def subexpressions_list(f, pars=None):
"""
Construct the lists with the intermediate steps on the evaluation of the
function.
INPUT:
- ``f`` -- a symbolic function of several components.
- ``pars`` -- a list of the parameters that appear in the function
this should be the symbolic constants that appear in f but are not
arguments.
OUTPUT:
- a list of the intermediate subexpressions that appear in the evaluation
of f.
- a list with the operations used to construct each of the subexpressions.
each element of this list is a tuple, formed by a string describing the
operation made, and the operands.
For the trigonometric functions, some extra expressions will be added.
These extra expressions will be used later to compute their derivatives.
EXAMPLES::
sage: from sage.interfaces.tides import subexpressions_list
sage: var('x,y')
(x, y)
sage: f(x,y) = [x^2+y, cos(x)/log(y)]
sage: subexpressions_list(f)
([x^2, x^2 + y, sin(x), cos(x), log(y), cos(x)/log(y)],
[('mul', x, x),
('add', y, x^2),
('sin', x),
('cos', x),
('log', y),
('div', log(y), cos(x))])
::
sage: f(a)=[cos(a), arctan(a)]
sage: from sage.interfaces.tides import subexpressions_list
sage: subexpressions_list(f)
([sin(a), cos(a), a^2, a^2 + 1, arctan(a)],
[('sin', a), ('cos', a), ('mul', a, a), ('add', 1, a^2), ('atan', a)])
::
sage: from sage.interfaces.tides import subexpressions_list
sage: var('s,b,r')
(s, b, r)
sage: f(t,x,y,z)= [s*(y-x),x*(r-z)-y,x*y-b*z]
sage: subexpressions_list(f,[s,b,r])
([-y,
x - y,
s*(x - y),
-s*(x - y),
-z,
r - z,
(r - z)*x,
-y,
(r - z)*x - y,
x*y,
b*z,
-b*z,
x*y - b*z],
[('mul', -1, y),
('add', -y, x),
('mul', x - y, s),
('mul', -1, s*(x - y)),
('mul', -1, z),
('add', -z, r),
('mul', x, r - z),
('mul', -1, y),
('add', -y, (r - z)*x),
('mul', y, x),
('mul', z, b),
('mul', -1, b*z),
('add', -b*z, x*y)])
::
sage: var('x, y')
(x, y)
sage: f(x,y)=[exp(x^2+sin(y))]
sage: from sage.interfaces.tides import *
sage: subexpressions_list(f)
([x^2, sin(y), cos(y), x^2 + sin(y), e^(x^2 + sin(y))],
[('mul', x, x),
('sin', y),
('cos', y),
('add', sin(y), x^2),
('exp', x^2 + sin(y))])
"""
from sage.functions.trig import sin, cos, arcsin, arctan, arccos
variables = f[0].arguments()
if not pars:
parameters = []
else:
parameters = pars
varpar = list(parameters) + list(variables)
F = symbolic_expression([i(*variables) for i in f]).function(*varpar)
lis = flatten([fast_callable(i,vars=varpar).op_list() for i in F], max_level=1)
stack = []
const =[]
stackcomp=[]
detail=[]
for i in lis:
if i[0] == 'load_arg':
stack.append(varpar[i[1]])
elif i[0] == 'ipow':
if i[1] in NN:
basis = stack[-1]
for j in range(i[1]-1):
a=stack.pop(-1)
detail.append(('mul', a, basis))
stack.append(a*basis)
stackcomp.append(stack[-1])
else:
detail.append(('pow',stack[-1],i[1]))
stack[-1]=stack[-1]**i[1]
stackcomp.append(stack[-1])
elif i[0] == 'load_const':
const.append(i[1])
stack.append(i[1])
elif i == 'mul':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('mul', a, b))
stack.append(a*b)
stackcomp.append(stack[-1])
elif i == 'div':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('div', a, b))
stack.append(b/a)
stackcomp.append(stack[-1])
elif i == 'add':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('add',a,b))
stack.append(a+b)
stackcomp.append(stack[-1])
elif i == 'pow':
a=stack.pop(-1)
b=stack.pop(-1)
detail.append(('pow', b, a))
stack.append(b**a)
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='log':
a=stack.pop(-1)
detail.append(('log', a))
stack.append(log(a))
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='exp':
a=stack.pop(-1)
detail.append(('exp', a))
stack.append(exp(a))
stackcomp.append(stack[-1])
elif i[0] == 'py_call' and str(i[1])=='sin':
a=stack.pop(-1)
