def test__povray( self ):
c0, s0, c1, s1 = sage_var( 'c0,s0,c1,s1' )
x, y, z = sage_var( 'x,y,z' )
r = 1
R = 2
pmz_AB_lst = [1, ( c0 * r + R ) * c1, ( c0 * r + R ) * s1, r * s0]
f_xyz = ( x ** 2 + y ** 2 + z ** 2 + R ** 2 - r ** 2 ) ** 2 - 4 * R ** 2 * ( x ** 2 + y ** 2 )
v0_lst = [ ( sage_QQ( i ) / 180 ) * sage_pi for i in range( 0, 360, 150 )]
v1_lst = [ ( sage_QQ( i ) / 180 ) * sage_pi for i in range( 0, 360, 300 )]
# set PovInput as container
# put very low quality for testing purposes
pin = PovInput()
pin.path = './' + get_time_str() + '_TEST_POVRAY_REMOVE_ME/'
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.light_radius = 5
pin.axes_dct['show'] = True
pin.axes_dct['len'] = 3
pin.width = 2
pin.height = 2
pin.quality = 1
pin.ani_delay = 1
pin.impl = f_xyz
pin.pmz_dct['A'] = ( pmz_AB_lst, 0 )
pin.pmz_dct['B'] = ( pmz_AB_lst, 1 )
pin.curve_dct['A'] = {'step0':v0_lst, 'step1': v1_lst, 'prec':1, 'width':0.08}
pin.curve_dct['B'] = {'step0':v0_lst, 'step1': v1_lst, 'prec':1, 'width':0.08}
pin.text_dct['A'] = [True, ( 0.5, 0.0, 0.0, 0.0 ), 'phong 0.2 phong_size 5' ]
pin.text_dct['B'] = [True, ( 0.2, 0.3, 0.2, 0.0 ), 'phong 0.2 phong_size 5' ]
pin.text_dct['SURF'] = [True, ( 0.2, 0.7, 0.3, 0.0 ), 'F_Glass10']
# raytrace image/animation
show_surf = True
ani = False
ft_lst = []
lst = create_pov( pin, ['A', 'B'], show_surf, ani, ft_lst )
lst = create_pov( pin, ['A'], False, True, [] )
def test__get_prj_S3(self):
x0, x1, x2, x3, x4 = sage_var('x0,x1,x2,x3,x4')
v = sage_vector([x1, x2, x3, x4, x0])
out = get_prj_S3(v)
print(out)
assert out == [-x1 / (x0 - x4), -x2 / (x0 - x4), -x3 / (x0 - x4)]
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_deg_dim( imp_lst ):
'''
Parameters
----------
imp_lst : list<OrbRing.R>
A list of homogenous polynomials in QQ[x0,...,x8]
representing a variety S in projective 8-space P^8.
Returns
-------
int[]
A 2-tuple of integers consisting of
the degree and the dimension of the variety S.
'''
# consider ideal in ring of the right dimension.
R = sage_PolynomialRing( sage_QQ, sage_var( 'y0,y1,y2,y3,y4,y5,y6,y7,y8' ), order = 'degrevlex' )
I = R.ideal( sage__eval( str( imp_lst ).replace( 'x', 'y' ), R.gens_dict() ) )
# compute Hilbert polynomial: (deg/dim!)*t^dim + ...
hpol = I.hilbert_polynomial()
dim = hpol.degree()
deg = hpol
for i in range( dim ):
deg = deg.diff()
OrbTools.p( 'hpol =', hpol, ' (deg, dim)=', ( deg, dim ) )
return deg, dim
def test__get_hp_P4(self):
c0, s0, c1, s1 = sage_var('c0,s0,c1,s1')
v = sage_vector([1, c0, s0, 0, 0])
w = sage_vector([1, c1, s1, 0, 0])
a12, a13, a23, a14, a24, a34 = [90, 0, 0, 0, 0, 0]
M = get_rot_S3([a12, a13, a23, a14, a24, a34])
out = get_hp_P4(v, M * w)
print(out)
assert str(out) == '[1, -c1*s0 - c0*s1, c0*c1 - s0*s1, 0, 0]'
def test__get_hp_S3(self):
a01, a02, a03, a12, a13, a23 = 5 * [0] + [2]
M = get_rot_S3(a01, a02, a03, a12, a13, a23)
c0, s0, c1, s1 = sage_var('c0,s0,c1,s1')
v = sage_vector([c0, s0, 0, 0, 1])
w = sage_vector([c1, s1, 0, 0, 1])
out = get_hp_S3(v, M * w)
print(out)
assert str(out) == '(c0*c1 - s0*s1, c1*s0 + c0*s1, 0, 0, 1)'
def get_pmz(A, B, prj):
'''
Computes parametrization of a stereographic projection of
the pointwise Hamiltonian product of circles in the sphere S^3
defined by transformations of the standard circle [1,cos(a),sin(a),0,0]
by A and B respectively.
