def test__get_pmz( self ):
pmat = get_pmat( False )
omat = sage_identity_matrix( 9 )
vmat = sage_identity_matrix( 9 )
pmz_lst, prj_pmz_lst = get_pmz( pmat, omat, vmat )
print( pmz_lst )
print( prj_pmz_lst )
assert str( pmz_lst ) == '[1, c1, s1, 0, 0, 0, 0, 0, 0]'
assert str( prj_pmz_lst ) == '[1, c1, s1, 0]'
def test__get_mat_1(self):
#
# Permute a block matrices to obtain rotation along x-axis.
#
print(get_mat('I', 'Oprpp', 'I'))
e_lst = [1, 4, 2, 3, 5, 6, 7, 8]
i_lst = sage_Permutation(e_lst).inverse()
A_str = 'E' + str(i_lst)
B_str = 'Oprpp'
C_str = 'E' + str(e_lst)
tup = (A_str, B_str, C_str)
print(tup)
mati = get_emat(A_str)
mate = get_emat(C_str)
assert mate * mati == sage_identity_matrix(9)
out = get_mat(*tup)
print(out)
print(list(out))
assert str(
list(out)
) == '[(1, 0, 0, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0, 0, 0), (0, 0, c0, -s0, 0, 0, 0, 0, 0), (0, 0, s0, c0, 0, 0, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 0, 0, 1)]'
def get_rot_S3(rot):
'''
Constructs a rotation matrix for S^3.
Parameters
----------
rot : list<int>
List of 6 rotation angles. Each angle is an integer in [0,360].
Returns
-------
sage_matrix
A 5x5 matrix that rotates the projective closure of S^3 in
projective 4-space along the angle parameters.
'''
a12, a13, a23, a14, a24, a34 = rot
M = sage_identity_matrix(5)
M *= get_rot_mat(5, 1, 2, a12)
M *= get_rot_mat(5, 1, 3, a13)
M *= get_rot_mat(5, 2, 3, a23)
M *= get_rot_mat(5, 1, 4, a14)
M *= get_rot_mat(5, 2, 4, a24)
M *= get_rot_mat(5, 3, 4, a34)
return sage_matrix(M)
def test__imp_pmz(self):
A = sage__eval(
'[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 119/169, -120/169, 0), (0, 0, 120/169, 119/169, 0), (0, 0, 0, 0, 1)]'
)
A = sage_matrix(A)
B = sage_identity_matrix(5)
baseA, baseB, pmzAB = get_pmz(A, B, 0)
dct = get_imp(A, B, 0, False, False)
key_lst = [
'Agreat', 'Bgreat', 'eqn_x', 'eqn_str', 'eqn_xyz', 'sng_lst'
]
Agreat, Bgreat, eqn_x, eqn_str, eqn_xyz, sng_lst = [
dct[key] for key in key_lst
]
x, y, z = sage_var('x,y,z')
seqn = eqn_xyz.subs({
x: pmzAB[0],
y: pmzAB[1],
z: pmzAB[2]
}).simplify_trig()
print(seqn)
assert seqn == 0
def get_omat(o_str):
'''
Parameters
----------
o_str : string
A string with format:
'O****'
where the *-symbol is a place holder for
one of the following characters: r,s,m,p,a.
