def usecase_orb_product_implicit_circle(num=10):
'''
Outputs "num" random surfaces in the projective n-sphere S^n,
that are obtained by rotating or translating an implicit circle.
The circle is obtained as a random hyperplane section.
The hyperplane section is obtained by applying
a random matrix in PGL(8+1) to the hyperplane
section { x | x3=..=x8=0=-x0^2+x1^2+...+x8^2=0 }.
Since the random matrix is not orthogonal
we cannot use the matrix to compute parametric
presentations for these examples.
Notice that we can recover the OrbInput object
from the short string output
(see "usecase_orb_product_investigate_example()")
Parameters
----------
num : int
Number of times a random surface should be computed.
'''
for idx in range(num):
try:
# compute random input
ch_str = ''.join([
OrbRing.random_elt(['r', 's', 'p', 'm', 'a']) for i in range(4)
])
pmat = list(sage_MatrixSpace(sage_ZZ, 9, 9).random_element())
p_tup = ('P1', 'I', 'I')
o_tup = ('I', 'O' + ch_str, 'I')
v_tup = ('M' + str(pmat), 'I', 'I')
input = OrbInput().set(p_tup, o_tup, v_tup)
# compute only few attributes
for key in input.do.keys():
input.do[key] = False
input.do['imp'] = True
input.do['dde'] = True
o = orb_product(input)
OrbTools.p('(deg, emb, dim ) =',
(o.deg, o.emb, o.dim), ' short string =',
o.get_short_str()) # output orbital product
except BaseException as e:
# continue with loop regardless of exceptions
OrbTools.p('Exception occurred: ', e)
OrbTools.p(input)
OrbTools.p(o)
def random(self, bnd=10):
'''
Sets attributes to random values.
Parameters
----------
coef_bnd : int
Positive integer.
Returns
-------
self
Sets self.rota, self.trna, self.sa and
self.rotb, self.trnb, self.sb with random
values.
'''
# OrbRing.random_int( coef_size )
q4 = sage_QQ(1) / 4
s_lst = [i * q4 for i in range(4 * bnd)]
s_lst += [-s for s in s_lst]
# circle A
self.rota = [OrbRing.random_elt(range(360)) for i in range(6)]
self.trna = [OrbRing.random_elt(range(360)) for i in range(3)]
self.sa = OrbRing.random_elt(s_lst)
# circle B
self.rotb = [OrbRing.random_elt(range(360)) for i in range(6)]
self.trnb = [OrbRing.random_elt(range(360)) for i in range(3)]
self.sb = OrbRing.random_elt(s_lst)
return self
def test__random_elt(self):
lst = range(10)
elt = OrbRing.random_elt(lst)
print(elt)
assert elt in lst
def random( self, coef_bnd = 3, random_pmat = True ):
'''
Sets (restricted) random values for "self.omat",
"self.vmat" and "self.pmat".
Parameters
----------
coef_bnd : int
Positive integer.
random_pmat : boolean
Returns
-------
self
Notes
-----
The translation coefficients of "self.vmat"
are in the interval: [-coef_bnd, +coef_bnd].
If "random_pmat== True", then "self.pmat"
is set to a random value (P1), and
otherwise "self.pmat" is set to a standard
projection (P0).
'''
ch_lst = ['r', 's', 'p', 'm', 'a']
#
# random self.pmat
#
Pn = {True:'P1', False:'P0'}[random_pmat]
self.pmat = get_mat( Pn, 'I', 'I' )
self.info_dct['pmat'] = ( Pn, 'I', 'I' )
#
# random self.omat
#
rnd = OrbRing.random_elt( [0, 1] )
if rnd == 0:
ch_str = ''.join( [OrbRing.random_elt( ch_lst ) for i in range( 4 ) ] )
B_str = 'O' + ch_str
rnd2 = OrbRing.random_elt( [0, 1] )
if rnd2 == 0:
A_str = 'I'
C_str = 'I'
else:
p = sage_Permutations( range( 1, 8 + 1 ) )
rp = p.random_element()
rpI = sage_Permutation( rp ).inverse()
A_str = 'E' + str( list( rpI ) )
C_str = 'E' + str( list( rp ) )
elif rnd == 1:
A_str = 'I'
B_str = 'tT'
C_str = 'I'
self.omat = get_mat( A_str, B_str, C_str )
self.info_dct['omat'] = ( A_str, B_str, C_str )
#
# random self.vmat
#
# All the coefficients are bound by coef_size-n for some integer n.
coef_size = OrbRing.random_elt( range( 1, coef_bnd + 1 ) )
# A -- self.vmat
t_lst = [OrbRing.random_int( coef_size ) for i in range( 7 )]
A_str = 'T' + str( t_lst )
# B -- self.vmat"
# note that small angles means small coefficients
angle_lst = [ OrbRing.random_elt( range( 0, 360 ) ) for i in range( 4 ) ]
ch_str = ''.join( [OrbRing.random_elt( ch_lst ) for i in range( 4 ) ] )
B_str = 'R' + ch_str + str( angle_lst )
# C -- self.vmat
rnd = OrbRing.random_elt( [0, 1, 2] )
if rnd == 0:
C_str = 'T' + str( [-t_lst[i] for i in range( 7 )] ) # inverse
elif rnd == 1:
C_str = 'T' + str( [OrbRing.random_int( coef_size ) for i in range( 7 )] )
elif rnd == 2:
C_str = 'I'
# A*B*C -- self.vmat
self.vmat = get_mat( A_str, B_str, C_str )
self.info_dct['vmat'] = ( A_str, B_str, C_str )
return self