Parameters
----------
A : sage_Matrix<sage_QQ>
Represents a linear transformation S^3--->S^3
B : sage_Matrix<sage_QQ>
Represents a linear transformation S^3--->S^3
prj : int
Choice for stereographic projection S^3--->P^3:
0: (x0:x1:x2:x3:x4) |--> (x0-x4:x1:x2:x3)
1: (x0:x1:x2:x3:x4) |--> (x0-x1:x4:x2:x3)
2: (x0:x1:x2:x3:x4) |--> (x0-x2:x1:x4:x3)
3: (x0:x1:x2:x3:x4) |--> (x0-x3:x1:x2:x4)
Returns
-------
tuple
Returns tuple (baseA, baseB, pmzAB) where
* baseA: Parametrization of projection of A in cos(a) and sin(a).
* baseB: Parametrization of projection of B in cos(b) and sin(b).
* pmzAB: Parametrization of projection of A*B.
'''
dct = {}
# Hamiltonian product of circles
a, b = sage_var('a,b')
u = list(A * sage_vector([1, sage_cos(a), sage_sin(a), 0, 0]))
v = list(B * sage_vector([1, sage_cos(b), sage_sin(b), 0, 0]))
p = get_hp_P4(u, v)
# stereographic projection
if prj == 0: j, i_lst = 4, [1, 2, 3]
if prj == 1: j, i_lst = 1, [4, 2, 3]
if prj == 2: j, i_lst = 2, [1, 4, 3]
if prj == 3: j, i_lst = 3, [1, 2, 4]
p = [p[i] / (p[0] - p[j]) for i in i_lst]
# put in dictionary
pmzAB = [elt.full_simplify() for elt in p]
baseA = [u[i] / (u[0] - u[j]) for i in i_lst]
baseB = [v[i] / (v[0] - v[j]) for i in i_lst]
return baseA, baseB, pmzAB
def clifford(sinp):
'''
Parameters
----------
sinp : SphereInput
Returns
-------
tuple
A tuple (plt,out) where
* plt : sage_Graphics object with plots of surface defined
by "sinp".
* out : Information string.
Notes
-----
See documentation at top of this file for possible usage of
this method.
'''
#
# compute matrices A and B
#
T = get_trn_S3(sinp.trna)
R = get_rot_S3(sinp.rota)
S = get_scale_S3(sinp.sa)
A = S * R * T
T = get_trn_S3(sinp.trnb)
R = get_rot_S3(sinp.rotb)
S = get_scale_S3(sinp.sb)
B = S * R * T
#
# Compute implicit and parametric form of a stereographic
# projection of the Hamiltonian product of A and B.
#
baseA, baseB, pmzAB = get_pmz(A, B, sinp.prj)
if sinp.imp:
dct = get_imp(A, B, sinp.prj, sinp.sng, sinp.snp)
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
]
#
# create graphics object
#
plt = sage_Graphics()
a, b = sage_var('a,b')
if sinp.pmz:
plt += sage_parametric_plot3d(pmzAB, (a, 0, 2 * sage_pi),
(b, 0, 2 * sage_pi),
color=sinp.col_pmz,
aspect_ratio=1,
plot_points=sinp.ppt,
opacity=sinp.opa)
if sinp.bas:
plt += sage_parametric_plot3d(baseA, (a, 0, 2 * sage_pi),
color=sinp.col_famA,
thickness=10,
aspect_ratio=1)
plt += sage_parametric_plot3d(baseB, (b, 0, 2 * sage_pi),
color=sinp.col_famB,
thickness=10,
aspect_ratio=1)
if sinp.mrk:
plt += sage_point3d((0, 0, 0), size=30, color=sage_Color("magenta"))
plt += sage_point3d((1, 0, 0), size=30, color=sage_Color("green"))
if sinp.fam:
if sinp.famA:
for b1 in range(0, 360, sinp.stp):
ps = [
pmzAB[i].subs({b: b1 * sage_pi / 180}) for i in [0, 1, 2]
]
plt += sage_parametric_plot3d(ps, (a, 0, 2 * sage_pi),
color=sage_Color("red"),
thickness=sinp.famt,
aspect_ratio=1,
plot_points=sinp.ppt)
if sinp.famB:
for a1 in range(0, 360, sinp.stp):
ps = [
pmzAB[i].subs({a: a1 * sage_pi / 180}) for i in [0, 1, 2]
]
plt += sage_parametric_plot3d(ps, (b, 0, 2 * sage_pi),
color=sage_Color("blue"),
thickness=sinp.famt,
aspect_ratio=1,
plot_points=sinp.ppt)
if sinp.imp:
rng = sinp.rng
x, y, z = sage_var('x,y,z')
plt += sage_implicit_plot3d(eqn_xyz, (x, -rng, rng), (y, -rng, rng),
(z, -rng, rng),
color=sinp.col_imp,
plot_points=sinp.ppt,
opacity=sinp.opa)
#
# create output string
#
out = ''
out += str(sinp)
out += '\n'
if sinp.imp:
out += '\neqn_str = ' + eqn_str
out += '\nAgreat = ' + str(Agreat)
out += '\nBgreat = ' + str(Bgreat)
out += '\nA = ' + str(list(A))
out += '\nB = ' + str(list(B))
out += '\npmzAB = ' + str(pmzAB)
if sinp.sng:
out += '\nsng_lst (long) ='
out += '\n-----'
for sng in sng_lst:
out += '\n' + str(sng)
out += '\n-----'
out += '\nsng_lst (short) ='
out += '\n-----'
for sng in sng_lst:
out += '\n' + str(sng[1])
out += '\n-----'
return plt, out
def veronese():
'''
Construct povray image of a 3-web of conics on the Veronese surface.