Returns
-------
sage_matrix
A 9x9 matrix over "OrbRing.num_field" of the shape
1 0 0 0 0 0 0 0 0
0 * * 0 0 0 0 0 0
0 * * 0 0 0 0 0 0
0 0 0 * * 0 0 0 0
0 0 0 * * 0 0 0 0
0 0 0 0 0 * * 0 0
0 0 0 0 0 * * 0 0
0 0 0 0 0 0 0 * *
0 0 0 0 0 0 0 * *
where each 2x2 matrix on the diagonal is of either
one of the following shapes:
'r': c0 -s0 's': -c0 s0
s0 c0 -s0 -c0
'p': 1 0 'm': -1 0 'a': 1 0
0 1 0 -1 0 -1
'''
# parse input
if o_str[0] != 'O' or len(o_str) != 5:
raise ValueError('Incorrect input string: ', o_str)
c0, s0 = OrbRing.coerce('c0,s0')
br = [[c0, -s0], [s0, c0]]
bs = [[-c0, s0], [-s0, -c0]]
bp = [[1, 0], [0, 1]]
bm = [[-1, 0], [0, -1]]
ba = [[1, 0], [0, -1]]
b_dct = {'r': br, 's': bs, 'p': bp, 'm': bm, 'a': ba}
bmat_lst = []
for ch in o_str[1:]:
bmat_lst += [b_dct.get(ch, 'error')]
omat = sage_identity_matrix(OrbRing.R, 9)
idx = 1
for bmat in bmat_lst:
omat.set_block(idx, idx, sage_matrix(OrbRing.R, bmat))
idx = idx + 2
return omat
def test__get_pmz(self):
A = sage__eval(
'[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 119/169, -120/169, 0), (0, 0, 120/169, 119/169, 0), (0, 0, 0, 0, 1)]'
)
A = sage_matrix(A)
B = sage_identity_matrix(5)
baseA, baseB, pmzAB = get_pmz(A, B, 0)
print(baseA)
print(baseB)
print(pmzAB)
assert len(baseA) == 3
assert len(baseB) == 3
assert len(pmzAB) == 3
def test__get_rot_S3(self):
a01, a02, a03, a12, a13, a23 = 6 * [0]
a23 = 2
c, s = get_cs(a23)
out = get_rot_S3(a01, a02, a03, a12, a13, a23)
print('out =')
print(out)
chk = sage_identity_matrix(sage_QQ, 5)
chk[2, 2] = c
chk[2, 3] = -s
chk[3, 2] = s
chk[3, 3] = c
print('chk =')
print(chk)
assert str(list(out)) == str(list(chk))
def test__get_imp(self):
A = sage__eval(
'[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 119/169, -120/169, 0), (0, 0, 120/169, 119/169, 0), (0, 0, 0, 0, 1)]'
)
A = sage_matrix(A)
B = sage_identity_matrix(5)
dct = get_imp(A, B, 0, False, False)
key_lst = [
'Agreat', 'Bgreat', 'eqn_x', 'eqn_str', 'eqn_xyz', 'sng_lst'
]
Agreat, Bgreat, eqn_x, eqn_str, eqn_xyz, sng_lst = [
dct[key] for key in key_lst
]
assert Agreat and Bgreat
assert eqn_x.total_degree() == 4
assert sng_lst == []
def blum_cyclide():
'''
Construct a povray image of 6 families of circles on a smooth Darboux cyclide.
This surface is also known as the Blum cyclide.
'''
# construct dct
a0 = PolyRing( 'x,y,v,w', True ).ext_num_field( 't^2 + 1' ).root_gens()[0] # i
bpt_1234 = BasePointTree( ['xv', 'xw', 'yv', 'yw'] )
bpt_1234.add( 'xv', ( -1 * a0, 1 * a0 ), 1 ) # e1
bpt_1234.add( 'xv', ( 1 * a0, -1 * a0 ), 1 ) # e2
bpt_1234.add( 'xw', ( -2 * a0, 2 * a0 ), 1 ) # e3
bpt_1234.add( 'xw', ( 2 * a0, -2 * a0 ), 1 ) # e4
bpt_12 = BasePointTree( ['xv', 'xw', 'yv', 'yw'] )
bpt_12.add( 'xv', ( -1 * a0, 1 * a0 ), 1 ) # e1
bpt_12.add( 'xv', ( 1 * a0, -1 * a0 ), 1 ) # e2
bpt_34 = BasePointTree( ['xv', 'xw', 'yv', 'yw'] )
bpt_34.add( 'xw', ( -2 * a0, 2 * a0 ), 1 ) # e3
bpt_34.add( 'xw', ( 2 * a0, -2 * a0 ), 1 ) # e4
