def test__approx_QQ_pol_lst(self):
ring = PolyRing('x,y,z', True).ext_num_field('t^2 - 3')
x, y, z = ring.gens()
a0 = ring.root_gens()[0]
q2 = sage_QQ(1) / 2
pol_lst = [x * y * a0 + z**2 + q2 * y**2 - q2 - a0 * x, a0, -a0 * x]
out_lst = OrbRing.approx_QQ_pol_lst(pol_lst, 1)
c = sage_QQ(3900231685776981) / 2251799813685248
print(out_lst)
assert out_lst[0] == x * y * c + z**2 + q2 * y**2 - q2 - c * x
assert out_lst[1] == c
assert out_lst[2] == -c * x
def CH1_cyclide():
'''
Creates povray image of a CH1 cyclide, which is
an inversion of a Circular Hyperboloid of 1 sheet.
'''
# Construct a trigonometric parametrization by rotating a circle.
r, R = 1, 1
c0, s0, c1, s1 = sage_var('c0,s0,c1,s1')
x, y, v, w, a0 = sage_var('x,y,v,w,a0')
q2 = sage_QQ(1) / 2
MX = sage_matrix([(1, 0, 0), (0, c1, s1), (0, -s1, c1)])
MXc = MX.subs({c1: a0, s1: a0}) # a0=1/sqrt(2)=cos(pi/4)=sin(pi/4)
MZ = sage_matrix([(c1, s1, 0), (-s1, c1, 0), (0, 0, 1)])
V = sage_vector([r * c0, 0, r * s0])
V = MXc * V
V[0] = V[0] + R
pmz_AB_lst = list(MZ * V)
OrbTools.p('V =', V)
OrbTools.p('pmz_AB_lst =', pmz_AB_lst)
for pmz in pmz_AB_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# Convert the trigonometric parametrization to a rational parametrization
# We convert via the following formulas,
#
# cos(s) = (y^2-x^2) / (y^2+x^2)
# sin(s) = 2*x*y / (y^2+x^2)
# y=1; x = arctan( s/2 )
#
C0 = (y**2 - x**2) / (y**2 + x**2)
S0 = 2 * x * y / (y**2 + x**2)
C1 = (w**2 - v**2) / (w**2 + v**2)
S1 = 2 * v * w / (w**2 + v**2)
den = (y**2 + x**2) * (w**2 + v**2)
dct = {c0: C0, s0: S0, c1: C1, s1: S1}
pmz_lst = [den] + [(elt.subs(dct) * den).simplify_full()
for elt in list(MZ * V)]
OrbTools.p('pmz_lst =', pmz_lst)
for pmz in pmz_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# do a basepoint analysis on the rational parametrization
# The True argument is for resetting the number field to QQ!
ring = PolyRing('x,y,v,w', True).ext_num_field('t^2-1/2')
ls = LinearSeries([str(pmz) for pmz in pmz_lst], ring)
OrbTools.p(ls.get_bp_tree())
# construct linear series for families of conics
ring = PolyRing(
'x,y,v,w') # construct polynomial ring over new ground field
OrbTools.p(ring)
x, y, v, w = ring.gens()
a0, a1 = ring.root_gens()
p1 = ['xv', (0, 2 * a0 * a1)]
p2 = ['xv', (0, -2 * a0 * a1)]
p3 = ['xv', (a1, 2 * a0 * a1)]
p4 = ['xv', (-a1, -2 * a0 * a1)]
bpt_1234 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_1234.add(p1[0], p1[1], 1)
bpt_1234.add(p2[0], p2[1], 1)
bpt_1234.add(p3[0], p3[1], 1)
bpt_1234.add(p4[0], p4[1], 1)
bpt_12 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_12.add(p1[0], p1[1], 1)
bpt_12.add(p2[0], p2[1], 1)
bpt_34 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_34.add(p3[0], p3[1], 1)
bpt_34.add(p4[0], p4[1], 1)
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)
# compute reparametrization from the linear series of families
ring = PolyRing(
'x,y,v,w,c0,s0,c1,s1') # construct polynomial ring with new generators
OrbTools.p(ring)
x, y, v, w, c0, s0, c1, s1 = ring.gens()
a0, a1 = ring.root_gens()
pmz_AB_lst = [1] + ring.coerce(pmz_AB_lst)
