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 usecase_orb_product(num=10):
'''
Outputs "num" random surfaces in the projective n-sphere S^n,
that are obtained by rotating or translating a parametrized
circle.
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:
input = OrbInput().random(3, False) # random input
# only compute dimension and degree
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 test__random(self):
coef_bnd = 3
in1 = OrbInput().random(3, False)
print(in1)
print(in1.info_dct['vmat'][0])
print(in1.info_dct['vmat'][2])
s = ''
s += in1.info_dct['vmat'][0][1:]
if in1.info_dct['vmat'][2] != 'I':
s += '+'
s += in1.info_dct['vmat'][2][1:]
for elt in sage__eval(s):
assert elt >= -coef_bnd and elt <= coef_bnd
in2 = OrbInput().set(in1.info_dct['pmat'], in1.info_dct['omat'],
in1.info_dct['vmat'])
assert in1.pmat == in2.pmat
assert in1.omat == in2.omat
assert in1.vmat == in2.vmat
def test__get_pmz_verify__perseus( self ):
o = OrbOutput( OrbInput() )
o.input.do['pmz'] = True
o.input.do['imp'] = True
o.prj_pmz_lst = '[-2*s1 + 3, 4/5*c0*c1 - 3/5*s0*c1 + 7/5*c0*s1 + 6/5*s0*s1 - 12/5*c0 - 11/5*s0 - 2*s1 + 3, 3/5*c0*c1 + 4/5*s0*c1 - 6/5*c0*s1 + 7/5*s0*s1 + 11/5*c0 - 12/5*s0, -2*s1 + 2]'
o.prj_pmz_lst = OrbRing.coerce( o.prj_pmz_lst )
o.prj_pol = '25*x0^4 + 100*x0^3*x1 + 50*x0^2*x1^2 - 100*x0*x1^3 + 25*x1^4 - 50*x0^2*x2^2 - 100*x0*x1*x2^2 + 50*x1^2*x2^2 + 25*x2^4 - 24*x0^3*x3 - 40*x0^2*x1*x3 + 20*x0*x1^2*x3 + 20*x0*x2^2*x3 - 41*x0^2*x3^2 - 100*x0*x1*x3^2 + 50*x1^2*x3^2 + 50*x2^2*x3^2 + 20*x0*x3^3 + 25*x3^4'
o.prj_pol = OrbRing.coerce( o.prj_pol )
o.gen = 1
print( o )
tst = get_pmz_verify( o )
print( tst )
assert tst == True
def usecase_orb_product_investigate_example():
'''
Here we investigate an example obtained
by "usecase_orb_product"
'''
s = "['@(4,3)=(deg,emb)', {'pmat': ('P0', 'I', 'I'), 'omat': ('T[1, 0, 0, 0, 0, 0, 0]', 'Orppp', 'T[-1, 0, 0, 0, 0, 0, 0]'), 'vmat': ('T[0, 1, 1, 0, 0, 0, 0]', 'Rrrrs[37, 0, 0, 0]', 'T[0, -1, -1, 0, 0, 0, 0]')}]"
input = OrbInput().set_short_str(s)
input.do['pmz'] = True
input.do['bpt'] = False
input.do['imp'] = True
input.do['dde'] = True
input.do['prj'] = True
input.do['fct'] = True # requires Maple
input.do['gen'] = True # requires Maple
input.do['sng'] = True # requires Magma
input.do['tst'] = True
o = orb_product(input)
OrbTools.p(o)
def test__orb_product__43_perseus( self ):
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/maple/link/bin'
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/magma/link'
# perseus cyclide
s = "['@(4,3)=(deg,emb)', {'pmat': ('P0', 'I', 'I'), 'omat': ('T[1, 0, 0, 0, 0, 0, 0]', 'Orppp', 'T[-1, 0, 0, 0, 0, 0, 0]'), 'vmat': ('T[0, 1, 1, 0, 0, 0, 0]', 'Rrrrs[37, 0, 0, 0]', 'T[0, -1, -1, 0, 0, 0, 0]')}]"
input = OrbInput().set_short_str( s )
input.do['pmz'] = True
input.do['bpt'] = False
input.do['imp'] = True
input.do['dde'] = True
input.do['prj'] = True
input.do['fct'] = True
input.do['gen'] = True
input.do['sng'] = True
input.do['tst'] = True
print( input )
o = orb_product( input )
print( o )
assert ( o.deg, o.emb, o.dim ) == ( 4, 3, 2 )
def test__orb_product__65_sing( self ):
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/maple/link/bin'
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/magma/link'
s = "['@(6,5)=(deg,emb)', {'pmat': ('P1', 'I', 'I'), 'omat': ('T[0, 0, 0, 1, 1, -1, -1]', 'Oprps', 'I'), 'vmat': ('T[0, -1, 0, -1, 0, 0, 0]', 'Rspps[148, 344, 284, 304]', 'T[0, 1, 0, 1, 0, 0, 0]')}]"
input = OrbInput().set_short_str( s )
input.do['pmz'] = True
input.do['bpt'] = True
input.do['imp'] = True
input.do['dde'] = True
input.do['prj'] = True
input.do['fct'] = False