ls_22 = LinearSeries.get( [2, 2], bpt_1234 ) # |2(l1+l2)-e1-e2-e3-e4|
ls_21 = LinearSeries.get( [2, 1], bpt_1234 )
ls_12 = LinearSeries.get( [1, 2], bpt_1234 )
ls_11a = LinearSeries.get( [1, 1], bpt_12 )
ls_11b = LinearSeries.get( [1, 1], bpt_34 )
OrbTools.p( 'linear series 22 =\n', ls_22 )
OrbTools.p( 'linear series 21 =\n', ls_21 )
OrbTools.p( 'linear series 12 =\n', ls_12 )
OrbTools.p( 'linear series 11a =\n', ls_11a )
OrbTools.p( 'linear series 11b =\n', ls_11b )
sig = ( 4, 1 )
pol_lst = ls_22.get_implicit_image()
# determine signature
x_lst = sage_PolynomialRing( sage_QQ, [ 'x' + str( i ) for i in range( sum( sig ) )] ).gens()
for pol in pol_lst:
if pol.degree() == 2:
M = sage_invariant_theory.quadratic_form( pol, x_lst ).as_QuadraticForm().matrix()
D, V = sage_matrix( sage_QQ, M ).eigenmatrix_right() # D has first all negative values on diagonal
cur_sig = ( len( [ d for d in D.diagonal() if d < 0 ] ), len( [ d for d in D.diagonal() if d > 0 ] ) )
else:
cur_sig = '[no signature]'
OrbTools.p( '\t\t', pol, cur_sig )
# obtain surface in sphere
coef_lst = [0, -1, -1]
dct = get_surf( ls_22, sig, coef_lst )
# construct projection matrix P
U, J = dct['UJ']
U.swap_rows( 0, 4 )
J.swap_columns( 0, 4 )
J.swap_rows( 0, 4 )
assert dct['M'] == approx_QQ( U.T * J * U )
approxU = approx_QQ( U )
P = sage_identity_matrix( 5 ).submatrix( 0, 0, 4, 5 )
P[0, 4] = -1;
P = P * approxU
OrbTools.p( ' approx_QQ( U ) =', list( approx_QQ( U ) ) )
OrbTools.p( ' approx_QQ( J ) =', list( approx_QQ( J ) ) )
OrbTools.p( ' P =', list( P ) )
# call get_proj
f_xyz, pmz_AB_lst = get_proj( dct['imp_lst'], dct['pmz_lst'], P )
f_xyz_deg_lst = [f_xyz.degree( sage_var( v ) ) for v in ['x', 'y', 'z']]
# compute reparametrization
ring = PolyRing( 'x,y,v,w,c0,s0,c1,s1' ) # construct polynomial ring with new generators
p_lst = ring.coerce( ls_22.pol_lst )
x, y, v, w, c0, s0, c1, s1 = ring.gens()
X = 1 - s0; Y = c0; # see get_S1xS1_pmz()
V = 1 - s1; W = c1;
CB_dct = { x:X, y:Y, v:X * W + Y * V, w: X * V - Y * W }
DB_dct = { x:X, y:Y, v:4 * X * W - Y * V, w: X * V + Y * W }
EB_dct = { x:X, y:Y, v:40 * W * X ** 2 + 25 * W * Y ** 2 + 24 * V * X * Y, w:40 * V * X ** 2 + 16 * V * Y ** 2 - 15 * W * X * Y }
AF_dct = { x:-10 * Y * V ** 2 - 25 * Y * W ** 2 + 9 * X * V * W, y:15 * X * V ** 2 + 24 * X * W ** 2 - 15 * Y * V * W, v:V, w:W }
pmz_CB_lst = list( P * sage_vector( [ p.subs( CB_dct ) for p in p_lst] ) )
pmz_DB_lst = list( P * sage_vector( [ p.subs( DB_dct ) for p in p_lst] ) )
pmz_EB_lst = list( P * sage_vector( [ p.subs( EB_dct ) for p in p_lst] ) )
pmz_AF_lst = list( P * sage_vector( [ p.subs( AF_dct ) for p in p_lst] ) )
# output
OrbTools.p( 'f_xyz =', f_xyz_deg_lst, '\n', f_xyz )
OrbTools.p( 'pmz_AB_lst =\n', pmz_AB_lst )
OrbTools.p( 'pmz_CB_lst =\n', pmz_CB_lst )
OrbTools.p( 'pmz_DB_lst =\n', pmz_DB_lst )
OrbTools.p( 'pmz_EB_lst =\n', pmz_EB_lst )
OrbTools.p( 'pmz_AF_lst =\n', pmz_AF_lst )
# mathematica
pmz_lst = [ ( pmz_AB_lst, 'AB' ),