'''
#############################################
# Construct projection of Veronese surface. #
#############################################
c0, s0, c1, s1, t0 = sage_var('c0,s0,c1,s1,t0')
x, y = sage_var('x,y')
pmz_A_lst = [1, c0 * s0 * s1, c0 * s0 * c1, c0 * c0 * c1 * s1]
P1 = c0 / (s0 - 1)
P2 = c1 / (s1 - 1)
P3 = (s0 / c0) * (c1 / (s1 - 1))
dct_CD = {x: P1, y: P2}
den_CD = (s0 - 1)**2 * (s1 - 1)**2
dct_ED = {x: P3, y: P2}
den_ED = c0**2 * (s1 - 1)**2
pmz_lst = [x**2 + y**2 + 1, -x, -x * y, y]
pmz_B_lst = [(pmz.subs(dct_CD) * den_CD).expand() for pmz in pmz_lst]
pmz_C_lst = [(pmz.subs(dct_ED) * den_ED).expand() for pmz in pmz_lst]
# parametrization of circles
#
pmz_C1_lst = [pmz.subs({x: t0, y: -t0 - 1}) for pmz in pmz_lst]
pmz_C2_lst = [pmz.subs({x: t0, y: -t0 + 1}) for pmz in pmz_lst]
pmz_C3_lst = [pmz.subs({x: t0, y: t0 + 1}) for pmz in pmz_lst]
pmz_C4_lst = [pmz.subs({x: t0, y: t0 - 1}) for pmz in pmz_lst]
# output
#
lst_lst = [('A', pmz_A_lst), ('B', pmz_B_lst), ('C', pmz_C_lst)]
lst_lst += [
('C1', pmz_C1_lst),
('C2', pmz_C2_lst),
('C3', pmz_C3_lst),
('C4', pmz_C4_lst),
]
for A, pmz_lst in lst_lst:
OrbTools.p('pmz_' + A + '_lst =', pmz_lst)
for pmz in pmz_lst:
OrbTools.p('\t\t', sage_factor(pmz))
#############################
# PovInput Veronese surface #
#############################
pin = PovInput()
pin.path = './' + get_time_str() + '_veronese/'
pin.fname = 'orb'
pin.scale = 1
pin.cam_dct['location'] = (0, -1.2, 0)
pin.cam_dct['lookat'] = (0, 0, 0)
pin.cam_dct['rotate'] = (35, 0, 45)
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'] = 0.5
pin.height = 400
pin.width = 800
pin.quality = 11
pin.ani_delay = 10
pin.impl = None
pin.pmz_dct['A'] = (pmz_A_lst, 0)
pin.pmz_dct['B'] = (pmz_B_lst, 1)
pin.pmz_dct['C'] = (pmz_C_lst, 1)
pin.pmz_dct['FA'] = (pmz_A_lst, 0)
pin.pmz_dct['FB'] = (pmz_B_lst, 1)
pin.pmz_dct['FC'] = (pmz_C_lst, 1)
pin.pmz_dct['FA2'] = (pmz_A_lst, 0)
pin.pmz_dct['FB2'] = (pmz_B_lst, 1)
pin.pmz_dct['FC2'] = (pmz_C_lst, 1)
pin.pmz_dct['C1'] = (pmz_C1_lst, 0)
pin.pmz_dct['C2'] = (pmz_C2_lst, 0)
pin.pmz_dct['C3'] = (pmz_C3_lst, 0)
pin.pmz_dct['C4'] = (pmz_C4_lst, 0)
v0_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 5)]
v1_A_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 9)]
v1_B_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 18)]
v1_C_lst = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 9)]
v1_lst_F = [(sage_QQ(i) / (3 * 180)) * sage_pi
for i in range(0, 3 * 360, 1)]
v1_lst_F2 = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 2)]
v0_lst_CC = [sage_QQ(i) / 10 for i in range(-100, 100, 1)]
prec = 50
pin.curve_dct['A'] = {
'step0': v0_lst,
'step1': v1_A_lst,
'prec': prec,
'width': 0.01
}
pin.curve_dct['B'] = {
'step0': v0_lst,
'step1': v1_B_lst,