detail.append(('sin', a))
detail.append(('cos', a))
stackcomp.append(sin(a))
stackcomp.append(cos(a))
stack.append(sin(a))
elif i[0] == 'py_call' and str(i[1])=='cos':
a=stack.pop(-1)
detail.append(('sin', a))
detail.append(('cos', a))
stackcomp.append(sin(a))
stackcomp.append(cos(a))
stack.append(cos(a))
elif i[0] == 'py_call' and str(i[1])=='tan':
a=stack.pop(-1)
b = sin(a)
c = cos(a)
detail.append(('sin', a))
detail.append(('cos', a))
detail.append(('div', b, c))
stackcomp.append(b)
stackcomp.append(c)
stackcomp.append(b/c)
stack.append(b/c)
elif i[0] == 'py_call' and str(i[1])=='arctan':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('add', 1, a*a))
detail.append(('atan', a))
stackcomp.append(a*a)
stackcomp.append(1+a*a)
stackcomp.append(arctan(a))
stack.append(arctan(a))
elif i[0] == 'py_call' and str(i[1])=='arcsin':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('mul', -1, a*a))
detail.append(('add', 1, -a*a))
detail.append(('pow', 1- a*a, 0.5))
detail.append(('asin', a))
stackcomp.append(a*a)
stackcomp.append(-a*a)
stackcomp.append(1-a*a)
stackcomp.append(sqrt(1-a*a))
stackcomp.append(arcsin(a))
stack.append(arcsin(a))
elif i[0] == 'py_call' and str(i[1])=='arccos':
a=stack.pop(-1)
detail.append(('mul', a, a))
detail.append(('mul', -1, a*a))
detail.append(('add', 1, -a*a))
detail.append(('pow', 1- a*a, 0.5))
detail.append(('mul', -1, sqrt(1-a*a)))
detail.append(('acos', a))
stackcomp.append(a*a)
stackcomp.append(-a*a)
stackcomp.append(1-a*a)
stackcomp.append(sqrt(1-a*a))
stackcomp.append(-sqrt(1-a*a))
stackcomp.append(arccos(a))
stack.append(arccos(a))
elif i[0] == 'py_call' and 'sqrt' in str(i[1]):
a=stack.pop(-1)
detail.append(('pow', a, 0.5))
stackcomp.append(sqrt(a))
stack.append(sqrt(a))
elif i == 'neg':
a = stack.pop(-1)
detail.append(('mul', -1, a))
stack.append(-a)
stackcomp.append(-a)
return stackcomp,detail
first_vector=first_point-center
second_vector=second_point-center
radius=norm(first_vector)
if norm(second_vector)!=radius:
raise ValueError("Ellipse not implemented")
first_unit_vector=first_vector/radius
second_unit_vector=second_vector/radius
normal_vector=second_vector-(second_vector*first_unit_vector)*first_unit_vector
if norm(normal_vector)==0:
print (first_point,second_point)
return
normal_unit_vector=normal_vector/norm(normal_vector)
scalar_product=first_unit_vector*second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle=arccos(scalar_product)
var('t')
return parametric_plot3d(center+first_vector*cos(t)+radius*normal_unit_vector*sin(t),(0,angle),**kwds)
def _arc(p,q,s,**kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p,q,s = map( lambda x: vector(x), [p,q,s])
def angle(self, other): # UHP
r"""
Return the angle between any two given completed geodesics if
they intersect.
INPUT:
- ``other`` -- a hyperbolic geodesic in the UHP model
OUTPUT:
- the angle in radians between the two given geodesics
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(3, 3 + I)
sage: g.angle(h)
1/2*pi
sage: numerical_approx(g.angle(h))
1.57079632679490
If the geodesics are identical, return angle 0::
sage: g.angle(g)
0
It is an error to ask for the angle of two geodesics that do not
intersect::
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(5, 7)
sage: g.angle(h)
Traceback (most recent call last):
...
ValueError: geodesics do not intersect
"""
if self.is_parallel(other):
raise ValueError("geodesics do not intersect")
# Make sure the segments are complete or intersect
if (not(self.is_complete() and other.is_complete())
and not self.intersection(other)):
print("Warning: Geodesic segments do not intersect. "
"The angle between them is not defined.\n"
"Returning the angle between their completions.")