( pmz_CB_lst, 'CB' ),
( pmz_DB_lst, 'DB' ),
( pmz_EB_lst, 'EB' ),
( pmz_AF_lst, 'AF' )]
OrbTools.p( 'Mathematica input for ParametricPlot3D:' )
for pmz, AB in pmz_lst:
s = 'pmz' + AB + '=' + str( pmz )
s = s.replace( '[', '{' ).replace( ']', '}' )
print( s )
# PovInput for Blum cyclide
#
pin = PovInput()
pin.path = './' + get_time_str() + '_blum_cyclide/'
pin.fname = 'orb'
pin.scale = 1
pin.cam_dct['location'] = ( 0, -7, 0 )
pin.cam_dct['lookat'] = ( 0, 0, 0 )
pin.cam_dct['rotate'] = ( 20, 180, 20 )
pin.shadow = True
pin.light_lst = [( 0, 0, -5 ), ( 0, -5, 0 ), ( -5, 0, 0 ),
( 0, 0, 5 ), ( 0, 5, 0 ), ( 5, 0, 0 ),
( -5, -5, -5 ), ( 5, -5, 5 ), ( -5, -5, 5 ), ( 5, -5, -5 ) ]
pin.axes_dct['show'] = False
pin.axes_dct['len'] = 1.2
pin.height = 400
pin.width = 800
pin.quality = 11
pin.ani_delay = 10
pin.impl = None
start0 = sage_QQ( 1 ) / 10 # step0=10 step1=15
v0_lst = [ start0 + ( sage_QQ( i ) / 180 ) * sage_pi for i in range( 0, 360, 10 )]
v1_lst = [ ( sage_QQ( i ) / 180 ) * sage_pi for i in range( 0, 360, 15 )]
v1_lst_F = [ start0 + ( sage_QQ( i ) / 360 ) * sage_pi for i in range( 0, 720, 1 )]
v1_lst_WE = [1.8, 2.3, 2.7, 3.1, 3.5, 3.8, 4.134, 4.31, 4.532, 4.7, 4.9, 5.08, 5.25, 5.405, 5.553, 5.7, 5.84]
v1_lst_WF = [1.69, 1.87, 2.07, 2.26, 2.5, 2.72, 2.96, 3.2, 3.42, 3.65, 3.81]
v1_lst_WD = [ 5.44, 5.56, 5.68, 5.81, 5.95, 6.1, 6.27, 6.474] # [5.01, 5.12, 5.22, 5.32,
v1_lst_SA = [6.5]; v1_lst_SE = [5.4];
v1_lst_SB = [5.95]; v1_lst_SF = [2.28];
v1_lst_SC = [4.83]; v1_lst_SD = [5.55];
pin.pmz_dct['A'] = ( pmz_AB_lst, 0 )
pin.pmz_dct['B'] = ( pmz_AB_lst, 1 )
pin.pmz_dct['C'] = ( pmz_CB_lst, 0 )
pin.pmz_dct['D'] = ( pmz_DB_lst, 0 )
pin.pmz_dct['E'] = ( pmz_EB_lst, 0 )
pin.pmz_dct['F'] = ( pmz_AF_lst, 1 )
pin.pmz_dct['WD'] = ( pmz_DB_lst, 0 )
pin.pmz_dct['WE'] = ( pmz_EB_lst, 0 )
pin.pmz_dct['WF'] = ( pmz_AF_lst, 1 )
pin.pmz_dct['SA'] = ( pmz_AB_lst, 0 )
pin.pmz_dct['SB'] = ( pmz_AB_lst, 1 )
pin.pmz_dct['SC'] = ( pmz_CB_lst, 0 )
pin.pmz_dct['SD'] = ( pmz_DB_lst, 0 )
pin.pmz_dct['SE'] = ( pmz_EB_lst, 0 )
pin.pmz_dct['SF'] = ( pmz_AF_lst, 1 )
pin.pmz_dct['FA'] = ( pmz_AB_lst, 0 )
pin.pmz_dct['FB'] = ( pmz_AB_lst, 1 )
pin.pmz_dct['FC'] = ( pmz_CB_lst, 0 )
pin.pmz_dct['FD'] = ( pmz_DB_lst, 0 )
pin.pmz_dct['FE'] = ( pmz_EB_lst, 0 )
pin.pmz_dct['FF'] = ( pmz_AF_lst, 1 )
pin.curve_dct['A'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['B'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['C'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['D'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['E'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['F'] = {'step0':v0_lst, 'step1':v1_lst, 'prec':10, 'width':0.05}
pin.curve_dct['WD'] = {'step0':v0_lst, 'step1':v1_lst_WD, 'prec':10, 'width':0.05}
pin.curve_dct['WE'] = {'step0':v0_lst, 'step1':v1_lst_WE, 'prec':10, 'width':0.05}
pin.curve_dct['WF'] = {'step0':v0_lst, 'step1':v1_lst_WF, 'prec':10, 'width':0.05}
