def _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if w0.real()==0 and w3.real() == 0:
y = (w0.imag() + w0.imag())/2
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
self._graphics.add_primitive(BezierPath([(0,w0.imag()),CC(0,y), (0,w3.imag())],self._options))
return
z0,z3 = CC(z0), CC(z3)
if z0.imag() == 0 and z3.imag() == 0:
p = (z0.real()+z3.real())/2
r = abs(z0-p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
p = (abs(z0)*abs(z0)-abs(z3)*abs(z3))/(z0-z3).real()/2 # center of the circle in H
r = abs(z0-p) # radius of the circle in H
zm = ((z0+z3)/2-p)/abs((z0+z3)/2-p)*r+p
t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
zm = self._cayley_transform(zm)
w1 = self._cayley_transform(z1)
w2 = self._cayley_transform(z2)
c = self._cayley_transform(CC(p,0)) # center of circle on the unit disk.
r = abs(w1-c) # radius
theta = t
self._graphics.add_primitive(Arc((zm.real(),zm.imag()),r,sector=(-t,t),options=self._options))
def _geodesic_between_two_points_d(x1,y1,x2,y2,z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi=RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos,sin)
# First compute the points
if(y1<0 or y2<0 ):
raise ValueError,"Need points in the upper half-plane! Got y1=%s, y2=%s" %(y1,y2)
if(y1==infinity):
P1=CC(1 )
else:
P1=CC((x1+I*y1-z0)/(x1+I*y1-z0.conjugate()))
if(y2==infinity):
P2=CC(1 )
else:
P2=CC((x2+I*y2-z0)/(x2+I*y2-z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if(x1==x2):
a=CC((x1-z0)/(x1-z0.conjugate()))
b=CC(1 )
else:
c=RR(y1**2 -y2**2 +x1**2 -x2**2 )/RR(2 *(x1-x2))
r=RR(sqrt(y1**2 +(x1-c)**2 ))
a=c-r
b=c+r
a=CC((a-z0)/(a-z0.conjugate()))
b=CC((b-z0)/(b-z0.conjugate()))
if( abs(a+b) < 1E-10 ): # On a diagonal
return line([[P1.real(),P1.imag()],[P2.real(),P2.imag()]])
th_a=a.argument()
th_b=b.argument()
# Compute the center of the circle in the disc model
if( min(abs(b-1 ),abs(b+1 ))< 1E-10 and min(abs(a-1 ),abs(a+1 ))>1E-10 ):
c=b+I*(1 -b*cos(th_a))/sin(th_a)
elif( min(abs(b-1 ),abs(b+1 ))> 1E-10 and min(abs(a-1 ),abs(a+1 ))<1E-10 ):
c=a+I*(1 -a*cos(th_b))/RR(sin(th_b))
else:
cx=(sin(th_b)-sin(th_a))/sin(th_b-th_a)
c=cx+I*(1 -cx*cos(th_b))/RR(sin(th_b))
# First find the endpoints of the completed geodesic
r=abs(c-a)
t1=CC(P1-c).argument()
t2=CC(P2-c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1,t2,c,r)
def _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if self._verbose>0:
print "in plane z0,z3=",z0,z3
print "in disc: ",w0,w3
npts = self._npts
if z0 == infinity or z0==CC(infinity):
zm = [z3 + CC(0,j+0.5) for j in range(npts-2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w3]
pts.extend(wm)
pts.append(w0)
opt = self._options
opt['fill']=False
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
return
if z3 == infinity or z3==CC(infinity):
zm = [z0 + CC(0,j+0.5) for j in range(npts-2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w0]
pts.extend(wm)
pts.append(w3)
opt = self._options
opt['fill']=False
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
#self._graphics.add_primitive(Line([w1.real(),w2.real()],[w1.imag(),w2.imag()],self._options))
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
return
x0=z0.real(); y0 = z0.imag()
x3=z3.real(); y3 = z3.imag()
if y0 == 0 and y3 == 0:
p = (z0.real()+z3.real())/2
r = abs(z0-p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
if abs(x0-x3)<1e-10: ## on the same vertical line
xmid = (x0+x3)*0.5; h = y3-y0
zm = [ CC(xmid,y0+t*h/(npts-1)) for t in range(npts) ]
else:
p = RR((x0+x3)*(x3-x0)+(y0+y3)*(y3-y0))/(2*(x3-x0))
r = RR((p - x0)**2 + y0**2).sqrt() # radius of the circle in H
zm = ((z0+z3)/2-p)/abs((z0+z3)/2-p)*r+p # midpoint (at least approximately) of geodesic between z0 and z3
t0 = CC(z0 - p).argument()
t3 = CC(z3 - p).argument()
if self._verbose>1:
print "x0,x3=",x0,x3
print "t0,t3=",t0,t3
print "r=",r
print "opt=",self._options
if x0 <= x3:
zm = [p + r*CC(0,(t0+t*(t3-t0)/(npts-1))).exp() for t in range(npts)]
else:
zm = [p + r*CC(0,(t0+t*(t3-t0)/(npts-1))).exp() for t in range(npts)]
#print "zm=",zm
#zm.insert(0,z0)
#zm.append(z3)
pts = [self._cayley_transform(x) for x in zm]
opt = self._options
opt['fill']=False
#w0 = self._cayley_transform(z0)
#w3 = self._cayley_transform(z3)
if self._verbose>2:
print "pts=",pts
self._graphics.add_primitive(BezierPath([[(x.real(),x.imag()) for x in pts ]],opt))
return
#print "z0_test=",(p+r*exp(t0*I))
#print "z3_test=",(p+r*exp(t3*I))
