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 test__get_linear_series__6(self): ring = PolyRing('x,y,v,w', True) ring.ext_num_field('t^2 + 1') a0 = ring.root_gens()[0] bp_tree_1 = BasePointTree(['xv', 'xw', 'yv', 'yw']) bp = bp_tree_1.add('xv', (a0, -a0), 2) bp_tree_2 = LinearSeries.get([2, 2], bp_tree_1).get_bp_tree() bp_tree_2_str = bp_tree_2.alt_str() bp_tree_2_str = bp_tree_2_str.replace('(a0)', 'a0') bp_tree_2_str = bp_tree_2_str.replace('(-a0)', '-a0') assert self.equal_output_strings(bp_tree_1.alt_str(), bp_tree_2_str)
def test__subs( self ): ring = PolyRing( 'x,y,z', True ) ring.ext_num_field( 't^2 + 1' ) ring.ext_num_field( 't^3 + a0' ) a0, a1 = ring.root_gens() x, y, z = ring.gens() pol_lst = [x ** 2 + a0 * y * z, y + a1 * z + x ] ls = LinearSeries( pol_lst, ring ) assert str( ls.pol_lst ) == '[x^2 + a0*y*z, x + y + a1*z]' ls.subs( {x: x + 1} ) assert str( ls.pol_lst ) == '[x^2 + a0*y*z + 2*x + 1, x + y + a1*z + 1]'
def test__approx_QQ_coef__2(self): ring = PolyRing('x,y,v,w', True) ring.ext_num_field('t^2 - 3') ring.ext_num_field('t^2 + 1') ring.ext_num_field('t^2 + 2/7*a0*a1 + 3/7') ring.ext_num_field('t^2 - 2/7*a0*a1 + 3/7') a0, a1, a2, a3 = ring.root_gens() 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 ci_idx = 5 # index for complex embedding C1 = -(b + c - d - e) C2 = -(b + c + d + e) for lbl, elt in [('a0', a0), ('C1', C1), ('C2', C2)]: print(lbl + ' = ' + str(elt.abs())) for ci in elt.complex_embeddings(): print('\t\t' + str(ci)) s = str(elt.complex_embeddings()[ci_idx]) print(lbl + '.complex_embedding()[' + str(ci_idx) + '] = ' + s) # C1 print('--- C1 ---') out = OrbRing.approx_QQ_coef(C1, ci_idx) print(out) print(sage_n(out)) chk = -sage_QQ(3687885631267691) / 2251799813685248 assert out == chk # C2 print('--- C2 ---') out = OrbRing.approx_QQ_coef(C2, ci_idx) print(out) print(sage_n(out)) chk = sage_QQ(2749869658604311) / 4503599627370496 assert out == chk
def test__ext_num_field(self): ring = PolyRing('x,y,z', True) assert str(ring) == 'QQ[x, y, z]' ring.ext_num_field('t^2 + t + 1') assert str(ring) == 'QQ( <a0|t^2 + t + 1> )[x, y, z]' ring.ext_num_field('t^3 + t + a0 + 3') assert str( ring ) == 'QQ( <a0|t^2 + t + 1>, <a1|t^3 + t + a0 + 3>, <a2|t^2 + a1*t + a1^2 + 1> )[x, y, z]' a = ring.root_gens() x, y, z = ring.gens() pol = x**3 + x + a[0] + 3 assert str(sage_factor(pol)) == '(x - a2) * (x - a1) * (x + a2 + a1)' mat = list(pol.sylvester_matrix(y**2 + x**2, x)) assert str( mat ) == '[(1, 0, 1, a0 + 3, 0), (0, 1, 0, 1, a0 + 3), (1, 0, y^2, 0, 0), (0, 1, 0, y^2, 0), (0, 0, 1, 0, y^2)]'
def test__approx_QQ_coef__1(self): ring = PolyRing('x,y,z', True).ext_num_field('t^2 - 3') a0 = ring.root_gens()[0] ci_idx = 1 for lbl, elt in [('a0', a0), ('-a0', -a0)]: print(lbl + ' = ' + str(elt.abs())) for ci in elt.complex_embeddings(): print('\t\t' + str(ci)) s = str(elt.complex_embeddings()[ci_idx]) print(lbl + '.complex_embedding()[' + str(ci_idx) + '] = ' + s) print('--- a0 ---') out = OrbRing.approx_QQ_coef(a0, ci_idx) chk = sage_QQ(3900231685776981) / 2251799813685248 print(out) print(sage_n(out)) assert out == chk print('--- (-a0) ---') out = OrbRing.approx_QQ_coef(-a0, ci_idx) print(out) print(sage_n(out)) assert out == -chk
