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])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
if abs(det(matrix([p-s,q-s])))<0.01:
return line((p,q),**kwds)
(cx,cy)=var('cx','cy')
equations=[
2*cx*(s[0]-p[0])+2*cy*(s[1]-p[1]) == s[0]**2+s[1]**2-p[0]**2-p[1]**2,
2*cx*(s[0]-q[0])+2*cy*(s[1]-q[1]) == s[0]**2+s[1]**2-q[0]**2-q[1]**2
]
c = vector( [solve( equations, (cx,cy), solution_dict=True )[0][i] for i in [cx,cy]] )
r = norm(p-c)
a_p,a_q,a_s = map( _to_angle, [p-c,q-c,s-c])
angles = [a_p,a_q,a_s]
angles.sort()
if a_s == angles[0]:
return arc( c, r, angle=angles[2], sector=(0,2*pi-angles[2]+angles[1]), **kwds)
if a_s == angles[1]:
return arc( c, r, angle=angles[0], sector=(0,angles[2]-angles[0]), **kwds)
if a_s == angles[2]:
return arc( c, r, angle=angles[1], sector=(0,2*pi-angles[1]+angles[0]), **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])
# to avoid running into division by 0 we set to be colinear vectors that are
# almost colinear
if abs(det(matrix([p - s, q - s]))) < 0.01:
return line((p, q), **kwds)
(cx, cy) = var('cx', 'cy')
equations = [
2 * cx * (s[0] - p[0]) + 2 * cy * (s[1] - p[1]) == s[0]**2 + s[1]**2 -
p[0]**2 - p[1]**2, 2 * cx * (s[0] - q[0]) + 2 * cy *
(s[1] - q[1]) == s[0]**2 + s[1]**2 - q[0]**2 - q[1]**2
]
c = vector([
solve(equations, (cx, cy), solution_dict=True)[0][i] for i in [cx, cy]
])
r = norm(p - c)
a_p, a_q, a_s = map(lambda x: atan2(x[1], x[0]), [p - c, q - c, s - c])
a_p, a_q = sorted([a_p, a_q])
if a_s < a_p or a_s > a_q:
return arc(c, r, angle=a_q, sector=(0, 2 * pi - a_q + a_p), **kwds)
return arc(c, r, angle=a_p, sector=(0, a_q - a_p), **kwds)
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: UHP.get_geodesic(I, 3+4*I).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: # on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2) / 2 # Circle center
radius = abs(bd_1 - bd_2) / 2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
pic = arc((real(center), imag(center)),
radius,
sector=(theta1, theta2),
**opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2) / 2
length = abs(shadow_1 - shadow_2)
bd_dict = {
'bd_min': midpoint - length,
'bd_max': midpoint + length
}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES::
sage: UHP = HyperbolicPlane().UHP()
sage: UHP.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: UHP.get_geodesic(I, 3+4*I).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
if (abs(real(end_1) - real(end_2)) < EPSILON) \
or CC(infinity) in [end_1, end_2]: # on same vertical line
# If one of the endpoints is infinity, we replace it with a
# large finite point
if end_1 == CC(infinity):
end_1 = (real(end_2), (imag(end_2) + 10))
end_2 = (real(end_2), imag(end_2))
elif end_2 == CC(infinity):
end_2 = (real(end_1), (imag(end_1) + 10))
end_1 = (real(end_1), imag(end_1))
pic = line((end_1, end_2), **opts)
if boundary:
cent = min(bd_1, bd_2)
bd_dict = {'bd_min': cent - 3, 'bd_max': cent + 3}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
else:
center = (bd_1 + bd_2) / 2 # Circle center
radius = abs(bd_1 - bd_2) / 2
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
if abs(theta1 - theta2) < EPSILON:
theta2 += pi
pic = arc((real(center), imag(center)), radius,
sector=(theta1, theta2), **opts)
if boundary:
# We want to draw a segment of the real line. The
# computations below compute the projection of the
# geodesic to the real line, and then draw a little
# to the left and right of the projection.
shadow_1, shadow_2 = [real(k) for k in [end_1, end_2]]
midpoint = (shadow_1 + shadow_2)/2
length = abs(shadow_1 - shadow_2)
bd_dict = {'bd_min': midpoint - length, 'bd_max': midpoint +
length}
bd_pic = self._model.get_background_graphic(**bd_dict)
pic = bd_pic + pic
return pic
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES:
First some lines::
sage: PD = HyperbolicPlane().PD()
sage: PD.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(0, 0.3+0.8*I).show()
Graphics object consisting of 2 graphics primitives
Then some generic geodesics::
sage: PD.get_geodesic(-0.5, 0.3+0.4*I).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(-1, exp(3*I*pi/7)).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(exp(2*I*pi/11), exp(1*I*pi/11)).show(thickness=6, color="orange")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
invdet = RR.one() / (real(bd_1) * imag(bd_2) -
real(bd_2) * imag(bd_1))
centerx = (imag(bd_2) - imag(bd_1)) * invdet
centery = (real(bd_1) - real(bd_2)) * invdet
center = centerx + I * centery
radius = RR(abs(bd_1 - center))
# Now we calculate the angles for the arc
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
theta1, theta2 = sorted([theta1, theta2])
# Make sure the sector is inside the disk
if theta2 - theta1 > pi:
theta1 += 2 * pi
pic = arc((centerx, centery),
radius,
sector=(theta1, theta2),
**opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
def show(self, boundary=True, **options):
r"""
Plot ``self``.
EXAMPLES:
First some lines::
sage: PD = HyperbolicPlane().PD()
sage: PD.get_geodesic(0, 1).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(0, 0.3+0.8*I).show()
Graphics object consisting of 2 graphics primitives
Then some generic geodesics::
sage: PD.get_geodesic(-0.5, 0.3+0.4*I).show()
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(-1, exp(3*I*pi/7)).show(linestyle="dashed", color="red")
Graphics object consisting of 2 graphics primitives
sage: PD.get_geodesic(exp(2*I*pi/11), exp(1*I*pi/11)).show(thickness=6, color="orange")
Graphics object consisting of 2 graphics primitives
"""
opts = {'axes': False, 'aspect_ratio': 1}
opts.update(self.graphics_options())
opts.update(options)
end_1, end_2 = [CC(k.coordinates()) for k in self.endpoints()]
bd_1, bd_2 = [CC(k.coordinates()) for k in self.ideal_endpoints()]
# Check to see if it's a line
if abs(bd_1 + bd_2) < EPSILON:
pic = line([end_1, end_2], **opts)
else:
# If we are here, we know it's not a line
# So we compute the center and radius of the circle
invdet = RR.one() / (real(bd_1)*imag(bd_2) - real(bd_2)*imag(bd_1))
centerx = (imag(bd_2) - imag(bd_1)) * invdet
centery = (real(bd_1) - real(bd_2)) * invdet
center = centerx + I * centery
radius = RR(abs(bd_1 - center))
# Now we calculate the angles for the arc
theta1 = CC(end_1 - center).arg()
theta2 = CC(end_2 - center).arg()
theta1, theta2 = sorted([theta1, theta2])
# Make sure the sector is inside the disk
if theta2 - theta1 > pi:
theta1 += 2 * pi
pic = arc((centerx, centery), radius,
sector=(theta1, theta2), **opts)
if boundary:
pic += self._model.get_background_graphic()
return pic