input.do['gen'] = False
input.do['sng'] = False
input.do['tst'] = False
print( input )
o = orb_product( input )
print( o )
assert ( o.deg, o.emb, o.dim ) == ( 6, 5, 2 )
assert 'chart=xv, depth=0, mult=1, sol=(a1, 0)' in str( o.bp_tree )
assert 'chart=xv, depth=0, mult=1, sol=(-a1, 0)' in str( o.bp_tree )
def test__orb_product__65_smooth( self ):
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/maple/link/bin'
os.environ['PATH'] += os.pathsep + '/home/niels/Desktop/n/app/magma/link'
s = "['@(6,5)=(deg,emb)', {'pmat': ('P1', 'I', 'I'), 'omat': ('T[1, -1, 1, 1, -1, 0, 0]', 'Opsms', 'I'), 'vmat': ('T[1, 0, -1, 0, 0, 0, 0]', 'Rramr[340, 225, 264, 320]', 'T[-1, 0, 1, 0, 0, 0, 0]')}]"
input = OrbInput().set_short_str( s )
input.do['pmz'] = True
input.do['bpt'] = False
input.do['imp'] = True
input.do['dde'] = True
input.do['prj'] = True
input.do['fct'] = False
input.do['gen'] = False
input.do['sng'] = False
input.do['tst'] = False
o = orb_product( input )
print( o )
assert ( o.deg, o.emb, o.dim ) == ( 6, 5, 2 )
def dp6_sing():
'''
Creates povray image of the projection of a sextic weak del Pezzo surface
wdP6 in S^5. This surface contains 2 families of conics. One of these
families has a base point at the isolated singularity of wdP6
'''
# init OrbInput
#
pmat = []
pmat += [[1, 0, 0, 0] + [0, 0, 0, -1, -1]]
pmat += [[0, 1, 0, 0] + [0, 0, 0, 0, 0]]
pmat += [[0, 0, 1, 0] + [0, 0, 0, 0, 0]]
pmat += [[0, 0, 0, 1] + [-1, 0, 0, 0, 0]]
p_tup = ('M' + str(pmat), 'I', 'I')
o_tup = ('I', 'Oprpr', 'I')
v_tup = ('T[0, 1, 1, 0, 0, 0, 0]', 'Rrppr[37,0,0,90]',
'T[0, -1, -1, 0, 0, 0, 0]')
input = OrbInput().set(p_tup, o_tup, v_tup)
input.do['pmz'] = True
input.do['bpt'] = False
input.do['imp'] = True
input.do['dde'] = True
input.do['prj'] = True
input.do['fct'] = False
input.do['gen'] = False
input.do['sng'] = False
input.do['tst'] = False
# init OrbOutput.prj_pmz_lst
#
o = orb_product(input)
pmz_AB_lst = o.prj_pmz_lst
# init PovInput
#
pin = PovInput()
pin.path = './' + get_time_str() + '_dp6_sing/'
pin.fname = 'orb'
pin.scale = 1
pin.cam_dct['location'] = (0, 0, -3)
pin.cam_dct['lookat'] = (0, 0, 0)
pin.cam_dct['rotate'] = (250, 330, 0)
pin.shadow = True
pin.light_lst = [(0, 0, -5), (0, -5, 0), (-5, 0, 0), (0, 0, 5), (0, 5, 0),
(5, 0, 0)]
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
v0_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 5)]
v1_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 10)]
v1_F_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 2)]
pin.pmz_dct['A'] = (pmz_AB_lst, 0)
pin.pmz_dct['B'] = (pmz_AB_lst, 1)
pin.pmz_dct['FA'] = (pmz_AB_lst, 0)
pin.pmz_dct['FB'] = (pmz_AB_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['FA'] = {
'step0': v0_lst,
'step1': v1_F_lst,
'prec': 10,
'width': 0.01
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_F_lst,
'prec': 10,
'width': 0.01
}
col_A = (0.4, 0.0, 0.0, 0.0)
col_B = (0.2, 0.3, 0.2, 0.0)
colFF = (0.1, 0.1, 0.1, 0.0)
pin.text_dct['A'] = [True, col_A, 'phong 0.2 phong_size 5']
pin.text_dct['B'] = [True, col_B, 'phong 0.2 phong_size 5']
pin.text_dct['FA'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['FB'] = [True, colFF, 'phong 0.2 phong_size 5']
if True:
# raytrace image/animation
#
create_pov(pin, ['A', 'B'])
create_pov(pin, ['A', 'B', 'FA', 'FB'])
return
else:
# very low quality for fast debugging
#
pin.width = 100
pin.height = 75
pin.quality = 4
v_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 2 * 36)]
pin.curve_dct['A'] = {
'step0': v_lst,
'step1': v_lst,
'prec': 5,
'width': 0.02
}
pin.curve_dct['B'] = {
'step0': v_lst,
'step1': v_lst,
'prec': 5,
'width': 0.02
}
create_pov(pin, ['A', 'B'])
return