def usecase__roman_circles(): ''' We compute circles on a Roman surface. ''' # parametrization of the Roman surface # p_lst = '[ z^2+x^2+y^2, -z*x, -x*y, z*y ]' # we consider the stereographic projection from # S^3 = { x in P^4 | -x0^2+x1^2+x2^2+x3^2+x4^2 = 0 } # where the center of projection is (1:0:0:0:1): # (x0:x1:x2:x3:x4) |---> (x0-x4:x1:x2:x3) # inverse stereographic projection into 3-sphere # s_lst = '[ y0^2+y1^2+y2^2+y3^2, 2*y0*y1, 2*y0*y2, 2*y0*y3, -y0^2+y1^2+y2^2+y3^2 ]' # compose p_lst with s_lst # ring = PolyRing('x,y,z,y0,y1,y2,y3') x, y, z, y0, y1, y2, y3 = ring.gens() p_lst = ring.coerce(p_lst) s_lst = ring.coerce(s_lst) dct = {y0: p_lst[0], y1: p_lst[1], y2: p_lst[2], y3: p_lst[3]} sp_lst = [s.subs(dct) for s in s_lst] NSTools.p('sp_lst =') for sp in sp_lst: NSTools.p('\t\t', sage_factor(sp)) NSTools.p('gcd(sp_lst) =', sage_gcd(sp_lst)) # determine base points # ring = PolyRing('x,y,z', True) sp_lst = ring.coerce(sp_lst) ls = LinearSeries(sp_lst, ring) NSTools.p(ls.get_bp_tree()) # We expect that the basepoints come from the intersection # of the Roman surface with the absolute conic: # A = { (y0:y1:y2:y3) in P^3 | y0=y1^2+y2^2+y3^2 = 0 } # # Circles are the image via p_lst of lines that pass through # complex conjugate points. # ring = PolyRing('x,y,z', False) # reinitialize ring with updated numberfield a0, a1, a2, a3 = ring.root_gens() # a0=(1-I*sqrt(3)) with conjugate a0-1 and minimal polynomial t^2-t+1 # we compute candidate classes of circles # h = Div.new('4e0-e1-e2-e3-e4-e5-e6-e7-e8') div_lst = get_divs(h, 2, -2, False) + get_divs(h, 2, -1, False) NSTools.p('Classes of circles up to permutation:') for c in div_lst: NSTools.p('\t\t', c) # We recover the preimages of circles in the Roman surface # under the map p_lst, by constructing for each candidate # class the corresponding linear series. # 2e0-e1-e2-e3-e4-e5-e6-e7-e8 b = [(a0 - 1, -a0), (-a0, a0 - 1)] b += [(-a0 + 1, a0), (a0, -a0 + 1)] b += [(a0 - 1, a0), (-a0, -a0 + 1)] b += [(-a0 + 1, -a0), (a0, a0 - 1)] bp_tree = BasePointTree() for i in range(6): bp_tree.add('z', b[i], 1) NSTools.p('basepoints =', b) NSTools.p(LinearSeries.get([2], bp_tree)) # e0-e1-e2 b = [(a0 - 1, -a0), (-a0, a0 - 1)] bp_tree = BasePointTree() bp = bp_tree.add('z', b[0], 1) bp = bp_tree.add('z', b[1], 1) NSTools.p('basepoints =', b) NSTools.p(LinearSeries.get([1], bp_tree)) # e0-e3-e4 b = [(-a0 + 1, a0), (a0, -a0 + 1)] bp_tree = BasePointTree() bp = bp_tree.add('z', b[0], 1) bp = bp_tree.add('z', b[1], 1) NSTools.p('basepoints =', b) NSTools.p(LinearSeries.get([1], bp_tree)) # e0-e5-e6 b = [(a0 - 1, a0), (-a0, -a0 + 1)] bp_tree = BasePointTree() bp = bp_tree.add('z', b[0], 1) bp = bp_tree.add('z', b[1], 1) NSTools.p('basepoints =', b) NSTools.p(LinearSeries.get([1], bp_tree)) # e0-e7-e8 b = [(-a0 + 1, -a0), (a0, a0 - 1)] bp_tree = BasePointTree() bp = bp_tree.add('z', b[0], 1) bp = bp_tree.add('z', b[1], 1) NSTools.p('basepoints =', b) NSTools.p(LinearSeries.get([1], bp_tree)) return
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'])