'prec': prec,
'width': 0.01
}
pin.curve_dct['C'] = {
'step0': v0_lst,
'step1': v1_C_lst,
'prec': prec,
'width': 0.01
}
pin.curve_dct['FA'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.001
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.001
}
pin.curve_dct['FC'] = {
'step0': v0_lst,
'step1': v1_lst_F,
'prec': prec,
'width': 0.001
}
pin.curve_dct['FA2'] = {
'step0': v0_lst,
'step1': v1_lst_F2,
'prec': prec,
'width': 0.001
}
pin.curve_dct['FB2'] = {
'step0': v0_lst,
'step1': v1_lst_F2,
'prec': prec,
'width': 0.001
}
pin.curve_dct['FC2'] = {
'step0': v0_lst,
'step1': v1_lst_F2,
'prec': prec,
'width': 0.001
}
pin.curve_dct['C1'] = {
'step0': v0_lst_CC,
'step1': [0],
'prec': prec,
'width': 0.01
}
pin.curve_dct['C2'] = {
'step0': v0_lst_CC,
'step1': [0],
'prec': prec,
'width': 0.01
}
pin.curve_dct['C3'] = {
'step0': v0_lst_CC,
'step1': [0],
'prec': prec,
'width': 0.01
}
pin.curve_dct['C4'] = {
'step0': v0_lst_CC,
'step1': [0],
'prec': prec,
'width': 0.01
}
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)
colCC = (0.6, 0.0, 0.0, 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']
pin.text_dct['FA2'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['FB2'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['FC2'] = [True, colFF, 'phong 0.2 phong_size 5']
pin.text_dct['C1'] = [True, colCC, 'phong 0.2 phong_size 5']
pin.text_dct['C2'] = [True, colCC, 'phong 0.2 phong_size 5']
pin.text_dct['C3'] = [True, colCC, 'phong 0.2 phong_size 5']
pin.text_dct['C4'] = [True, colCC, 'phong 0.2 phong_size 5']
############################
# raytrace image/animation #
############################
# four circles on projection Veronese surface
pin.cam_dct['location'] = (0, -1.5, 0)
pin.cam_dct['rotate'] = (60, 10, 45)
create_pov(pin, ['FA2', 'FB2', 'FC2'])
create_pov(pin, ['C1', 'C2', 'C3', 'C4'] + ['FA2', 'FB2', 'FC2'])
# hexagonal web on Veronese surface
pin.cam_dct['location'] = (0, -1.2, 0)
pin.cam_dct['rotate'] = (35, 0, 45)
create_pov(pin, ['A', 'B', 'C'])
create_pov(pin, ['A', 'B', 'C', 'FA', 'FB', 'FC'])
create_pov(pin, ['FA2', 'FB2', 'FC2'])
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 get_orb_bp_tree( pmz_lst ):
'''
Parameters
----------
pmz_lst : list
A list of 9 elements p0,...,p8 in QQ[c0,s0,c1,s1]
such that -p0^2+p1^2+...+p8^2==0.
Some of the polynomials can be equal to zero.
The list should represent a parametrization:
S^1xS^1--->S^7.
Here (c0,s0) is a points on S^1 such that
thus c0^2+s0^2-1==0. Similarly for (c1,s1).
Returns
-------
linear_series.BasePointTree
Base points of a parametrizing map given by the composition:
P^1xP^1---->S^1xS^1--->S^7--->S^n
with 2<=n<=7. The composition of the latter two maps
are defined by omitting the zero polynomials from "pmz_lst".
