def get_minmax_data(self, start=None, end=None):
"""
return the min and max of x and y for the graph of `self`
and the parameter between `start` and `end`
INPUT:
- ``start,end`` - interval on which we are considering the function.
OUTPUT:
A dictionary
"""
x_list = [
numerical_approx(P.x, prec=30)
for P in self.representative_points()
]
y_list = [
numerical_approx(P.y, prec=30)
for P in self.representative_points()
]
d = {}
d['xmin'] = min(x_list)
d['xmax'] = max(x_list)
d['ymin'] = min(y_list)
d['ymax'] = max(y_list)
return d
def are_almost_equal(a, b, epsilon=0.0001):
# \brief Says if `a` and `b` are equal up to epsilon
aN = numerical_approx(a)
bN = numerical_approx(b)
if abs(aN - bN) < 0.0001: # epsilon
return True
return False
def check_too_large(obj,pspict=None):
try:
bb=obj.bounding_box(pspict)
mx=bb.xmin
my=bb.ymin
Mx=bb.xmax
My=bb.ymax
except AttributeError:
print "Object {0} has no method bounding_box.".format(obj)
mx=obj.mx
my=obj.my
Mx=obj.Mx
My=obj.My
if pspict:
from Exceptions import TooLargeBBException
# In some circumstances, the comparison
# mx<pspict.mx_acceptable_BB
# provokes a MemoryError.
n_Mx=numerical_approx(Mx)
n_mx=numerical_approx(mx)
n_My=numerical_approx(My)
n_my=numerical_approx(my)
if n_mx<pspict.mx_acceptable_BB :
raise TooLargeBBException(obj=obj,faulty="xmin",acceptable=pspict.mx_acceptable_BB,got=n_mx)
if n_my<pspict.my_acceptable_BB :
raise TooLargeBBException(obj=obj,faulty="ymin",acceptable=pspict.my_acceptable_BB,got=n_my)
if n_Mx>pspict.Mx_acceptable_BB :
raise TooLargeBBException(obj=obj,faulty="xmax",acceptable=pspict.Mx_acceptable_BB,got=n_Mx)
if n_My>pspict.My_acceptable_BB :
raise TooLargeBBException(obj=obj,faulty="ymax",acceptable=pspict.My_acceptable_BB,got=n_My)
def equation(self):
"""
return the equation of the line under the form
x + by + c = 0
Coefficients 'b' and 'c' are numerical approximations.
See Utilities.Intersection
EXAMPLES::
sage: from yanntricks import *
sage: Segment(Point(0,0),Point(1,1)).equation
x - y == 0
sage: Segment(Point(1,0),Point(0,1)).equation
x + y - 1 == 0
"""
if self.is_vertical:
self.coefs = [1, 0, -self.I.x]
if self.is_horizontal:
self.coefs = [0, 1, -self.I.y]
if not (self.is_vertical or self.is_horizontal):
self.coefs = [1, -1/self.slope, self.independent/self.slope]
x, y = var('x,y')
Ix = numerical_approx(self.I.x)
Iy = numerical_approx(self.I.y)
Fx = numerical_approx(self.F.x)
Fy = numerical_approx(self.F.y)
coefs = [numerical_approx(s) for s in self.coefs]
return coefs[0]*x+coefs[1]*y+coefs[2] == 0
def assert_almost_equal(e1, e2, epsilon=0.0001, failure_message=""):
"""
Raise a FailedAssertException if the two expressions 'e1' and 'e2'
are not equal up to 'epsilon'
First try to use
```
e1.is_almost_equal(e2,epsilon)
```
because some objects in 'yanntricks' have that method.