pmz_lst = ring.coerce(pmz_lst)
X = 1 - s0
Y = c0
V = 1 - s1
W = c1
CB_dct = {
x: X,
y: Y,
v: W * X - 2 * a0 * V * Y,
w: V * X + 2 * a0 * W * Y
}
pmz_CB_lst = [pmz.subs(CB_dct) for pmz in pmz_lst] # CB 11b
# output
OrbTools.p('pmz_AB_lst =\n', pmz_AB_lst)
OrbTools.p('pmz_CB_lst =\n', pmz_CB_lst)
# approximate by map defined over rational numbers
ci_idx = 0 # index defining the complex embedding
OrbTools.p('complex embeddings =')
for i in range(len(a0.complex_embeddings())):
a0q = OrbRing.approx_QQ_coef(a0, i)
OrbTools.p('\t\t' + str(i) + ' =', a0q, sage_n(a0q))
pmz_AB_lst = OrbRing.approx_QQ_pol_lst(pmz_AB_lst, ci_idx)
pmz_CB_lst = OrbRing.approx_QQ_pol_lst(pmz_CB_lst, ci_idx)
# mathematica input
ms = ''
for pmz, AB in [(pmz_lst, 'ZZ'), (pmz_AB_lst, 'AB'), (pmz_CB_lst, 'CB')]:
s = 'pmz' + AB + '=' + str(pmz) + ';'
s = s.replace('[', '{').replace(']', '}')
ms += '\n' + s
OrbTools.p('Mathematica input =', ms)
# PovInput ring cyclide
#
pin = PovInput()
pin.path = './' + get_time_str() + '_CH1_cyclide/'
pin.fname = 'orb'
pin.scale = 1
pin.cam_dct['location'] = (0, -5, 0)
pin.cam_dct['lookat'] = (0, 0, 0)
pin.cam_dct['rotate'] = (20, 0, 0)
pin.shadow = True
pin.light_lst = [(1, 0, 0), (0, 1, 0), (0, 0, 1), (-1, 0, 0), (0, -1, 0),
(0, 0, -1), (10, 0, 0), (0, 10, 0), (0, 0, 10),
(-10, 0, 0), (0, -10, 0), (0, 0, -10)]
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
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['FA'] = (pmz_AB_lst, 0)
pin.pmz_dct['FB'] = (pmz_AB_lst, 1)
pin.pmz_dct['FC'] = (pmz_CB_lst, 0)
v0_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 10)]
v1_lst_A = [(sage_QQ(i) / 180) * sage_pi for i in range(180, 360, 10)]
v1_lst_B = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 10)]
v1_lst_C = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 10)]
v1_lst_FA = [(sage_QQ(i) / 180) * sage_pi for i in range(180, 360, 2)]
v1_lst_FB = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 2)]
v1_lst_FC = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 2)]
prec = 50
pin.curve_dct['A'] = {
'step0': v0_lst,
'step1': v1_lst_A,
'prec': prec,
'width': 0.03
}
pin.curve_dct['B'] = {
'step0': v0_lst,
'step1': v1_lst_B,
'prec': prec,
'width': 0.03
}
pin.curve_dct['C'] = {
'step0': v0_lst,
'step1': v1_lst_C,
'prec': prec,
'width': 0.03
}
pin.curve_dct['FA'] = {
'step0': v0_lst,
'step1': v1_lst_FA,
'prec': prec,
'width': 0.02
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_lst_FB,
'prec': prec,
'width': 0.02
}
pin.curve_dct['FC'] = {
'step0': v0_lst,
'step1': v1_lst_FC,
'prec': prec,
'width': 0.02
}
col_A = (0.6, 0.4, 0.1, 0.0)
col_B = (0.1, 0.15, 0.0, 0.0)
col_C = (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['C'] = [True, col_C, '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']
pin.text_dct['FC'] = [True, colFF, 'phong 0.2 phong_size 5']
# raytrace image/animation
create_pov(pin, ['A', 'B', 'C'])
create_pov(pin, ['A', 'B', 'C', 'FA', 'FB', 'FC'])
create_pov(pin, ['A', 'B', 'FA', 'FB'])
create_pov(pin, ['B', 'C', 'FA', 'FB'])
def perseus_cyclide():
'''
Creates povray image of the Perseus cyclide.