'''
# setup dictionary for reparametrization'
#
c0, s0, c1, s1 = OrbRing.coerce( 'c0,s0,c1,s1' )
dct1 = {}
dct1[c0] = '2*t0*t1/(t0^2+t1^2)'
dct1[s0] = '(-t0^2+t1^2)/(t0^2+t1^2)'
dct1[c1] = '2*v0*v1/(v0^2+v1^2)'
dct1[s1] = '(-v0^2+v1^2)/(v0^2+v1^2)'
for key in dct1: dct1[key] = OrbRing.coerce( dct1[key] )
# apply reparametrization and multiply out denominators
# where we only consider non-zero entries
#
ps_lst = [ pmz for pmz in pmz_lst if pmz != 0 ]
gcm1 = OrbRing.coerce( '(t0^2+t1^2)*(v0^2+v1^2)' )
ps_lst = [ OrbRing.coerce( ps.subs( dct1 ) * gcm1 ) for ps in ps_lst ]
# ensure that polynomials are co-prime
#
gcd1 = sage_gcd( ps_lst )
ps_lst = [ OrbRing.coerce( ps / gcd1 ) for ps in ps_lst ]
OrbTools.p( 'gcd =', gcd1 )
OrbTools.p( 'ps_lst =', ps_lst )
# Verify whether "ps_lst" represents a map P^1xP^1--->S^n
# where "n==len(ps_lst)".
#
sum1 = sum( [-ps_lst[0] ** 2] + [ ps ** 2 for ps in ps_lst[1:] ] )
OrbTools.p( 'sum =', sum1 )
if sum1 != 0:
warnings.warn( 'Warning: Not parametrization of surface in S^7: ' + str( sum1 ), )
# set coordinates x,y,v,w
#
t0, t1, v0, v1 = OrbRing.coerce( 't0,t1,v0,v1' )
dct2 = {}
dct2[t0] = sage_var( 'x' )
dct2[t1] = sage_var( 'y' )
dct2[v0] = sage_var( 'v' )
dct2[v1] = sage_var( 'w' )
xyvw_lst = [ str( ps.subs( dct2 ) ) for ps in ps_lst ]
#
# Compute base point tree using "linear_series" package
#
ls = LinearSeries( xyvw_lst, PolyRing( 'x,y,v,w', True ) )
bp_tree = ls.get_bp_tree()
OrbTools.p( ls )
OrbTools.p( bp_tree )
return bp_tree
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'])
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 ring_cyclide():
'''
Creates povray image of 4 families of circles on a ring cyclide.
'''
# We construct a trigonometric parametrization of the ring cyclide,
# by rotating a circle of radius r along a circle of radius R.
R = 2
r = 1
x, y, v, w, c0, s0, c1, s1 = sage_var('x,y,v,w,c0,s0,c1,s1')
V = sage_vector([r * c0 + R, 0, r * s0])
M = sage_matrix([(c1, -s1, 0), (s1, c1, 0), (0, 0, 1)])
pmz_AB_lst = [1] + list(M * V)
OrbTools.p('pmz_AB_lst =', pmz_AB_lst)
for pmz in pmz_AB_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# convert pmz_AB_lst to rational parametrization pmz_lst
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(M * V)]
OrbTools.p('pmz_lst =', pmz_lst)
# find basepoints
ls = LinearSeries(pmz_lst, PolyRing('x,y,v,w', True))
OrbTools.p(ls.get_bp_tree())
# construct linear series for families of conics
a0, a1 = PolyRing('x,y,v,w').ext_num_field('t^2+1/3').ext_num_field(
't^2+1').root_gens()
p1 = ['xv', (-a0, a1)]
p2 = ['xv', (a0, -a1)]
p3 = ['xv', (-a0, -a1)]
p4 = ['xv', (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
ring = PolyRing(
'x,y,v,w,c0,s0,c1,s1') # construct polynomial ring with new generators
pmz_lst = ring.coerce(pmz_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
q = sage_n(sage_sqrt(3)).exact_rational() # approximation of sqrt(3)
CB_dct = {x: X, y: Y, v: W * X + q * V * Y, w: V * X - q * W * Y}
DB_dct = {x: X, y: Y, v: W * X - q * V * Y, w: V * X + q * W * Y}
pmz_CB_lst = [pmz.subs(CB_dct) for pmz in pmz_lst]
pmz_DB_lst = [pmz.subs(DB_dct) for pmz in pmz_lst]
# 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)
# mathematica
for pmz, AB in [(pmz_AB_lst, 'AB'), (pmz_CB_lst, 'CB'),
(pmz_DB_lst, 'DB')]:
s = 'pmz' + AB + '=' + str(pmz) + ';'
s = s.replace('[', '{').replace(']', '}')
print(s)
# PovInput ring cyclide
#
pin = PovInput()
pin.path = './' + get_time_str() + '_ring_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'] = (55, 0, 0) # 45
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.width = 800
pin.height = 400
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['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['WA'] = (pmz_AB_lst, 0)
pin.pmz_dct['WB'] = (pmz_AB_lst, 1)
pin.pmz_dct['WC'] = (pmz_CB_lst, 0)
pin.pmz_dct['WD'] = (pmz_DB_lst, 0)
v0_lst = [(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, 24)]
v1_lst_A = [