pin.curve_dct['SA'] = {'step0':v0_lst, 'step1':v1_lst_SA, 'prec':10, 'width':0.05}
pin.curve_dct['SB'] = {'step0':v0_lst, 'step1':v1_lst_SB, 'prec':10, 'width':0.05}
pin.curve_dct['SC'] = {'step0':v0_lst, 'step1':v1_lst_SC, 'prec':10, 'width':0.05}
pin.curve_dct['SD'] = {'step0':v0_lst, 'step1':v1_lst_SD, 'prec':10, 'width':0.06}
pin.curve_dct['SE'] = {'step0':v0_lst, 'step1':v1_lst_SE, 'prec':10, 'width':0.05}
pin.curve_dct['SF'] = {'step0':v0_lst, 'step1':v1_lst_SF, 'prec':10, 'width':0.05}
pin.curve_dct['FA'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
pin.curve_dct['FB'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
pin.curve_dct['FC'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
pin.curve_dct['FD'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
pin.curve_dct['FE'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
pin.curve_dct['FF'] = {'step0':v0_lst, 'step1':v1_lst_F, 'prec':10, 'width':0.01}
col_A = rgbt2pov( ( 28, 125, 154, 0 ) ) # blue
col_B = rgbt2pov( ( 74, 33, 0, 0 ) ) # brown
col_C = rgbt2pov( ( 75, 102, 0, 0 ) ) # green
col_E = col_A
col_F = col_B
col_D = col_C
colFF = rgbt2pov( ( 179, 200, 217, 0 ) ) # light blue
pin.text_dct['A'] = [True, col_A, 'phong 0.2' ]
pin.text_dct['B'] = [True, col_B, 'phong 0.2' ]
pin.text_dct['C'] = [True, col_C, 'phong 0.2' ]
pin.text_dct['E'] = [True, col_E, 'phong 0.2' ]
pin.text_dct['F'] = [True, col_F, 'phong 0.2' ]
pin.text_dct['D'] = [True, col_D, 'phong 0.2' ]
pin.text_dct['WE'] = [True, col_E, 'phong 0.2' ]
pin.text_dct['WF'] = [True, col_F, 'phong 0.2' ]
pin.text_dct['WD'] = [True, col_D, 'phong 0.2' ]
pin.text_dct['SA'] = [True, col_A, 'phong 0.2' ]
pin.text_dct['SB'] = [True, col_B, 'phong 0.2' ]
pin.text_dct['SC'] = [True, col_C, 'phong 0.2' ]
pin.text_dct['SE'] = [True, col_E, 'phong 0.2' ]
pin.text_dct['SF'] = [True, col_F, 'phong 0.2' ]
pin.text_dct['SD'] = [True, col_D, 'phong 0.2' ]
pin.text_dct['FA'] = [True, colFF, 'phong 0.2' ]
pin.text_dct['FB'] = [True, colFF, 'phong 0.2' ]
pin.text_dct['FC'] = [True, colFF, 'phong 0.2' ]
pin.text_dct['FE'] = [True, colFF, 'phong 0.2' ]
pin.text_dct['FF'] = [True, colFF, 'phong 0.2' ]
pin.text_dct['FD'] = [True, colFF, 'phong 0.2' ]
# raytrace image/animation
F_lst = ['FA', 'FB', 'FC']
S_lst = ['SA', 'SB', 'SC', 'SD', 'SE', 'SF']
create_pov( pin, ['A', 'B', 'C'] )
create_pov( pin, ['A', 'B', 'C'] + F_lst )
create_pov( pin, ['WD', 'WE', 'WF'] )
create_pov( pin, ['WD', 'WE', 'WF'] + F_lst )
create_pov( pin, S_lst + F_lst )
# ABC - EFD
create_pov( pin, ['A', 'B'] + F_lst )
create_pov( pin, ['E', 'F'] + F_lst )
def get_mat(A_str, B_str, C_str):
'''
Parameters
----------
A_str : string
A string of either one of the following forms:
'O****'
'tT'
'T[#,#,#,#,#,#,#]'
'R****[%,%,%,%]'
'I'
'P1'
'P0'
'M<list of a 9x9 matrix>'
'E[$,$,$,$,$,$,$,$]'
'E'
'X[%,%,%]'
where:
*-symbol is a place holder for one of the following characters:
r,s,m,p,a
#-symbol denotes an element of "OrbRing.num_field".