'''
# We first construct a trigonometric parametrization
# by rotating a circle.
# cos(pi/3) = 1/2
# sin(pi/3) = sqrt(3)/2
#
r, R = 1, 2
c0, s0, c1, s1 = sage_var('c0,s0,c1,s1')
x, y, v, w, a0 = sage_var('x,y,v,w,a0')
q2 = sage_QQ(1) / 2
MZ = sage_matrix([(c1, s1, 0), (-s1, c1, 0), (0, 0, 1)])
MZc = MZ.subs({c1: q2, s1: q2 * a0})
V = sage_vector([r * c0, 0, r * s0])
V = MZc * V
V[0] = V[0] + R
pmz_AB_lst = list(MZ * V)
OrbTools.p('V =', V)
OrbTools.p('pmz_AB_lst =', pmz_AB_lst)
for pmz in pmz_AB_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# We convert the trigonometric parametrization to a
# rational parametrization, via the following formulas:
#
# cos(s) = (y^2-x^2) / (y^2+x^2)
# sin(s) = 2*x*y / (y^2+x^2)
# y=1; x = arctan( s/2 )
#
C0 = (y**2 - x**2) / (y**2 + x**2)
S0 = 2 * x * y / (y**2 + x**2)
C1 = (w**2 - v**2) / (w**2 + v**2)
S1 = 2 * v * w / (w**2 + v**2)
den = (y**2 + x**2) * (w**2 + v**2)
dct = {c0: C0, s0: S0, c1: C1, s1: S1}
pmz_lst = [den] + [(elt.subs(dct) * den).simplify_full()
for elt in list(MZ * V)]
OrbTools.p('pmz_lst =', pmz_lst)
for pmz in pmz_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# do a basepoint analysis on the rational parametrization
#
ring = PolyRing('x,y,v,w', True).ext_num_field('t^2-3')
ls = LinearSeries([str(pmz) for pmz in pmz_lst], ring)
OrbTools.p(ls.get_bp_tree())
# construct linear series for families of conics
#
ring = PolyRing(
'x,y,v,w') # construct polynomial ring over new ground field
OrbTools.p(ring)
x, y, v, w = ring.gens()
a0, a1, a2, a3 = ring.root_gens()
p1 = ['xv', (-a3, a1)]
p2 = ['xv', (-a2, -a1)]
p3 = ['xv', (a3, a1)]
p4 = ['xv', (a2, -a1)]
bpt_1234 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_1234.add(p1[0], p1[1], 1)
bpt_1234.add(p2[0], p2[1], 1)
bpt_1234.add(p3[0], p3[1], 1)
bpt_1234.add(p4[0], p4[1], 1)
bpt_12 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_12.add(p1[0], p1[1], 1)
bpt_12.add(p2[0], p2[1], 1)
bpt_34 = BasePointTree(['xv', 'xw', 'yv', 'yw'])
bpt_34.add(p3[0], p3[1], 1)
bpt_34.add(p4[0], p4[1], 1)
ls_22 = LinearSeries.get([2, 2], bpt_1234)
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)
# compute reparametrization from the linear series of families
ring = PolyRing('x,y,v,w,c0,s0,c1,s1')
OrbTools.p(ring)
x, y, v, w, c0, s0, c1, s1 = ring.gens()
a0, a1, a2, a3 = ring.root_gens()
pmz_AB_lst = [1] + ring.coerce(pmz_AB_lst)
pmz_lst = ring.coerce(pmz_lst)
q2 = sage_QQ(1) / 2
a = 2 * a0 / 3
b = (-a0 * a1 / 3 - q2) * a3
c = (a0 * a1 / 3 - q2) * a2
d = (a1 / 2 - a0 / 3) * a3
e = (-a1 / 2 - a0 / 3) * a2
bc = b + c
de = d + e
X = 1 - s0
Y = c0
V = 1 - s1
W = c1
CB_dct = {
x: X,
y: Y,
v: W * X + bc * W * Y - de * V * Y,
w: V * X + bc * V * Y + de * W * Y
}
DB_dct = {
x: X,
y: Y,
v: W * X - bc * W * Y + de * V * Y,
w: V * X - bc * V * Y - de * W * Y
}
EB_dct = {
x: X,
y: Y,
v: W * X**2 + W * Y**2 - a * V * Y**2,
w: V * X**2 + V * Y**2 + a * W * Y**2
}
pmz_CB_lst = [pmz.subs(CB_dct) for pmz in pmz_lst] # CB 11a
pmz_DB_lst = [pmz.subs(DB_dct) for pmz in pmz_lst] # CB 11b
pmz_EB_lst = [pmz.subs(EB_dct) for pmz in pmz_lst] # CB 21
# output
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)