If `e1` has not that method :
- assume that `e1` and `e2` are numerical
- compute a numerical approximation
- fail if the absolute value of the difference
is larger than `epsilon`
"""
if hasattr(e1, 'is_almost_equal'):
if not e1.is_almost_equal(e2, epsilon):
from yanntricks.testing.unit_tests.Exceptions import FailedAssertException
raise FailedAssertException(
str(e1) + " is not equal to " + str(e2) + " up to " +
str(epsilon))
return True
v1 = numerical_approx(e1)
v2 = numerical_approx(e2)
d = abs(v1 - v2)
if not d < epsilon:
from yanntricks.testing.unit_tests.Exceptions import FailedAssertException
raise FailedAssertException(
str(e1) + " is not equal to " + str(e2) + " up to " + str(epsilon))
def ERPMooZibfNOiU():
pspict, fig = SinglePicture("ERPMooZibfNOiU")
pspict.dilatation(1)
t0 = 40
a = 2.3
b = 1.3
O = Point(0, 0)
lp = Segment(O, Circle(O, 5).getPoint(t0))
P = lp.midpoint()
fun = lambda t: Point(a * cos(t), b * sin(t)).rotation(t0) + P
decal = fun(pi / 2) - P
Gamma = NonAnalyticPointParametricCurve(lambda x: fun(x) + decal, 0,
2 * pi)
Gamma.parameters.plotpoints = 20
pt = Gamma(3 * pi / 2 - 0.15)
xx = pt.x
v = AffineVector(P, pt).normalize(2)
v.parameters.color = "blue"
pspict.DrawGraphs(v)
F = v.F
Fx = F.x
a = 7.73542889062775 * cos(11 / 9 * pi +
1.30951587282752) - 7.55775391156456 * cos(
5 / 18 * pi) + 2.5 * cos(2 / 9 * pi)
print(numerical_approx(a))
print(numerical_approx(a, digits=5))
fig.no_figure()
fig.conclude()
fig.write_the_file()
def distance_sq(P,Q,numerical=False):
if not numerical :
return (P.x-Q.x)**2+(P.y-Q.y)**2
Px=numerical_approx(P.x)
Qx=numerical_approx(Q.x)
Qy=numerical_approx(Q.y)
Py=numerical_approx(P.y)
return (Px-Qx)**2+(Py-Qy)**2
def points_list(self):
l = []
import numpy
ai = numerical_approx(self.angleI)
af = numerical_approx(self.angleF)
angles = numpy.linspace(ai, af, self.linear_plotpoints)
for a in angles:
l.append(self.get_point(a))
return l
def PointToPolaire(P=None,x=None,y=None,origin=None,numerical=True):
"""
Return the polar coordinates of a point.
INPUT:
- ``P`` - (default=None) a point
- ``x,y`` - (defautl=None) the coordinates of the points
EXAMPLES:
You can provide a point::
sage: from phystricks import Point
sage: from phystricks.SmallComputations import *
sage: print PointToPolaire(Point(1,1))
PolarCoordinates, r=sqrt(2),degree=45,radian=1/4*pi
or directly the coordinates ::
sage: print PointToPolaire(x=1,y=1)
PolarCoordinates, r=sqrt(2),degree=45,radian=1/4*pi
"""
from Numerical import numerical_is_negative
if origin:
Ox=origin.x
Oy=origin.y
if not origin:
Ox=0
Oy=0
if P:
Px=P.x
Py=P.y
else :
Px=x
Py=y
Qx=Px-Ox
Qy=Py-Oy
if numerical:
Qx=numerical_approx(Qx)
Qy=numerical_approx(Qy)
r=sqrt( Qx**2+Qy**2 )
if abs(Qx)<0.001: # epsilon
if Qy>0:
radian=pi/2
else :
radian=3*pi/2
else :
radian=arctan(Qy/Qx)
if Qx<0:
if Qy>0:
radian=radian+pi
if Qy<=0:
radian=pi+radian
# Only positive values (February 11, 2015)
if numerical_is_negative(radian):
radian=radian+2*pi
return PolarCoordinates(r,value_radian=radian)
def numerical_max(x, y, epsilon=None):
# Same as `numerical_min` with ad-hoc changes
nx = numerical_approx(x)
ny = numerical_approx(y)
if epsilon is not None:
if abs(nx, ny) > epsilon:
raise ValueError
return max(nx, ny)
def is_almost_orthogonal(self, other, epsilon=0.001):
if self.is_vertical:
return other.is_horizontal
if self.is_horizontal:
return other.is_vertical
s_slope = numerical_approx(self.slope)
o_slope = numerical_approx(other.slope)
if abs(s_slope + 1 / o_slope) < epsilon:
return True
return False
def test_imaginary_part(z,epsilon=0.0001):
"""
Return a tuple '(isreal,w)' where 'isreal' is a boolean saying if 'z' is real (in the sense that it is real and does not contain 'I' in its string representation) and 'w' is 'z' when the imaginary part is larger than epsilon and an 'numerical_approx' of 'z' when its imaginary part is smaller than 'epsilon'
With the collateral effect that it returns a numerical approximation.