# Make sure both are in the same model
if other._model is not self._model:
other = other.to_model(self._model)
(p1, p2) = sorted([k.coordinates()
for k in self.ideal_endpoints()], key=str)
(q1, q2) = sorted([k.coordinates()
for k in other.ideal_endpoints()], key=str)
# if the geodesics are equal, the angle between them is 0
if (abs(p1 - q1) < EPSILON
and abs(p2 - q2) < EPSILON):
return 0
elif p2 != infinity: # geodesic not a straight line
# So we send it to the geodesic with endpoints [0, oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, (p1 + p2) / 2, p2)
else:
# geodesic is a straight line, so we send it to the geodesic
# with endpoints [0,oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, p1 + 1, p2)
# b1 and b2 are the endpoints of the image of other
b1, b2 = [mobius_transform(T, k) for k in [q1, q2]]
# If other is now a straight line...
if (b1 == infinity or b2 == infinity):
# then since they intersect, they are equal
return 0
return real(arccos((b1 + b2) / abs(b2 - b1)))
def angle(self, other): # UHP
r"""
Return the angle between any two given completed geodesics if
they intersect.
INPUT:
- ``other`` -- a hyperbolic geodesic in the UHP model
OUTPUT:
- the angle in radians between the two given geodesics
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(3, 3 + I)
sage: g.angle(h)
1/2*pi
sage: numerical_approx(g.angle(h))
1.57079632679490
If the geodesics are identical, return angle 0::
sage: g.angle(g)
0
It is an error to ask for the angle of two geodesics that do not
intersect::
sage: g = UHP.get_geodesic(2, 4)
sage: h = UHP.get_geodesic(5, 7)
sage: g.angle(h)
Traceback (most recent call last):
...
ValueError: geodesics do not intersect
"""
if self.is_parallel(other):
raise ValueError("geodesics do not intersect")
# Make sure the segments are complete or intersect
if (not (self.is_complete() and other.is_complete())
and not self.intersection(other)):
print("Warning: Geodesic segments do not intersect. "
"The angle between them is not defined.\n"
"Returning the angle between their completions.")
# Make sure both are in the same model
if other._model is not self._model:
other = other.to_model(self._model)
(p1, p2) = sorted([k.coordinates() for k in self.ideal_endpoints()],
key=str)
(q1, q2) = sorted([k.coordinates() for k in other.ideal_endpoints()],
key=str)
# if the geodesics are equal, the angle between them is 0
if (abs(p1 - q1) < EPSILON and abs(p2 - q2) < EPSILON):
return 0
elif p2 != infinity: # geodesic not a straight line
# So we send it to the geodesic with endpoints [0, oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, (p1 + p2) / 2, p2)
else:
# geodesic is a straight line, so we send it to the geodesic
# with endpoints [0,oo]
T = HyperbolicGeodesicUHP._crossratio_matrix(p1, p1 + 1, p2)
# b1 and b2 are the endpoints of the image of other
b1, b2 = [mobius_transform(T, k) for k in [q1, q2]]
# If other is now a straight line...
if (b1 == infinity or b2 == infinity):
# then since they intersect, they are equal
return 0
return real(arccos((b1 + b2) / abs(b2 - b1)))
second_vector = second_point - center
radius = norm(first_vector)
if norm(second_vector) != radius:
raise ValueError("Ellipse not implemented")
first_unit_vector = first_vector / radius
second_unit_vector = second_vector / radius
normal_vector = second_vector - (second_vector *
first_unit_vector) * first_unit_vector
if norm(normal_vector) == 0:
print(first_point, second_point)
return
normal_unit_vector = normal_vector / norm(normal_vector)
scalar_product = first_unit_vector * second_unit_vector
if abs(scalar_product) == 1:
raise ValueError("The points are alligned")
angle = arccos(scalar_product)
var('t')
return parametric_plot3d(
center + first_vector * cos(t) + radius * normal_unit_vector * sin(t),
(0, angle), **kwds)
def _arc(p, q, s, **kwds):
#rewrite this to use polar_plot and get points to do filled triangles
from sage.misc.functional import det
from sage.plot.line import line
from sage.misc.functional import norm
from sage.symbolic.all import pi
from sage.plot.arc import arc
p, q, s = map(lambda x: vector(x), [p, q, s])