sage_pi / 2 + (sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 12)
]
v1_lstFF = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 1)]
v1_lst_WA = [
0.1, 0.52, 0.94, 1.36, 1.78, 2.2, 2.61, 3.04, 3.45, 3.88, 4.3, 4.712,
5.13, 5.55, 5.965
]
v1_lst_WB = [
0, 0.7, 1.31, 1.8, 2.18, 2.5, 2.77, 3.015, 3.26, 3.51, 3.78, 4.099,
4.49, 4.97, 5.579
]
v1_lst_WD = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 24)]
v1_lst_WC = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 360, 24)]
pin.curve_dct['A'] = {
'step0': v0_lst,
'step1': v1_lst_A,
'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['FA'] = {
'step0': v0_lst,
'step1': v1_lstFF,
'prec': 10,
'width': 0.02
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_lstFF,
'prec': 10,
'width': 0.02
}
pin.curve_dct['FC'] = {
'step0': v0_lst,
'step1': v1_lstFF,
'prec': 10,
'width': 0.02
}
pin.curve_dct['FD'] = {
'step0': v0_lst,
'step1': v1_lstFF,
'prec': 10,
'width': 0.02
}
pin.curve_dct['WA'] = {
'step0': v0_lst,
'step1': v1_lst_WA,
'prec': 10,
'width': 0.05
}
pin.curve_dct['WB'] = {
'step0': v0_lst,
'step1': v1_lst_WB,
'prec': 10,
'width': 0.05
}
pin.curve_dct['WC'] = {
'step0': v0_lst,
'step1': v1_lst_WC,
'prec': 10,
'width': 0.05
}
pin.curve_dct['WD'] = {
'step0': v0_lst,
'step1': v1_lst_WD,
'prec': 10,
'width': 0.05
}
# A = | rotated circle
# B = - horizontal circle
# C = / villarceau circle
# D = \ villarceau circle
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_D = rgbt2pov((187, 46, 0, 0)) # red/orange
colFF = rgbt2pov((179, 200, 217, 0)) # light blue
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['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['WA'] = [True, col_A, 'phong 0.2 phong_size 5']
pin.text_dct['WB'] = [True, col_B, 'phong 0.2 phong_size 5']
pin.text_dct['WC'] = [True, col_C, 'phong 0.2 phong_size 5']
pin.text_dct['WD'] = [True, col_D, 'phong 0.2 phong_size 5']
# raytrace image/animation
create_pov(pin, ['A', 'C', 'D'])
create_pov(pin, ['A', 'C', 'D'] + ['FA', 'FC', 'FD'])
create_pov(pin, ['WA', 'WB', 'WC', 'WD'])
create_pov(pin, ['WA', 'WB', 'WC', 'WD'] + ['FA', 'FC', 'FD'])
create_pov(pin, ['WA', 'WB', 'WD'])
create_pov(pin, ['WA', 'WB', 'WD'] + ['FA', 'FC', 'FD'])
def get_project( pol_lst, pmat ):
'''
Parameters
----------
pol_lst : list<OrbRing.R>
A list of homogeneous polynomials in QQ[x0,...,x8].
pmat : sage_matrix
A matrix defined over the rationals QQ.
Returns
-------
tuple
A 2-tuple of polynomials:
* a homogeneous polynomial F in QQ[x0,x1,x2,x3].
* F(1,x,y,z) in QQ[x,y,z] (affine polynomial)
'''
Ry = sage_PolynomialRing( sage_GF( 2 ), sage_var( 'y0,y1,y2,y3,y4,y5,y6,y7,y8' ), order = 'degrevlex' )
v = OrbRing.coerce( '[v0,v1,v2,v3,v4,v5,v6,v7,v8]' )
x = OrbRing.coerce( '[x0,x1,x2,x3,x4,x5,x6,x7,x8]' )
vx_dct = {v[i]:x[i] for i in range( 9 )}
OrbTools.p( "\n" + str( pmat ) )
tries = 0
projected = False
while not projected:
# obtain the linear equations of the projection map
pmat = sage_matrix( OrbRing.R, list( pmat ) )
leq_lst = list( pmat * sage_vector( x ) )
# compute the image of this projection map
proj_lst = [ v[i] - leq_lst[i] for i in range( len( leq_lst ) ) ]
p_lst = sage_ideal( pol_lst + proj_lst ).elimination_ideal( x ).gens()
# obtain a polynomial in x0,...,x8
p_lst = [p.subs( vx_dct ) for p in p_lst]
fx = p_lst[0]
tries += 1
if len( fx.variables() ) < 4 or len( p_lst ) != 1:
pmat = get_pmat( True )
if tries % 100 == 0:
OrbTools.p( 'tries =', tries, p_lst )
if tries == 1000:
return -1
else:
projected = True
w0, w1, w2, w3 = fx.variables()
fx = fx.subs( {w0:x[0], w1:x[1], w2:x[2], w3:x[3]} )
x0, x1, x2, x3 = OrbRing.coerce( 'x0,x1,x2,x3' )
x, y, z = sage_var( 'x,y,z' )
fxyz = fx.subs( {x0:1, x1:x, x2:y, x3:z} )
OrbTools.p( fx )
OrbTools.p( fxyz )
return fx, fxyz
def get_imp(A, B, prj, sng, snp):
'''
Computes implicit equation of a stereographic projection S of
the pointwise Hamiltonian product of circles in the sphere S^3
defined by transformations of the standard circle [1,cos(a),sin(a),0,0]
by A and B respectively.