"""
if is_real(z) and "I" not in str(z):
return True,z
k=numerical_approx(z)
if is_real(k):
return True,k
if abs( k.imag_part() )<epsilon:
return True,numerical_approx( z.real_part() )
print("It seems that an imaginary part is not so small.")
return False,z
def psi(x, k, prec):
if len(x) == 0 or (x[0]**k) > 10**prec:
return sg.Rational(0.0)
x_power = x.apply_map(lambda y: y**k)
return sg.sum(
x_power.apply_map(lambda y: sg.numerical_approx(1 / y, digits=prec)))
def distance_sq(P, Q, numerical=False):
"""
Return the squared distance between P and Q.
@param {bool} `numerical`
If True, use numerical approximations and return a
numerical approximation.
"""
if not numerical:
return (P.x - Q.x)**2 + (P.y - Q.y)**2
Px = numerical_approx(P.x)
Qx = numerical_approx(Q.x)
Qy = numerical_approx(Q.y)
Py = numerical_approx(P.y)
return (Px - Qx)**2 + (Py - Qy)**2
def __init__(self, f, mx, Mx):
ObjectGraph.__init__(self, self)
self.f = f
self.mx = mx
self.Mx = Mx
self.I = self.get_point(mx)
self.F = self.get_point(Mx)
self.parameters.plotpoints = 100
from numpy import linspace
if self.mx is not None and self.Mx is not None:
self.drawpoints = linspace(numerical_approx(self.mx), numerical_approx(
self.Mx), self.parameters.plotpoints, endpoint=True)
self._curve = None
self.mode = None
def LaTeX_lines(self):
"""
Return the lines to be included in your LaTeX file.
"""
a = []
a.append(self.comments())
if self.figure_environment:
a.append(
"The result is on figure \\ref{"+self.name+"}. % From file "+self.script_filename)
# The pseudo_caption is changed to the function name later.
a.append("\\newcommand{"+self.caption+"}{"+pseudo_caption+"}")
a.append("\\input{%s}" % (self.filename.from_main()))
else:
text = r"""\\begin{center}
INCLUSION
\end{center}""".replace("INCLUSION", "\\input{%s}" % (self.filename.from_main()))
if len(self.record_pspicture) == 1:
pspict = self.record_pspicture[0]
# By the way, this is a reason why we cannot do this before to have
visual_xsize = pspict.visual_xsize()
# concluded the picture.
text = text.replace("WIDTH", str(
numerical_approx(visual_xsize, digits=3))+"cm")
a.append(text)
text = "\n".join(a)
return text
def number_to_string(x,digits):
from Numerical import is_almost_zero
nx=numerical_approx(x)
# Avoid something like "0.125547e-6" (LaTeX will not accept).
if is_almost_zero(nx,0.001):
if digits==1:
return "0"
return "0."+"0"*(digits-1)
sx=str(nx)
# in a definitive release, this test can be removed.
# this is only for my culture; I guess that it never happens
if "." not in sx:
print(x)
print(sx)
raise
sx=sx+"0"*(digits+1) # be sure not to lack digits
if nx<0:
sx=sx[0:digits+2] # +1 for the decimal dot, +1 for the minus
else :
sx=sx[0:digits+1]
if sx.endswith("."):
sx=sx[:-1]
return sx
def __call__(self, xe, numerical=False):
"""
return the value of the function at given point
INPUT:
- ``xe`` - a number. The point at which we want to evaluate the function
- ``numerical`` (boolean, default=False) If True, return a numerical_approximation
EXAMPLES::
sage: from yanntricks import *
sage: x=var('x')
sage: f=phyFunction(cos(x))
sage: f(1)
cos(1)
sage: f(1,numerical=True)
0.540302305868140
"""
if numerical:
return numerical_approx(self.sageFast(xe))
else:
try:
return self.sage(x=xe)
except TypeError: # Happens when one has a distribution function
try:
return self.sage(xe)
except TypeError:
print("ooMHAQooMbDokI")
print(self, type(self))
print(xe, type(xe))
raise
def inner_product(v, w, numerical=False):
"""
Return the inner product of vectors `v` and `w`
@param v a vector
@param w a vector
@param numerical a boolean
If `numerical` is true, the computations are done on
numerical approximations of the coordinates.