#t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
# I have no idea what these points should represent....
#z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
#z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
wm = [self._cayley_transform(x) for x in zm]
pp = self._cayley_transform(CC(p,0))
w1 = self._cayley_transform(z0)
w2 = self._cayley_transform(z3)
c = self._cayley_transform(CC(p,0)) # center of circle on the unit disk.
if self._verbose>2:
print "p,r=",p,r
print "zm=",zm
#print "t=",t
#print "tt=",(8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
print "C(c)=",pp
print "C(zm)=",wm
print "C(z0)=",w1
print "C(z3)=",w2
#print "z2=",z2
r = abs(w1-c) # radius
t0 = CC(w0 - pp).argument()
t3 = CC(w3 - pp).argument()
t = abs(t0-t3)
if self._verbose>0:
print "adding a:rc ",zm.real(),zm.imag(),r,r,t,t0,t3
self._graphics.add_primitive(Line([w1.real(),w2.real(),wm.real()],[w1.imag(),w2.imag(),wm.imag()],{'thickness':2,'alpha':1,
'rgbcolor':'blue',
'legend_label':""}))
self._graphics.add_primitive(Point([w1.real(),w2.real(),wm.real()],[w1.imag(),w2.imag(),wm.imag()],{'size':10,'alpha':1,
'faceted':True,
'rgbcolor':'red',
'legend_label':""}))
def _geodesic_between_two_points_d(x1, y1, x2, y2, z0=I):
r""" Geodesic path between two points represented in the unit disc
by the map w = (z-I)/(z+I)
INPUTS:
- ''(x1,y1)'' -- starting point (0<y1<=infinity)
- ''(x2,y2)'' -- ending point (0<y2<=infinity)
- ''z0'' -- (default I) the point in the upper corresponding
to the point 0 in the disc. I.e. the transform is
w -> (z-I)/(z+I)
OUTPUT:
- ''ca'' -- a polygonal approximation of a circular arc centered
at c and radius r, starting at t0 and ending at t1
EXAMPLES::
sage: l=_geodesic_between_two_points_d(0.1,0.2,0.0,0.5)
"""
pi = RR.pi()
from sage.plot.plot import line
from sage.functions.trig import (cos, sin)
# First compute the points
if (y1 < 0 or y2 < 0):
raise ValueError, "Need points in the upper half-plane! Got y1=%s, y2=%s" % (
y1, y2)
if (y1 == infinity):
P1 = CC(1)
else:
P1 = CC((x1 + I * y1 - z0) / (x1 + I * y1 - z0.conjugate()))
if (y2 == infinity):
P2 = CC(1)
else:
P2 = CC((x2 + I * y2 - z0) / (x2 + I * y2 - z0.conjugate()))
# First find the endpoints of the completed geodesic in D
if (x1 == x2):
a = CC((x1 - z0) / (x1 - z0.conjugate()))
b = CC(1)
else:
c = RR(y1**2 - y2**2 + x1**2 - x2**2) / RR(2 * (x1 - x2))
r = RR(sqrt(y1**2 + (x1 - c)**2))
a = c - r
b = c + r
a = CC((a - z0) / (a - z0.conjugate()))
b = CC((b - z0) / (b - z0.conjugate()))
if (abs(a + b) < 1E-10): # On a diagonal
return line([[P1.real(), P1.imag()], [P2.real(), P2.imag()]])
th_a = a.argument()
th_b = b.argument()
# Compute the center of the circle in the disc model
if (min(abs(b - 1), abs(b + 1)) < 1E-10
and min(abs(a - 1), abs(a + 1)) > 1E-10):
c = b + I * (1 - b * cos(th_a)) / sin(th_a)
elif (min(abs(b - 1), abs(b + 1)) > 1E-10
and min(abs(a - 1), abs(a + 1)) < 1E-10):
c = a + I * (1 - a * cos(th_b)) / RR(sin(th_b))
else:
cx = (sin(th_b) - sin(th_a)) / sin(th_b - th_a)
c = cx + I * (1 - cx * cos(th_b)) / RR(sin(th_b))
# First find the endpoints of the completed geodesic
r = abs(c - a)
t1 = CC(P1 - c).argument()
t2 = CC(P2 - c).argument()
#print "t1,t2=",t1,t2
return _circ_arc(t1, t2, c, r)
def _hyperbolic_arc_d(self, z0, z3, first=False):
"""
Function to construct Bezier path as an approximation to
the hyperbolic arc between the complex numbers z0 and z3 in the
hyperbolic plane.