Parameters
----------
A : sage_Matrix<sage_QQ>
Represents a linear transformation S^3--->S^3
B : sage_Matrix<sage_QQ>
Represents a linear transformation S^3--->S^3
prj : int
Choice for stereographic projection S^3--->P^3:
0: (x0:x1:x2:x3:x4) |--> (x0-x4:x1:x2:x3)
1: (x0:x1:x2:x3:x4) |--> (x0-x1:x4:x2:x3)
2: (x0:x1:x2:x3:x4) |--> (x0-x2:x1:x4:x3)
3: (x0:x1:x2:x3:x4) |--> (x0-x3:x1:x2:x4)
sng : boolean
If true computes singular locus of S. Needs Magma path set
in os.environ['PATH']. Otherwise the empty-list is returned.
snp : boolean
If true and if sng is True, then the singular locus is
computed with a probablistic method, which is faster but
the correctness of the output is not guaranteed.
Returns
-------
dict
{
'Agreat' : boolean
If True, then circle A is great.
'Bgreat' : boolean
If True, then circle B is great.
'eqn_x' : sage_PolynomialRing
Equation of S in x0,...,x3
'eqn_str':
Formatted equation in x0,...,x3 of the
form f(x1:x2:x3)+x0*g(x0:x1:x2:x3).
'eqn_xyz':
Equation of S in x,y,z
'sng_lst':
Empty-list if Magma is not installed or
list of singularities of S otherwise.
}
'''
dct = {} # output
# create polynomial ring
#
R = sage_PolynomialRing(sage_QQ,
'x0,x1,x2,x3,a0,a1,a2,a3,a4,b0,b1,b2,b3,b4')
x0, x1, x2, x3, a0, a1, a2, a3, a4, b0, b1, b2, b3, b4 = R.gens()
# construct ideal for A
#
sv = [0, 0, 0, 1, 0]
tv = [0, 0, 0, 0, 1]
u0, u1, u2, u3, u4 = list(A * sage_vector(sv))
v0, v1, v2, v3, v4 = list(A * sage_vector(tv))
eqA = [-a0**2 + a1**2 + a2**2 + a3**2 + a4**2]
eqA += [u0 * a0 + u1 * a1 + u2 * a2 + u3 * a3 + u4 * a4]
eqA += [v0 * a0 + v1 * a1 + v2 * a2 + v3 * a3 + v4 * a4]
dct['Agreat'] = u0 == v0 == 0
# construct ideal for B
#
u0, u1, u2, u3, u4 = list(B * sage_vector(sv))
v0, v1, v2, v3, v4 = list(B * sage_vector(tv))
eqB = [-b0**2 + b1**2 + b2**2 + b3**2 + b4**2]
eqB += [u0 * b0 + u1 * b1 + u2 * b2 + u3 * b3 + u4 * b4]
eqB += [v0 * b0 + v1 * b1 + v2 * b2 + v3 * b3 + v4 * b4]
dct['Bgreat'] = u0 == v0 == 0
# stereographic projection
#
if prj == 0: j, i_lst = 4, [1, 2, 3]
if prj == 1: j, i_lst = 1, [4, 2, 3]
if prj == 2: j, i_lst = 2, [1, 4, 3]
if prj == 3: j, i_lst = 3, [1, 2, 4]
# construct equation of for projection of A*B
#
c = c0, c1, c2, c3, c4 = get_hp_P4([a0, a1, a2, a3, a4],
[b0, b1, b2, b3, b4])
x = [x0, x1, x2, x3]
i1, i2, i3 = i_lst
id = [x[0] -
(c[0] - c[j]), x[1] - c[i1], x2 - c[i2], x3 - c[i3]] + eqA + eqB
dct['eqn_x'] = eqn_x = R.ideal(id).elimination_ideal(
[a0, a1, a2, a3, a4, b0, b1, b2, b3, b4]).gens()[0]
# get equation in string form
#
f = eqn_x.subs({x0: 0})
dct['eqn_str'] = str(sage_factor(f)) + '+' + str(sage_factor(eqn_x - f))
xs, ys, zs = sage_var('x,y,z')
dct['eqn_xyz'] = sage_SR(eqn_x.subs({x0: 1, x1: xs, x2: ys, x3: zs}))
# compute singular locus
#
dct['sng_lst'] = []
if sng:
dct['sng_lst'] = get_sing_lst(OrbRing.coerce(eqn_x), snp)
return dct
def spindle_cyclide():
'''
Constructs a povray image of a spindle cyclide. The spindle cyclide is
an inversion of a circular cylinder.