"""
if numerical:
if not v.I.is_almost_equal(w.I):
raise OperationNotPermitedException(
"I only compute inner products "
"of vectors based on the same point.")
if not numerical:
if v.I != w.I:
raise OperationNotPermitedException(
"I only compute inner products "
"of vectors based on the same point.")
s = v.Dx * w.Dx + v.Dy * w.Dy
if numerical:
return numerical_approx(s)
return s
def __init__(self, a, b):
self.x = SR(a)
self.y = SR(b)
ObjectGraph.__init__(self, self)
self.point = self.obj
self.add_option("PointSymbol=*")
self._advised_mark_angle = None
try:
ax = abs(numerical_approx(self.x))
if ax < 0.00001 and ax > 0:
self.x = 0
ay = abs(numerical_approx(self.y))
if ay < 0.00001 and ay > 0:
self.y = 0
except TypeError:
pass
def roundingMinMax(d):
"""
From a dictionary of "xmin,..." return the dictionary of three-digit rounded values
"""
new = {}
for p in ["xmax", "xmin", "ymax", "ymin"]:
s = numerical_approx(d[p], digits=3)
new[p] = s
return new
def extrapolate(matrix, Psi, prec=None):
v = Psi[1][-1, :]
B = create_B(matrix)
a = sum(sum(Psi[1]))
b = sum(sum(B*v))
if prec:
return sg.numerical_approx(a + b, digits=prec)
else:
return a + b
def equation(self, numerical=False):
"""
Return the equation of `self`.
OUTPUT:
an equation.
EXAMPLES::
sage: from yanntricks import *
sage: circle=Circle(Point(0,0),1)
sage: circle.equation()
x^2 + y^2 - 1 == 0
::
sage: circle=CircleOA(Point(-1,-1),Point(0,0))
sage: circle.equation()
(x + 1)^2 + (y + 1)^2 - 2 == 0
If 'numerical' is True, return numerical approximations of the coefficients.
"""
if numerical == True and self._numerical_equation is not None:
return self._numerical_equation
if numerical == False and self._equation is not None:
return self._equation
x, y = var('x,y')
if not self.visual:
cx = self.center.x
cy = self.center.y
cr = self.radius
self._equation = (x - cx)**2 + (y - cy)**2 - cr**2 == 0
if not numerical:
return self._equation
if numerical:
cx = numerical_approx(cx)
cy = numerical_approx(cy)
cr = numerical_approx(cr)
self._numerical_equation = (x - cx)**2 + (y -
cy)**2 - cr**2 == 0
return self._numerical_equation
Rx = self.radius / self.pspict.xunit
Ry = self.radius / self.pspict.yunit
if numerical == False:
self._equation = (x-self.center.x)**2/Rx**2 + \
(y-self.center.y)**2/Ry**2-1 == 0
return self._equation
if numerical == True:
Rx = numerical_approx(Rx)
Ry = numerical_approx(Ry)
cx = numerical_approx(self.center.x)
cy = numerical_approx(self.center.y)
self._numerical_equation = (x - cx)**2 / Rx**2 + (
y - cy)**2 / Ry**2 - 1 == 0
return self._numerical_equation
def point_to_box_intersection(P,box,pspict=None):
from phystricks.src.Utilities import distance_sq
A=Point(box.xmin,box.ymin)
B=Point(box.xmax,box.ymin)
C=Point(box.xmax,box.ymax)
D=Point(box.xmin,box.ymax)
# n'écrivez pas ça au tableau quand un inspecteur est dans la salle :
center=(A+B+C+D)/4
line=Segment(P,center)
edges=[Segment(A,B),Segment(B,C),Segment(C,D),Segment(D,A)]
inter=[]
for ed in edges:
c=Intersection(line,ed)
if len(c)>0:
S=c[0]