# approximate by map defined over rational numbers
ci_idx = 5 # index defining the complex embedding
pmz_AB_lst = OrbRing.approx_QQ_pol_lst(pmz_AB_lst, ci_idx)
pmz_CB_lst = OrbRing.approx_QQ_pol_lst(pmz_CB_lst, ci_idx)
pmz_DB_lst = OrbRing.approx_QQ_pol_lst(pmz_DB_lst, ci_idx)
pmz_EB_lst = OrbRing.approx_QQ_pol_lst(pmz_EB_lst, ci_idx)
# mathematica input
ms = ''
for pmz, AB in [(pmz_lst, 'ZZ'), (pmz_AB_lst, 'AB'), (pmz_CB_lst, 'CB'),
(pmz_DB_lst, 'DB'), (pmz_EB_lst, 'EB')]:
s = 'pmz' + AB + '=' + str(pmz) + ';'
s = s.replace('[', '{').replace(']', '}')
ms += '\n' + s
OrbTools.p('Mathematica input =', ms)
# PovInput ring cyclide
#
pin = PovInput()
pin.path = './' + get_time_str() + '_perseus_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'] = (45, 0, 0)
pin.shadow = True
pin.light_lst = [(0, 0, -10), (0, -10, 0), (-10, 0, 0), (0, 0, 10),
(0, 10, 0), (10, 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
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['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)
v0_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 10)]
v1_lst_A = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 10)] # 5
v1_lst_B = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 15)]
v1_lst_C = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 36)]
v1_lst_D = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 36)]
v1_lst_E = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 10)] # 5
v1_lst_F = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 1)]
prec = 50
pin.curve_dct['A'] = {
'step0': v0_lst,
'step1': v1_lst_A,
'prec': prec,
'width': 0.04
}
pin.curve_dct['B'] = {
'step0': v0_lst,
'step1': v1_lst_B,
'prec': prec,
'width': 0.04
}
pin.curve_dct['C'] = {
'step0': v0_lst,
'step1': v1_lst_C,
'prec': prec,
'width': 0.05
}
pin.curve_dct['D'] = {
'step0': v0_lst,
'step1': v1_lst_D,
'prec': prec,
'width': 0.05
}
pin.curve_dct['E'] = {
'step0': v0_lst,
'step1': v1_lst_E,
'prec': prec,
'width': 0.04
}
pin.curve_dct['FA'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.01
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.01
}
pin.curve_dct['FC'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.01
}
pin.curve_dct['FD'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.01
}
pin.curve_dct['FE'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.01
}
col_A = (0.6, 0.0, 0.0, 0.0) # red
col_B = (0.8, 0.6, 0.2, 0.0) # beige
col_C = (0.6, 0.0, 0.0, 0.0
) # red *** rgbt2pov( ( 74, 33, 0, 0 ) ) # brown
col_D = (0.2, 0.6, 0.0, 0.0
) # green *** rgbt2pov( ( 28, 125, 154, 0 ) ) # blue
col_E = (0.2, 0.6, 0.0, 0.0) # green
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['C'] = [True, col_C, 'phong 0.2 phong_size 5']
pin.text_dct['D'] = [True, col_D, 'phong 0.2 phong_size 5']
pin.text_dct['E'] = [True, col_E, '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']
pin.text_dct['FC'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['FD'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['FE'] = [True, colFF, 'phong 0.2 phong_size 5']
# raytrace image/animation
create_pov(pin, ['C', 'D', 'FC', 'FD'])
create_pov(pin, ['A', 'B', 'FC', 'FD'])
create_pov(pin, ['E', 'B', 'FC', 'FD'])