"""
w0 = self._cayley_transform(z0)
w3 = self._cayley_transform(z3)
if self._verbose > 0:
print "in plane z0,z3=", z0, z3
print "in disc: ", w0, w3
npts = self._npts
if z0 == infinity or z0 == CC(infinity):
zm = [z3 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w3]
pts.extend(wm)
pts.append(w0)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
if z3 == infinity or z3 == CC(infinity):
zm = [z0 + CC(0, j + 0.5) for j in range(npts - 2)]
wm = [self._cayley_transform(x) for x in zm]
pts = [w0]
pts.extend(wm)
pts.append(w3)
opt = self._options
opt['fill'] = False
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
#self._graphics.add_primitive(Line([w1.real(),w2.real()],[w1.imag(),w2.imag()],self._options))
#self.path.append([(0,w0.imag()),CC(0,y), (0,w3.imag())])
return
x0 = z0.real()
y0 = z0.imag()
x3 = z3.real()
y3 = z3.imag()
if y0 == 0 and y3 == 0:
p = (z0.real() + z3.real()) / 2
r = abs(z0 - p)
zm = CC(p, r)
self._hyperbolic_arc_d(z0, zm, first)
self._hyperbolic_arc_d(zm, z3)
return
else:
if abs(x0 - x3) < 1e-10: ## on the same vertical line
xmid = (x0 + x3) * 0.5
h = y3 - y0
zm = [CC(xmid, y0 + t * h / (npts - 1)) for t in range(npts)]
else:
p = RR((x0 + x3) * (x3 - x0) + (y0 + y3) *
(y3 - y0)) / (2 * (x3 - x0))
r = RR((p - x0)**2 + y0**2).sqrt() # radius of the circle in H
zm = ((z0 + z3) / 2 - p) / abs(
(z0 + z3) / 2 - p
) * r + p # midpoint (at least approximately) of geodesic between z0 and z3
t0 = CC(z0 - p).argument()
t3 = CC(z3 - p).argument()
if self._verbose > 1:
print "x0,x3=", x0, x3
print "t0,t3=", t0, t3
print "r=", r
print "opt=", self._options
if x0 <= x3:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
else:
zm = [
p + r * CC(0, (t0 + t * (t3 - t0) / (npts - 1))).exp()
for t in range(npts)
]
#print "zm=",zm
#zm.insert(0,z0)
#zm.append(z3)
pts = [self._cayley_transform(x) for x in zm]
opt = self._options
opt['fill'] = False
#w0 = self._cayley_transform(z0)
#w3 = self._cayley_transform(z3)
if self._verbose > 2:
print "pts=", pts
self._graphics.add_primitive(
BezierPath([[(x.real(), x.imag()) for x in pts]], opt))
return
#print "z0_test=",(p+r*exp(t0*I))
#print "z3_test=",(p+r*exp(t3*I))
#t = (8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
# I have no idea what these points should represent....
#z1 = z0 + t*CC(z0.imag(), (p-z0.real()))
#z2 = z3 - t*CC(z3.imag(), (p-z3.real()))
wm = [self._cayley_transform(x) for x in zm]
pp = self._cayley_transform(CC(p, 0))
w1 = self._cayley_transform(z0)
w2 = self._cayley_transform(z3)
c = self._cayley_transform(CC(
p, 0)) # center of circle on the unit disk.
if self._verbose > 2:
print "p,r=", p, r
print "zm=", zm
#print "t=",t
#print "tt=",(8*zm-4*(z0+z3)).imag()/3/(z3-z0).real()
print "C(c)=", pp
print "C(zm)=", wm
print "C(z0)=", w1
print "C(z3)=", w2
#print "z2=",z2
r = abs(w1 - c) # radius
t0 = CC(w0 - pp).argument()
t3 = CC(w3 - pp).argument()
t = abs(t0 - t3)
if self._verbose > 0:
print "adding a:rc ", zm.real(), zm.imag(), r, r, t, t0, t3
self._graphics.add_primitive(
Line([w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'thickness': 2,
'alpha': 1,
'rgbcolor': 'blue',
'legend_label': ""
}))
self._graphics.add_primitive(
Point(
[w1.real(), w2.real(), wm.real()],
[w1.imag(), w2.imag(), wm.imag()], {
'size': 10,
'alpha': 1,
'faceted': True,
'rgbcolor': 'red',
'legend_label': ""
}))