# We deal with the case in which the line travers the corner.
# In this case, the line passes trough the other one.
if S==A:
inter=[A,C]
if S==B:
inter=[B,D]
if S==C:
inter=[A,C]
if S==D:
inter=[B,D]
# The last two tests are to know if S lies between ed.I and ed.F
# We use numerical approximations in order to avoid some
# OverflowError: Python int too large to convert to C long
elif numerical_approx( (S.x-ed.I.x)*(S.x-ed.F.x) )<0:
inter.append(S)
elif numerical_approx( (S.y-ed.I.y)*(S.y-ed.F.y) )<0:
inter.append(S)
if len(inter)==2:
inter.sort(key=lambda Q:distance_sq(Q,P,numerical=True))
if pspict:
for i,S in enumerate(inter):
S.put_mark(0.2,angle=None,added_angle=0,text=str(i),pspict=pspict)
pspict.DrawGraphs(inter,line,center,box)
return inter
def visual_angleIF(self, pspict):
from yanntricks.src.point import Point
from yanntricks.src.AngleMeasure import AngleMeasure
aI1 = visual_polar_coordinates(
Point(cos(self.angleI.radian), sin(self.angleI.radian)),
pspict).measure
aF1 = visual_polar_coordinates(
Point(cos(self.angleF.radian), sin(self.angleF.radian)),
pspict).measure
a = numerical_approx(aI1.degree)
b = numerical_approx(aF1.degree)
if a > b:
a = a - 360
aI2 = AngleMeasure(value_degree=a)
else:
aI2 = aI1
aF2 = aF1
return aI2, aF2
def numerical_min(x, y, epsilon=None):
##
# \brief return the minimum of `x` and `y`
#
# Compute numerical approximations of `x` and `y` and return the min
#
# If `epsilon` is given, raise an exception if the difference is
# smaller than `epsilon`.
#
# The reason is that Sage cannot always determine the min or the max of
# expressions like ```1000``` of type `int` and ```cos(0.0823552493237255*pi)``` of type `sage.symbolic.expression.Expression`
nx = numerical_approx(x)
ny = numerical_approx(y)
if epsilon is not None:
if abs(nx, ny) > epsilon:
raise ValueError
return min(nx, ny)
def solve_one_var(self, eqs, var):
"""
Solve the equations with respect to the given variable
Returns a list of numerical values.
"""
liste = solve(eqs, var, explicit_solutions=True)
a = []
for soluce in liste:
a.append(numerical_approx(soluce.rhs()))
return a
def is_almost_equal(self, other, epsilon=0.0001):
##
# return true if `self` and `other` have coordinates difference
# lower than `epsilon`
#
from yanntricks.src.NoMathUtilities import logging
if not isinstance(other, Point):
logging("We are comparing " + type(self) + " with " + type(other) +
". We continue, but this is strange.")
sx = numerical_approx(self.x)
sy = numerical_approx(self.y)
ox = numerical_approx(other.x)
oy = numerical_approx(other.y)
if abs(sx - ox) > epsilon:
return False
if abs(sy - oy) > epsilon:
return False
return True
def length(self):
"""
return (a numerical approximation of) the length of the segment
EXAMPLES::
sage: from phystricks import *
sage: Segment(Point(1,1),Point(2,2)).length
sqrt(2)
"""
return numerical_approx(self.exact_length)
def is_almost_orthogonal(self, other, epsilon=0.001):
"""
Return true is `self` and `other` are orthogonal segments
The answer is based on numerical approximations of the slopes.
If \f$ k \f$ is the slope of `self`, check if the
slope of the other is \f$ -1/k \f$ up to `epsilon`.
See `is_orthogonal`
"""
if self.is_vertical:
return other.is_horizontal
if self.is_horizontal:
return other.is_vertical
s_slope = numerical_approx(self.slope)
o_slope = numerical_approx(other.slope)
if abs(s_slope+1/o_slope) < epsilon:
return True
return False