%-symbol denotes an integer in [0,360].
$-symbol denotes an integer in [1,8] such that each
integer occurs only once. For example 'E[2,1,3,4,5,6,7,8]'
is correct, but 'E[1,1,3,4,5,6,7,8]' is incorrect.
B_str : string
Same specs as "A_str".
C_str : string
Same specs as "A_str".
Returns
-------
sage_matrix
Let matrix A over QQ[c0,s0,c1,s1] be defined as follows,
where we make a case distinction on "A_str":
"A_str[0] == 'O' " : "A = get_omat( A_str )".
"A_str[0] == 'T' " : "A = get_tmat( A_str )".
"A_str[0] == 'E' " : "A = get_emat( A_str )".
"A_str[0] == 'X' " : "A = get_xmat( A_str )".
"A_str == 'tT'" : "A = get_tmat('tT')".
"A_str[0] == 'R' " : "A = get_rmat( A_str )".
"A_str == 'I' " : "A = sage_identity_matrix(9,9)".
"A_str == 'P1'" : "A = get_pmat( True )".
"A_str == 'P0'" : "A = get_pmat( False )".
"A_str[0] == 'M' " : "A = sage_matrix(<list of a matrix>)".
Similarly, we obtain matrices B and C.
We return the matrix A*B*C.
'''
M_lst = []
for M_str in [A_str, B_str, C_str]:
if M_str[0] == 'O': M_lst += [get_omat(M_str)]
elif M_str[0] == 'T': M_lst += [get_tmat(M_str)]
elif M_str[0] == 'E': M_lst += [get_emat(M_str)]
elif M_str[0] == 'X': M_lst += [get_xmat(M_str)]
elif M_str == 'tT': M_lst += [get_tmat('tT')]
elif M_str[0] == 'R': M_lst += [get_rmat(M_str)]
elif M_str == 'I':
M_lst += [sage_identity_matrix(OrbRing.num_field, 9, 9)]
elif M_str == 'P1':
M_lst += [get_pmat(True)]
elif M_str == 'P0':
M_lst += [get_pmat(False)]
elif M_str[0] == 'M':
mat_lst = OrbRing.coerce(M_str[1:])
M_lst += [sage_matrix(mat_lst)]
return M_lst[0] * M_lst[1] * M_lst[2]
def get_tmat(t_str=None):
'''
Parameters
----------
t_str : string
A String with either one of the following 3 formats:
* A string with format:
'T[#,#,#,#,#,#,#]'
where # are in "OrbRing.num_field".
* 'tT'.
* "None".
Returns
-------
sage_matrix
A 9x9 matrix defined over "OrbRing.num_field" representing an
Euclidean translation of S^7. This map is obtained as the composition
of a stereographic projection with center
(1:0:0:0:0:0:0:0:1),
an Euclidean translation in R^7, and the inverse stereographic projection.
If "t_str==None", then the entries of the translation matrix are indeterminates
t1,...,t7 in "OrbRing.R".
If "t_str=='tT'", then the indeterminates are set to [c0,s0,0,0,0,0,0].
Thus the translations along a circle.
'''
t = OrbRing.coerce('[t1,t2,t3,t4,t5,t6,t7]')
# construct a translation matrix with undetermined
# translations in t1,...,t7
a = (sage_QQ(1) / 2) * sum([ti**2 for ti in t])
mat = []
mat += [[1 + a] + list(t) + [-a]]
for i in range(0, 7):
mat += [[t[i]] + sage_identity_matrix(OrbRing.R, 7).row(i).list() +
[-t[i]]]
mat += [[a] + list(t) + [1 - a]]
if t_str == None:
# return matrix with indeterminates
return sage_matrix(OrbRing.R, mat)
elif t_str == 'tT':
# translations along a circle
c0, s0 = OrbRing.coerce('c0,s0')
q = [c0, s0, 0, 0, 0, 0, 0]
else:
# Substitute coordinates [#,#,#,#,#,#,#]
# for the t0,...,t7 variables.
q = sage__eval(t_str[1:])
if len(q) != 7:
raise ValueError('Expect 7 translation coordinates: ', t_str)
# substitute q for t
smat = []
for row in mat:
srow = []
for col in row:
srow += [col.subs({t[i]: q[i] for i in range(7)})]
smat += [srow]
if t_str == 'tT':
return sage_matrix(OrbRing.R, smat)
else:
return sage_matrix(OrbRing.num_field, smat)