'''
# We construct a trigonometric parametrization
# of the cyclide by rotating a circle.
#
r = 1
R = 1
# radii of circles
x, y, v, w = sage_var('x,y,v,w')
c0, s0, c1, s1 = sage_var('c0,s0,c1,s1')
V = sage_vector([r * c0 + R, 0, r * s0])
M = sage_matrix([(c1, -s1, 0), (s1, c1, 0), (0, 0, 1)])
pmz_AB_lst = [1] + list(M * V)
OrbTools.p('pmz_AB_lst =', pmz_AB_lst)
for pmz in pmz_AB_lst:
OrbTools.p('\t\t', sage_factor(pmz))
# PovInput spindle cyclide
#
pin = PovInput()
pin.path = './' + get_time_str() + '_spindle_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'] = (45, 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['FA'] = (pmz_AB_lst, 0)
pin.pmz_dct['FB'] = (pmz_AB_lst, 1)
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, 270, 15)]
v1_lst_B = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 15)]
v1_lstFA = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 270 - 15, 1)]
v1_lstFB = [(sage_QQ(i) / 180) * sage_pi for i in range(0, 180, 1)]
pin.curve_dct['A'] = {
'step0': v0_lst,
'step1': v1_lst_A,
'prec': 10,
'width': 0.03
}
pin.curve_dct['B'] = {
'step0': v0_lst,
'step1': v1_lst_B,
'prec': 10,
'width': 0.03
}
pin.curve_dct['FA'] = {
'step0': v0_lst,
'step1': v1_lstFA,
'prec': 10,
'width': 0.02
}
pin.curve_dct['FB'] = {
'step0': v0_lst,
'step1': v1_lstFB,
'prec': 10,
'width': 0.02
}
col_A = (0.6, 0.4, 0.1, 0.0)
col_B = (0.1, 0.15, 0.0, 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']
# raytrace image/animation
create_pov(pin, ['A', 'B'])
create_pov(pin, ['A', 'B', 'FA', 'FB'])
def get_sing_lst( pol, probable = True ):
'''
The method requires that the magma-command
is in "os.environ['PATH']".
See [http://magma.maths.usyd.edu.au/magma/].
Parameters
----------
pol : OrbRing.R
A homogeneous polynomial in QQ[x0,x1,x2,x3].
probable : boolean
If True, performs a non-deterministic version of a
radical decomposition algorithm, which is faster.
The correctness of output is not guaranteed in this case.
Returns
-------
list
Suppose that X is the surface defined by the zero-set of "pol".
The output is a list
[ (<I>, <H>), ... ]
where <I> is the ideal of a component in singular locus of X,
and <H> is the Hilbert polynomial of this ideal.
If Magma is not accessible, then the empty-list [] is returned.
'''
x0, x1, x2, x3 = OrbRing.coerce( 'x0,x1,x2,x3' )
df_str = str( [sage_diff( pol, x0 ), sage_diff( pol, x1 ), sage_diff( pol, x2 ), sage_diff( pol, x3 )] )[1:-1]
OrbTools.p( df_str )
mi = ''
mi += 'P<x0,x1,x2,x3> := PolynomialRing(RationalField(), 4);\n'
mi += 'MI := ideal< P |' + df_str + '>;\n'
if probable:
mi += 'MD := ProbableRadicalDecomposition( MI );\n'
else:
mi += 'MD := RadicalDecomposition( MI );\n'
mi += '#MD;\n'
try:
mo1 = int( sage_magma.eval( mi ) )
except:
return []
sing_lst = []
Ry = sage_PolynomialRing( OrbRing.num_field, sage_var( 'y0,y1,y2,y3' ), order = 'degrevlex' )
for idx in range( mo1 ):
comp = str( sage_magma.eval( 'Basis(MD[' + str( idx + 1 ) + ']);\n' ) )
comp = comp.replace( '\n', '' )
# compute hilbert polynomial of component in singular locus
compy = sage__eval( comp.replace( 'x', 'y' ), Ry.gens_dict() )
idy = Ry.ideal( compy )
hpol = idy.hilbert_polynomial()
sing_lst += [( comp, hpol )]
OrbTools.p( idx, sing_lst[-1] )
OrbTools.p( sing_lst )
return sing_lst