [a, b, c, d]) 

return A